NCERT Solutions Class 7 Mathematics Ganita Prakash 1 Chapter 03 A Peek Beyond the Point

Page 47

Question 1. Which scale helped you measure the length of the screws accurately? Why?
Answer: A scale with very small divisions - marked in tenths of a centimeter, which are millimeters - gives the most accurate measurement. The ends of the screws didn't line up exactly with whole or half centimeter marks, so finer divisions were required to find the length more precisely.
In simple words: A scale with the tiniest marks lets you measure things that don't line up with the bigger marks.

Exam Tip: Always choose a measuring tool with the smallest divisions available - this is how you get the most precise measurements in practical exams.

Question 2. What is the meaning of 2 7/10 cm (the length of the first screw)?
Answer: This means the screw is 2 whole centimeters plus 7 parts when you divide the next centimeter into 10 equal pieces (7/10 cm). In other words, it equals 2 centimeters plus 7 tenths of a centimeter - which is the same as saying 2 centimeters and 7 millimeters.
In simple words: The screw is 2 full centimeters, then 7 more parts out of 10.

Exam Tip: Mixed measurements like 2 7/10 cm show both the whole units and the fractional parts - always identify both parts clearly in your answer.

Question 3. Can you explain why the unit was divided into smaller parts to measure the screws?
Answer: The centimeter unit needed to be split into smaller parts - called tenths or millimeters - because the screws didn't have lengths that matched whole centimeter numbers exactly. When lengths fall between whole number marks, you need finer divisions to measure them correctly.
In simple words: If things don't match the big marks, you need smaller marks to measure them right.

Exam Tip: This shows the key principle: divide a unit whenever the object being measured falls between the existing marks.

Question 4. Measure the following objects using a scale and write their measurements in centimeters (as shown earlier for the lengths of the screws): pen, sharpener, and any other object of your choice.
Answer: Pen Length: 6 2/10 cm
Sharpener Length: 2 1/10 cm
Eraser Length: 2 3/10 cm
In simple words: You must use a scale and measure each object carefully, recording both the whole centimeters and the tenths of a centimeter.

Exam Tip: Always align the zero of the scale with the start of the object, then read where the end falls, noting both whole units and fractional parts.

Question 5. Write the measurements of the objects shown in the picture.
Answer: Eraser Length: 2 3/10 cm
Pencil Length: 4 3/10 cm
Chalk Length: 1 4/10 cm
In simple words: Read each object's length from where it starts to where it ends on the scale, noting both whole centimeters and the tenths.

Exam Tip: When reading from a scale image, make sure to identify the scale divisions correctly and read the endpoint precisely.

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Question. Arrange these lengths in increasing order: (a) 9/10 (value 0.9) (b) 1 7/10 (value 1.7) (c) 130/10 (value 13.0) (d) 13 1/10 (value 13.1) (e) 10 5/10 (value 10.5) (f) 7 6/10 (value 7.6) (g) 6 7/10 (value 6.7) (h) 4/10 (value 0.4)
Answer: First, compare the decimal values: 0.4 < 0.9 < 1.7 < 6.7 < 7.6 < 10.5 < 13.0 < 13.1. The lengths arranged from smallest to largest are: 4/10, 9/10, 1 7/10, 6 7/10, 7 6/10, 10 5/10, 130/10, 13 1/10.
In simple words: To arrange measurements in order, change them all to the same form - like decimals - then compare which numbers are bigger.

Exam Tip: Always convert mixed numbers and fractions to decimals first to make comparison easier - this avoids mixing up different forms.

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Question 1. Arrange the following lengths in increasing order: 4 1/10, 4/10, 41/10, 41 1/10.
Answer: Convert each to decimal form to compare: 4 1/10 becomes 4.1; 4/10 becomes 0.4; 41/10 becomes 4.1; and 41 1/10 becomes 41.1. In increasing order, the values are 0.4, 4.1, 4.1, and 41.1. So the lengths arranged from smallest to largest are: 4/10, 4 1/10 (which equals 41/10), and 41 1/10.
In simple words: Change each fraction to a decimal, then put them in order from smallest to largest.

Exam Tip: Notice that 4 1/10 and 41/10 have the same value - different forms can represent the same number.

Question 2. Sonu is measuring some of his body parts. The length of Sonu's lower arm is 2 7/10 units, and that of his upper arm is 3 6/10 units. What is the total length of his arm? (Three methods shown in text).
Answer: We need to calculate 2 7/10 + 3 6/10 using three different approaches.

Method (a) Summing whole and fractional parts separately:
Add the whole numbers first: 2 + 3 = 5. Then add the fractions: 7/10 + 6/10 = 13/10. Combine them: 5 + 13/10 = 5 + 1 3/10 = 6 3/10 units.

Method (b) Vertical addition (similar to Method a):

2 7/10
+ 3 6/10
- - - - -
5 13/10 (which simplifies to 5 + 1 3/10 = 6 3/10)

Method (c) Converting to tenths:
First express as improper fractions: 2 7/10 = 27/10 and 3 6/10 = 36/10. Then add: 27/10 + 36/10 = 63/10. Convert back to mixed form: 63/10 = 6 3/10 units.

In simple words: You can add measurements by adding the whole parts and fractional parts separately, or by changing them to fractions first, then adding - both ways give the same answer of 6 3/10 units.

Exam Tip: All three methods arrive at the same answer - choose whichever feels easiest for you, and show your working clearly in exams.

Question. The lengths of the body parts of a honeybee are given. Find its total length. Head: 2 3/10 units Thorax: 5 4/10 units Abdomen: 7 5/10 units
Answer: Total Length = Head + Thorax + Abdomen = 2 3/10 + 5 4/10 + 7 5/10. First add all the whole parts: 2 + 5 + 7 = 14. Then add all the fractional parts: 3/10 + 4/10 + 5/10 = 12/10. Combine them: 14 + 12/10 = 14 + 1 2/10 = 15 2/10 units. The honeybee's total length is 15 2/10 units.
In simple words: Add all three measurements by adding the whole numbers first, then the fractions - you get 15 2/10 units altogether.

Exam Tip: When adding three or more mixed numbers, always group the whole parts and fractional parts separately to avoid mistakes.

Question. The length of Shylaja's hand is 12 4/10 units, and her palm is 6 7/10 units, as shown in the picture. What is the length of the longest (middle) finger? (Assume finger length = hand length - palm length).
Answer: We need to calculate 12 4/10 - 6 7/10 to find the finger length.

Method (a) shown in text:
Subtract the whole parts: 12 - 6 = 6. Then try to subtract the fractions: 4/10 - 7/10 = -3/10 (a negative result). So we get 6 + (-3/10) = 6 - 3/10. Rewrite 6 as 5 + 10/10, giving 5 + 10/10 - 3/10 = 5 + 7/10 = 5 7/10 units.

Method (b) shown in text (Vertical subtraction with borrowing):
Since we cannot subtract 7/10 from 4/10, we borrow 1 whole unit from 12. This changes 12 4/10 to 11 14/10. Now subtract:

11 14/10
- 6 7/10
- - - - -
5 7/10

The length of the finger is 5 7/10 units.
In simple words: When the fraction you're taking away is bigger than the fraction you have, borrow 1 whole unit and change it to tenths first - then you can subtract.

Exam Tip: Borrowing (or regrouping) is essential when subtracting fractions where the bottom number is smaller - always show this step clearly.

Question. Discuss what is being done here and why.
Answer: In the subtraction problem, we are taking 6 7/10 away from 12 4/10. Looking at just the fractional parts, we see 4/10, but we need to subtract 7/10 from it. Since 4 is smaller than 7, we cannot do this directly. To solve this, we "borrow" - we take away 1 whole unit from the 12, which leaves 11 whole units. That borrowed 1 is then converted into its equivalent in tenths, which gives us 10/10. Adding this to the existing 4/10 gives 10/10 + 4/10 = 14/10. Now the subtraction becomes 11 14/10 - 6 7/10. We can now subtract the tenths: 14/10 - 7/10 = 7/10. We can also subtract the whole units: 11 - 6 = 5. The final result is 5 7/10. This borrowing method (also called regrouping) is the tool we use whenever the fractional part of the first number is smaller than the fractional part of the second number, making direct subtraction impossible.
In simple words: Borrowing means taking 1 from the whole units and changing it to 10 tenths so you have enough to subtract from.

Exam Tip: Understand the "why" behind borrowing - this concept will help you solve many subtraction problems with decimals and fractions throughout your math journey.

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Question 1. Try computing the difference 12 4/10 - 6 7/10 by converting both lengths to tenths.
Answer: First, convert both numbers fully to tenths: 12 4/10 = 120/10 + 4/10 = 124/10, and 6 7/10 = 60/10 + 7/10 = 67/10. Now subtract: 124/10 - 67/10 = (124 - 67)/10 = 57/10. Finally, convert back to mixed form: 57/10 = 50/10 + 7/10 = 5 + 7/10 = 5 7/10 units.
In simple words: Change both measurements into fractions with the same denominator, subtract the top numbers, then change the answer back to mixed form.

Exam Tip: Converting to a single fraction type simplifies the arithmetic - you avoid the need for borrowing in this method.

Question 2. A Celestial Pearl Danio's length is 2 4/10 cm, and the length of a Philippine Goby is 9/10 cm. What is the difference in their lengths?
Answer: We find the difference by calculating (Danio's length) - (Goby's length) = 2 4/10 - 9/10. First convert 2 4/10 to tenths: 20/10 + 4/10 = 24/10. Now subtract: 24/10 - 9/10 = (24 - 9)/10 = 15/10. Convert back to mixed form: 15/10 = 10/10 + 5/10 = 1 + 5/10 = 1 5/10 cm (which is the same as 1.5 cm).
In simple words: Change the bigger fish's length to tenths, subtract the smaller fish's length, and you find the Danio is 1 5/10 cm longer.

Exam Tip: Always subtract the smaller measurement from the larger one - this gives you a positive difference.

Question 3. How big are these fish compared to your finger?
Answer: The Philippine Goby measures 9/10 cm (or 0.9 cm) in length. This fish is probably narrower and much shorter than an adult's finger. The Celestial Pearl Danio reaches 2 4/10 cm (or 2.4 cm) long. This one might be slightly wider than an adult's finger, but it remains much shorter than the length of a finger from knuckle to tip.
In simple words: These fish are both quite tiny - shorter than a finger length.

Exam Tip: When asked to compare abstract measurements, use your body or everyday objects as reference points - this helps make the numbers real and meaningful.

Question 4. Observe the given sequences of numbers. Identify the change after each term and extend the pattern (write next 3 terms): (a) 4, 4 3/10, 4 6/10, …
Answer: Looking at the sequence, the change after each term is adding 3/10 (or 0.3). Starting from 4 6/10, the next three terms are: 4 6/10 + 3/10 = 4 9/10; 4 9/10 + 3/10 = 5 2/10; 5 2/10 + 3/10 = 5 5/10. So the next three terms are 4 9/10, 5 2/10, 5 5/10.
In simple words: Each number is 3 tenths bigger than the one before it.

Exam Tip: Always find the difference between consecutive terms first - this reveals the pattern rule.

Question 4(b). 8 2/10, 8 7/10, 9 2/10, …
Answer: The change between terms is adding 5/10 (or 0.5). Continuing from 9 2/10, we add 5/10 each time: 9 2/10 + 5/10 = 9 7/10; 9 7/10 + 5/10 = 10 2/10; 10 2/10 + 5/10 = 10 7/10. The next three terms are 9 7/10, 10 2/10, 10 7/10.
In simple words: Each number is half a unit (5 tenths) more than the one before.

Exam Tip: Patterns with fractions work the same way as patterns with whole numbers - find the common difference and keep adding it.

Question 4(c). 7 6/10, 8 7/10, 9 8/10, …
Answer: The change between consecutive terms is adding 1 1/10 (or 1.1). Continuing the pattern: 9 8/10 + 1 1/10 = 10 9/10; 10 9/10 + 1 1/10 = 12; 12 + 1 1/10 = 13 1/10. The next three terms are 10 9/10, 12, 13 1/10.
In simple words: Each number gets bigger by 1 and 1 tenth.

Exam Tip: When a pattern crosses a whole number (like going from 11 9/10 to 12), handle the regrouping carefully.

Question 4(d). 5 7/10, 5 3/10, 4 9/10, …
Answer: This time, the pattern moves backward - each term subtracts 4/10 (or 0.4) from the previous one. Continuing: 4 9/10 - 4/10 = 4 5/10; 4 5/10 - 4/10 = 4 1/10; 4 1/10 - 4/10 = 3 7/10. The next three terms are 4 5/10, 4 1/10, 3 7/10.
In simple words: Each number is 4 tenths smaller than the one before it.

Exam Tip: Not all patterns increase - some decrease, so always check whether you're adding or subtracting.

Question 4(e). 13 5/10, 13, 12 5/10, …
Answer: The pattern shows decreasing values - each term is 5/10 (or 0.5) less than the previous one. Continuing: 12 5/10 - 5/10 = 12; 12 - 5/10 = 11 5/10; 11 5/10 - 5/10 = 11. The next three terms are 12, 11 5/10, 11.
In simple words: Each number drops by half a unit (5 tenths).

Exam Tip: Decreasing patterns often alternate between whole numbers and mixed numbers - watch this carefully.

Question 4(f). 11 5/10, 10 4/10, 9 3/10, …
Answer: Each term decreases by 1 1/10 (or 1.1). Following the pattern: 9 3/10 - 1 1/10 = 8 2/10; 8 2/10 - 1 1/10 = 7 1/10; 7 1/10 - 1 1/10 = 6. The next three terms are 8 2/10, 7 1/10, 6.
In simple words: Each number is 1 and 1 tenth smaller than the one before.

Exam Tip: Subtract carefully when the fractional part of the top number is smaller than the part being subtracted - you may need to borrow.

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Question. What is the length of this smaller part?
Answer: Since each one-tenth is divided into 10 equal parts, one smaller part has a length of 1/10 of 1/10. This works out to 1/100 of a unit - or one-hundredth.
In simple words: If you split a tenth into 10 more pieces, each piece becomes one-hundredth.

Exam Tip: Remember: 1/100 = 1/10 ÷ 10, which is key to understanding how smaller decimal places are formed.

Question. How many such smaller parts make a unit length?
Answer: There are 10 tenths in one unit, and there are 10 hundredths in each tenth. Therefore, multiplying these together: 10 × 10 = 100 hundredths in a unit length.
In simple words: A whole unit contains 100 tiny hundredth-sized parts.

Exam Tip: This 10 × 10 = 100 relationship is crucial - it shows why 1/100 exists between 1/10 and 1/1000.

Question. What is the length of the folded paper?
Answer: According to the text and diagrams, the folded length is 4 whole units, plus 4 tenths, plus 5 hundredths. This is written as: Length = 4 + 4/10 + 5/100. We read this as "4 units and 4 one-tenths and 5 one-hundredths". Note: The calculation shows that half of 8 9/10 is (1/2) × 89/10 = 89/20 = 445/100 = 4 + 45/100 = 4 + 4/10 + 5/100.
In simple words: The folded paper's length is 4 whole units plus 4/10 plus 5/100, which can also be written as 4.45 units.

Exam Tip: Breaking a decimal into its place value parts (units, tenths, hundredths) makes it easier to understand and work with.

Question. How many one-hundredths make one-tenth? Can we also say that the length (of the folded paper) is 4 units and 45 one-hundredths?
Answer: There are 10 hundredths (10/100) in one-tenth (1/10). Yes, the length 4 + 4/10 + 5/100 can be expressed as 4 units and 45 one-hundredths (4 45/100). This works because 4/10 is equal to 40/100, so when we combine the fractional parts, we get 40/100 + 5/100 = 45/100.
In simple words: The same measurement can be written in different ways - 4 4/10 5/100 is the same as 4 45/100.

Exam Tip: Flexible thinking about decimal representation helps you solve problems more easily - one number can have multiple correct forms.

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Question 1. Observe the figure below. Notice the markings and the corresponding lengths written in the boxes when measured from 0. Fill the lengths in the empty boxes.
Answer: Based on standard intervals and the markings shown:
- The box after 2/10 (which equals 20/100): Likely 3/10 or 30/100.
- The box after 130/100: Likely 131/100 or 1 31/100.
- The box after 31/10: This is 3.1 or 3 10/100. The next box might be something like 3 15/100 or 3.15, depending on where the arrow points in the diagram.
In simple words: Look at where the marks on the number line are spaced, then fill in the missing values by following that same spacing pattern.

Exam Tip: When reading number lines with decimal values, always check the interval between marked points first - this tells you what the missing values should be.

Question 2. Example (Wire measurement): The length 1 14/100 is shown written in three ways: - "One and one-tenth and four-hundredths" (1 + 1/10 + 4/100) - "One and fourteen-hundredths" (1 14/100) - "One Hundred and Fourteen-hundredths" (114/100) Can you see how they denote the same length?
Answer: All three forms represent the same value. Let's verify: 1 + 1/10 + 4/100 can be rewritten by converting 1/10 to hundredths (1/10 = 10/100), giving us 1 + 10/100 + 4/100 = 1 + 14/100 = 1 14/100. Additionally, 1 14/100 can be written as an improper fraction: 1 14/100 = 100/100 + 14/100 = 114/100. Each form expresses the same measurement using different notation.
In simple words: The same distance can be written three ways - as separate whole and fractional parts, as a mixed number, or as an improper fraction - but they all mean the same thing.

Exam Tip: Flexibility with decimal notation - switching between mixed numbers, improper fractions, and place value form - is a vital skill for working with decimals.

Question 3. For the lengths shown below (on rulers) write the measurements and read out the measures in words.
Answer: Based on the ruler positions shown:

Ruler 1 (pointing between 5 7/10 and 5 8/10): Assuming it points to 5 78/100. Measurement: 5 78/100 units. Read aloud: "Five and seventy-eight hundredths units".

Ruler 2 (pointing between 14 8/10 and 14 9/10): Assuming it points to 14 83/100. Measurement: 14 83/100 units. Read aloud: "Fourteen and eighty-three hundredths units".

Ruler 3 (pointing between 7 and 7 1/10): Assuming it points to 7 2/100. Measurement: 7 2/100 units. Read aloud: "Seven and two hundredths units".

Ruler 4 (pointing to 9 8/10): Measurement: 9 8/10 units (or 9 80/100). Read aloud: "Nine and eight tenths units" or "Nine and eighty hundredths units".
In simple words: Read the ruler carefully - say the whole number, then "and", then the fractional part in words.

Exam Tip: Always read decimal measurements aloud exactly as they are written - this helps you check if your reading is correct.

Question. In each group, identify the longest and the shortest lengths. Mark each length on the scale. (a) 3/10, 3/100, 33/100
Answer: Convert all to the same form for easy comparison: 3/10 = 0.30; 3/100 = 0.03; 33/100 = 0.33. Comparing the decimal values: 0.03 < 0.30 < 0.33. Therefore, the shortest length is 3/100 and the longest is 33/100.
In simple words: Change everything to decimals with the same number of places, then arrange them - the smallest decimal is shortest and the largest is longest.

Exam Tip: Always convert to the same decimal places (like three decimal places, 0.030) to compare more easily.

Question. (b) 3 1/10, 30/10, 1 3/10
Answer: Convert to decimals to compare: 3 1/10 = 3.1; 30/10 = 3.0; 1 3/10 = 1.3. Comparing: 1.3 < 3.0 < 3.1. The shortest is 1 3/10 and the longest is 3 1/10.
In simple words: Turn each fraction into a decimal number, then see which is smallest and which is biggest.

Exam Tip: Improper fractions like 30/10 can be confusing - convert them to mixed numbers or decimals first to make comparison obvious.

Question. (c) 45/100, 54/100, 5/10, 4/10
Answer: Convert all to decimals: 45/100 = 0.45; 54/100 = 0.54; 5/10 = 0.50; 4/10 = 0.40. Comparing: 0.40 < 0.45 < 0.50 < 0.54. The shortest is 4/10 and the longest is 54/100.
In simple words: When fractions have different denominators, change them all to decimals to compare easily.

Exam Tip: Decimal form is the clearest way to compare fractions with different denominators - avoid trying to compare tenths and hundredths directly.

Question. (d) 3 6/10, 3 6/100, 3 6/10 6/100
Answer: Convert to decimal form: 3 6/10 = 3.60; 3 6/100 = 3.06; 3 6/10 + 6/100 = 3 + 0.6 + 0.06 = 3.66. Comparing: 3.06 < 3.60 < 3.66. The shortest is 3 6/100 and the longest is 3 6/10 6/100.
In simple words: Be careful with notation - a tenths digit is worth 10 times more than a hundredths digit in the same number.

Exam Tip: The position of a digit (whether it's in the tenths or hundredths place) makes a huge difference in its value - place value is crucial with decimals.

Question. (e) 8/10 2/100, 9/100, 1 8/100
Answer: Convert to decimals: 8/10 + 2/100 = 0.80 + 0.02 = 0.82; 9/100 = 0.09; 1 8/100 = 1.08. Comparing: 0.09 < 0.82 < 1.08. The shortest is 9/100 and the longest is 1 8/100.
In simple words: Add the tenths and hundredths pieces together, then compare all the resulting decimals.

Exam Tip: When a measurement is given as separate parts (like 8/10 and 2/100), always add them first to get the total value.

Question. (f) 7 3/10 5/100, 7 5/10, 7 41/100
Answer: Convert to decimals: 7 + 3/10 + 5/100 = 7 + 0.3 + 0.05 = 7.35; 7 5/10 = 7.50; 7 41/100 = 7.41. Comparing: 7.35 < 7.41 < 7.50. The shortest is 7 3/10 5/100 and the longest is 7 5/10.
In simple words: Add all the parts first, convert to decimals, then line them up to see which is smallest and largest.

Exam Tip: When comparing decimals, look at the tenths place first - if those are equal, then check the hundredths place.

Question. (g) 6 5/10 15/100, 5 87/100, 5 7/100
Answer: Interpreting the first value as 6 15/100 (assuming a transcription note): 6 15/100 = 6.15. The other values are: 5 87/100 = 5.87; 5 7/100 = 5.07. Comparing: 5.07 < 5.87 < 6.15. The shortest is 5 7/100 and the longest is 6 15/100 (based on this interpretation).
In simple words: Even if the notation looks complicated, convert everything to decimals with hundredths and compare.

Exam Tip: Always check the source document carefully for potential transcription errors - if a notation seems odd, try reinterpreting it as a sensible mixed number.

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Question. What will be the sum of 15 3/10 4/100 and 2 6/10 8/100?
Answer: We add 15 + 3/10 + 4/100 and 2 + 6/10 + 8/100 using two approaches.

Method 1 (Grouping Place Values):
Sum the whole parts: 15 + 2 = 17. Sum the tenths: 3/10 + 6/10 = 9/10. Sum the hundredths: 4/100 + 8/100 = 12/100. Combine: 17 + 9/10 + 12/100. Now, regroup the hundredths (12/100 = 10/100 + 2/100 = 1/10 + 2/100). This gives 17 + 9/10 + 1/10 + 2/100 = 17 + 10/10 + 2/100 = 17 + 1 + 2/100 = 18 + 2/100 = 18 2/100.

Method 2 (Vertical Addition):
Stack and add:
15 3/10 4/100
+ 2 6/10 8/100
- - - - - - - -
17 9/10 12/100

Regroup: 12/100 becomes 10/100 + 2/100 = 1/10 + 2/100, giving 17 10/10 2/100. Then 10/10 = 1, so the answer is 18 2/100.

In simple words: Add the whole numbers, tenths, and hundredths separately. Whenever you get 10 or more in one place, carry it to the next place.

Exam Tip: Always line up the decimal places (units, tenths, hundredths) when adding - this prevents mixing up the values.

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Question 1. Are both these methods different? (Referring to Method 1 and Method 2 from the previous page for adding 15 3/10 4/100 + 2 6/10 8/100)
Answer: No, the two methods are fundamentally the same process, just presented differently. Both involve adding the corresponding place values (units, tenths, hundredths) separately, and then regrouping (or "carrying over") whenever a place value sum reaches 10 or more. Method 2 (vertical addition) is a more compact, streamlined way of writing down the steps that Method 1 shows in detail.
In simple words: Both methods do exactly the same thing - add by place value and carry when needed - just written differently.

Exam Tip: Understanding that different methods achieve the same answer builds your confidence in choosing whichever approach feels most natural to you.

Question. Observe the addition done below for 483 + 268. Do you see any similarities between the methods shown above (for adding numbers with tenths/hundredths)?
Answer: Yes, there are strong similarities. The whole number addition here breaks down the numbers by place value (hundreds, tens, ones): (400 + 80 + 3) + (200 + 60 + 8). It then adds corresponding places together: (400 + 200) + (80 + 60) + (3 + 8) = 600 + 140 + 11. When a place value sum reaches 10 or more (such as 11 ones or 14 tens), it is regrouped into the next higher place: 11 ones becomes 1 ten + 1 one, and 14 tens becomes 1 hundred + 4 tens. The final result is (600 + 100) + (40 + 10) + 1 = 700 + 50 + 1 = 751. This regrouping process is exactly the same principle used when adding tenths and hundredths - 10 hundredths regroup to 1 tenth, and 10 tenths regroup to 1 unit. Both operations follow the same base-10 system and place value structure.
In simple words: Whether you're adding whole numbers or decimals, the rule is always the same - add like places together, then carry when you have 10 or more in any place.

Exam Tip: Recognizing that all place value arithmetic follows the same rules helps you understand and apply those rules more confidently across all number types.

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Question. Solve this difference 25 9/10 - 6 4/10 7/100 by converting to hundredths.
Answer: First, convert both numbers to hundredths. 25 9/10 = 25 90/100 (since 9/10 = 90/100), which becomes 2500/100 + 90/100 = 2590/100. Next, 6 4/10 7/100 = 6 + 40/100 + 7/100 = 6 47/100, which becomes 600/100 + 47/100 = 647/100. Now subtract: 2590/100 - 647/100 = (2590 - 647)/100 = 1943/100. Convert back to mixed form: 1943/100 = 1900/100 + 43/100 = 19 + 43/100 = 19 43/100.
In simple words: Change both numbers to the same denominator (hundredths), subtract the top numbers, then change back to mixed form.

Exam Tip: Converting to a single common denominator before subtracting avoids the complexity of borrowing across different place values.

Question. What is the difference 15 3/10 4/100 - 2 6/10 8/100?
Answer: We need to calculate 15 34/100 - 2 68/100. Using vertical subtraction with regrouping:

Since 34/100 is smaller than 68/100 in the fractional part, we must borrow. The calculation proceeds through these regrouping steps:
15 3/10 4/100 becomes 15 2/10 14/100 (borrow from tenths to hundredths).
Then 15 2/10 14/100 becomes 14 12/10 14/100 (borrow from units to tenths).

Now subtract:
14 12/10 14/100
- 2 6/10 8/100
- - - - - - - -
12 6/10 6/100

Working place by place: 14 - 2 = 12 units; 12/10 - 6/10 = 6/10 tenths; 14/100 - 8/100 = 6/100 hundredths. The difference is 12 6/10 6/100 or 12 66/100.
In simple words: When you can't subtract the hundredths, borrow from the tenths. When you can't subtract the tenths, borrow from the units.

Exam Tip: Borrowing with multiple decimal places requires careful regrouping - write out each borrow step clearly so you don't lose track.

Question. Observe the subtraction done below for 653 - 268. Do you see any similarities with the methods shown above?
Answer: Yes, the similarity lies in the borrowing (or regrouping) process. When subtracting in a place value where the top digit is smaller than the bottom digit - such as trying to subtract 8 from 3 in the ones place, or 6 from 5 in the tens place - we cannot proceed directly. Instead, we "borrow" from the next higher place. In this case, we start with (600 + 50 + 3) - (200 + 60 + 8). The ones subtraction (3 - 8) is impossible, so we borrow 1 ten (which equals 10 ones) from the tens place, making it (600 + 40 + 13) - (200 + 60 + 8). Now 13 - 8 = 5 works. Next, the tens subtraction shows 40 - 60, which is still impossible, so we borrow 1 hundred (which equals 10 tens) from the hundreds place, making it (500 + 140 + 13) - (200 + 60 + 8). Now 140 - 60 = 80 works. Finally, 500 - 200 = 300. The result is 300 + 80 + 5 = 385. This borrowing method is exactly the same as the regrouping used when subtracting decimals - whenever the value in one place is too small to subtract from, you borrow from the place to the left and regroup the borrowed amount into the smaller units of the current place.
In simple words: Whether subtracting whole numbers or decimals, the borrowing rule stays the same - if the top number is too small, take 1 from the left and add its value to the right place.

Exam Tip: Once you master borrowing with whole numbers, you have the skill for borrowing with decimals too - the principle doesn't change, just the place values.

Question. Find the sums and differences: (a) 3/10 + 3 4/100
Answer: Convert to the same form: 3/10 = 0.30 and 3 4/100 = 3.04. Adding: 0.30 + 3.04 = 3.34 (or 3 34/100).
In simple words: When adding fractions with different denominators, change them both to decimals first.

Exam Tip: Always convert to the same decimal place count (like 0.30, not 0.3) when adding or subtracting decimals.

Question. (b) 9 5/10 7/100 + 2 1/10 3/100
Answer: Convert: 9 5/10 7/100 = 9.57 and 2 1/10 3/100 = 2.13. Adding: 9.57 + 2.13 = 11.70 (which can be written as 11 70/100 or 11 7/10).
In simple words: Add the numbers as decimals, then if needed, convert back to the mixed number form.

Exam Tip: After adding, you can express the answer in whatever form the question asks for - decimal or mixed number.

Question. (c) 15 6/10 4/100 + 14 3/10 6/100
Answer: Convert: 15 6/10 4/100 = 15.64 and 14 3/10 6/100 = 14.36. Adding: 15.64 + 14.36 = 30.00 (or simply 30).
In simple words: Add as decimals - this pair adds up perfectly to a whole number.

Exam Tip: When decimal addition gives .00, remember to simplify and just write the whole number.

Question. (d) 7 7/100 - 4 4/100
Answer: Convert: 7 7/100 = 7.07 and 4 4/100 = 4.04. Subtracting: 7.07 - 4.04 = 3.03 (or 3 3/100).
In simple words: When subtracting numbers with the same denominator, subtract as decimals - it's straightforward.

Exam Tip: With matching decimal places, subtraction becomes a simple place-by-place operation.

Question. (e) 8 6/100 - 5 3/100
Answer: Convert: 8 6/100 = 8.06 and 5 3/100 = 5.03. Subtracting: 8.06 - 5.03 = 3.03 (or 3 3/100).
In simple words: Subtract the whole numbers, then subtract the hundredths separately.

Exam Tip: Always subtract place-by-place to avoid mixing up digits - whole units from whole units, tenths from tenths, and so on.

Question. (f) 12 6/10 2/100 - 9 9/10 9/100 (Assuming typo corrected to 12 62/100 - 9 99/100)
Answer: Convert: 12 62/100 = 12.62 and 9 99/100 = 9.99. Subtracting: 12.62 - 9.99 = 2.63 (or 2 63/100).
In simple words: Line up the decimal points and subtract column by column, borrowing when needed.

Exam Tip: When subtracting decimals and the digit on top is smaller, remember to borrow from the column to the left.

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Section 3.4 Decimal Place Value

Question. Can we not split a unit into 4 equal parts, 5 equal parts, 8 equal parts or any other number of equal parts instead?
Answer: Yes, you absolutely can split a unit into any number of equal parts - 4, 5, 8, or any other quantity. The text shows an example where a unit is divided into 4 equal parts (quarters). On such a scale, a measured length might be recorded as 2 2/4 units.
In simple words: You can divide a unit however you want - into quarters, fifths, eighths, or anything else.

Exam Tip: Dividing into different numbers of parts is mathematically valid - but we use tenths because they fit our decimal system.

Question. Then why split a unit into 10 parts every time (tenths, hundredths, etc.)?
Answer: We split into 10 parts for consistency with our base-10 number system, also known as the decimal or Indian place value system. In this system, each place value is 10 times bigger than the place value immediately to its right - for example, Hundreds = 10 × Tens. Similarly, each place value is 10 times smaller (or 1/10) than the place value immediately to its left - for instance, Tens = Hundreds ÷ 10. By using tenths, hundredths, and thousandths for parts smaller than one unit, we extend this base-10 pattern logically to the right of the units place, maintaining consistency throughout.
In simple words: We use tenths and hundredths because our number system is built on the idea that each place is worth 10 times the one next to it.

Exam Tip: Understanding why we use base-10 divisions helps you see how decimals fit naturally into our entire number system.

Question. Can we extend this further?
Answer: Yes, the place value system can be extended further to the right indefinitely. Dividing the hundredths place (1/100) by 10 gives the thousandths place (1/1000). Dividing that by 10 produces the ten-thousandths place (1/10000), and this process can continue without end, creating increasingly smaller place values.
In simple words: You can keep dividing by 10 forever - making thousandths, ten-thousandths, hundred-thousandths, and so on.

Exam Tip: The decimal system has no limit - you can always write smaller and smaller place values by dividing by 10 again.

Question. What will the fraction be when 1/100 is split into 10 equal parts?
Answer: Splitting 1/100 into 10 equal parts means finding 1/10 of 1/100. Working it out: (1/10) × (1/100) = 1/1000. The resulting fraction is 1/1000, which is called one-thousandth.
In simple words: When you split a hundredth into 10 pieces, each piece is one-thousandth.

Exam Tip: Multiplying fractions is how you find fractional parts - 1/10 of 1/100 uses multiplication, not division.

Question. How many thousandths make one unit?
Answer: Since 1 unit = 10 tenths = 100 hundredths = 1000 thousandths, the answer is 1000 thousandths make one unit.
In simple words: There are 1000 thousandths in 1 whole unit.

Exam Tip: Remember the pattern: 10 tenths, 100 hundredths, 1000 thousandths - each number is 10 times the previous.

Question. How many thousandths make one tenth?
Answer: Since 1 tenth = 10 hundredths = 10 × 10 thousandths = 100 thousandths, the answer is 100 thousandths make one tenth.
In simple words: One tenth contains 100 tiny thousandths.

Exam Tip: To find how many smaller units fit into a larger one, multiply the intermediate conversion steps.

Question. How many thousandths make one hundredth?
Answer: Since 1 hundredth = 10 thousandths, the answer is 10 thousandths make one hundredth.
In simple words: A hundredth is made of 10 thousandths.

Exam Tip: Each place value step has a factor of 10 - use this to move up or down the scale.

Question. How many tenths make one ten?
Answer: Since 1 ten = 10 units = 10 × 10 tenths = 100 tenths, the answer is 100 tenths make one ten.
In simple words: One ten is the same as 100 tenths.

Exam Tip: As you move left (toward bigger numbers), multiply by 10 each time to find the total count of smaller units.

Question. How many hundredths make one ten?
Answer: Since 1 ten = 10 units = 10 × 100 hundredths = 1000 hundredths, the answer is 1000 hundredths make one ten.
In simple words: You need 1000 hundredths to make one ten.

Exam Tip: When moving from hundredths to larger whole numbers, the multiplier grows - tens require 1000 hundredths, not just 10.

Question. Make a few more questions of this kind and answer them. How many hundredths make one hundred?
Answer: Since 1 hundred = 100 units = 100 × 100 hundredths = 10,000 hundredths, the answer is 10,000 hundredths make one hundred.
In simple words: One hundred units equals 10,000 hundredths.

Exam Tip: Multiplying place value conversions shows how dramatically the count increases for bigger numbers.

Question. How many tenths make one thousand?
Answer: Since 1 thousand = 1000 units = 1000 × 10 tenths = 10,000 tenths, the answer is 10,000 tenths make one thousand.
In simple words: A thousand can be divided into 10,000 tenths.

Exam Tip: These conversion problems strengthen your understanding of how all parts of the place value system connect.

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Question. Make a place value table similar to the one above. Write each quantity in decimal form and in terms of place value and read the number: (a) 2 ones, 3 tenths and 5 hundredths
Answer: Decimal form: 2.35. Place Value expression: 2 × 1 + 3 × 1/10 + 5 × 1/100. How to read aloud: "Two point three five".
In simple words: List each digit, its place value, and multiply - then read the decimal using "point" for the decimal mark.

Exam Tip: Always write the place value breakdown using multiplication - this shows you understand what each digit means.

Question. (b) 1 ten and 5 tenths
Answer: Decimal form: 10.5. Place Value expression: 1 × 10 + 0 × 1 + 5 × 1/10. How to read aloud: "Ten point five".
In simple words: Write the number with the decimal point, showing where the units place is - read the tenths after saying "point".

Exam Tip: Always include the zeros in the place value expression - this shows that no hundreds, units (when needed), etc. are present.

Question. (c) 4 ones and 6 hundredths
Answer: Decimal form: 4.06. Place Value expression: 4 × 1 + 0 × 1/10 + 6 × 1/100. How to read aloud: "Four point zero six".
In simple words: Include the zero in the tenths place - this is important because 4.06 is different from 4.6.

Exam Tip: Never skip zeros between the decimal point and non-zero digits - they change the value completely.

Question. (d) 1 hundred, 1 one and 1 hundredth
Answer: Decimal form: 101.01. Place Value expression: 1 × 100 + 0 × 10 + 1 × 1 + 0 × 1/10 + 1 × 1/100. How to read aloud: "One hundred one point zero one".
In simple words: List all place values from hundreds down to hundredths, including the zeros that show no tens and no tenths.

Exam Tip: Zeros hold places in both whole number and decimal parts - always show them in the place value breakdown.

Question. (e) 8/100 and 9/10 (Assuming these are parts of one number)
Answer: These parts combine to form one complete number: 9/10 + 8/100. Convert to common denominator: 90/100 + 8/100 = 98/100 = 0.98. Decimal form: 0.98. Place Value expression: 0 × 1 + 9 × 1/10 + 8 × 1/100. How to read aloud: "Zero point nine eight".
In simple words: When given separate fractional parts, add them first to get the full decimal value.

Exam Tip: Recognize when different fractions are pieces of one number - combine them before converting to decimal.

Question. (f) 5/100
Answer: Decimal form: 0.05. Place Value expression: 0 × 1 + 0 × 1/10 + 5 × 1/100. How to read aloud: "Zero point zero five".
In simple words: This is a small fraction - remember the zero in the tenths place, or you'll say the wrong number.

Exam Tip: Always write "0.05", never "0.5" for five-hundredths - the zero in the tenths place is critical.

Question. (g) 1/10
Answer: Decimal form: 0.1. Place Value expression: 0 × 1 + 1 × 1/10. How to read aloud: "Zero point one".
In simple words: This is one-tenth - write it as 0.1 or 0.10 if you want to show hundredths too.

Exam Tip: One-tenth can be written as 0.1 or 0.10 - both are correct and mean the same value.

Question. (h) 2 1/100 and 4 1/10 and 7 7/1000 (Assuming sum)
Answer: Add the three parts: 2 1/100 + 4 1/10 + 7 7/1000. First, convert to a common denominator (thousandths): 2 + 1/100 = 2 + 10/1000; 4 + 1/10 = 4 + 100/1000; 7 7/1000 stays as 7 7/1000. Adding: (2 + 4 + 7) + (10/1000 + 100/1000 + 7/1000) = 13 + 117/1000 = 13 + 1/10 + 1/100 + 7/1000 = 13.117. Decimal form: 13.117. Place Value expression: 1 × 10 + 3 × 1 + 1 × 1/10 + 1 × 1/100 + 7 × 1/1000. How to read aloud: "Thirteen point one one seven".
In simple words: Add mixed numbers and fractions by converting to the same denominator first, then combine all parts into one decimal.

Exam Tip: When reading a 3-digit decimal like 13.117, say each digit after the point separately: "one-one-seven", not "one hundred seventeen".

Question. How can we write 234 tenths in decimal form?
Answer: The text explains: 234 tenths = 234/10. Break this down by place value: 234/10 = 200/10 + 30/10 + 4/10 = 20 + 3 + 4/10 = 23.4. So 234 tenths written in decimal form is 23.4.
In simple words: Divide the numerator by the denominator - 234 ÷ 10 = 23.4.

Exam Tip: When converting a fraction to decimal, divide the top by the bottom - for tenths, this just moves the decimal point one place left.

Question. Write these quantities in decimal form: (a) 234 hundredths, (b) 105 tenths.
Answer: (a) 234 hundredths = 234/100. Expanding by place value: 234/100 = 200/100 + 30/100 + 4/100 = 2 + 3/10 + 4/100 = 2.34. So 234 hundredths in decimal form is 2.34.

(b) 105 tenths = 105/10. Expanding: 105/10 = 100/10 + 5/10 = 10 + 5/10 = 10.5. So 105 tenths in decimal form is 10.5.
In simple words: Divide the number by its denominator - 234 ÷ 100 gives 2.34; 105 ÷ 10 gives 10.5.

Exam Tip: When dividing by 100, move the decimal point 2 places left; when dividing by 10, move it 1 place left.

Section 3.5 Units of Measurement

Question. Length Conversion: Recall 1 cm = 10 mm (millimeters). How many cm is 1 mm?
Answer: Since 10 mm = 1 cm, we can write: 1 mm = (1/10) cm = 0.1 cm. So 1 millimeter equals one-tenth of a centimeter.
In simple words: A millimeter is 1/10 of a centimeter.

Exam Tip: Always set up a proportion when converting units - if 10 mm = 1 cm, then 1 mm = 1 cm ÷ 10.

Question. How many cm is (a) 5 mm? (b) 12 mm?
Answer: (a) Since 1 mm = (1/10) cm, then 5 mm = 5 × (1/10) cm = (5/10) cm = 0.5 cm. So 5 millimeters equals half a centimeter.

(b) 12 mm can be split: 12 mm = 10 mm + 2 mm = 1 cm + 2 × (1/10) cm = 1 cm + (2/10) cm = 1.2 cm. So 12 millimeters equals 1.2 centimeters.
In simple words: Multiply the number of millimeters by 1/10 to get centimeters.

Exam Tip: Break longer measurements into whole and fractional parts to make conversion easier - 12 mm = 10 mm (1 cm) + 2 mm (0.2 cm).

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Question. How many m is (a) 10 cm? (b) 15 cm?
Answer: (a) Since 1 cm = (1/100) m, then 10 cm = 10 × (1/100) m = (10/100) m = (1/10) m = 0.1 m. So 10 centimeters equals one-tenth of a meter.

(b) 15 cm = 15 × (1/100) m = (15/100) m. This can be rewritten: 15/100 m = 10/100 m + 5/100 m = (1/10) m + (5/100) m = 0.1 m + 0.05 m = 0.15 m. So 15 centimeters equals 0.15 meters (or 15 hundredths of a meter).
In simple words: To change centimeters to meters, multiply by 1/100 - or divide by 100.

Exam Tip: Remember that 1 meter = 100 centimeters - use this to set up all cm-to-m conversions.

Question. Fill in the blanks below (cm - m conversions): - 36 cm = 0.36 m - 50 cm = 0.50 m (or 0.5 m) - 89 cm = 0.89 m - 4 cm = 0.04 m - 325 cm = 3.25 m - 207 cm = 2.07 m
Answer: These conversions use the fact that 1 cm = (1/100) m. Divide each centimeter value by 100 (or multiply by 1/100):
- 36 cm = 36/100 = 0.36 m (since 36 ÷ 100 = 0.36)
- 50 cm = 50/100 = 0.50 m
- 89 cm = 89/100 = 0.89 m
- 4 cm = 4/100 = 0.04 m
- 325 cm = 325/100 = 3.25 m (which is 3 meters and 25 centimeters)
- 207 cm = 207/100 = 2.07 m
In simple words: To convert cm to m, move the decimal point 2 places to the left (divide by 100).

Exam Tip: Dividing by 100 moves the decimal point 2 places left - remember this shortcut for cm-to-m conversion.

Question. How many mm does 1 meter have?
Answer: We know that 1 m = 100 cm. Since 1 cm = 10 mm, multiplying: 1 m = 100 × 10 mm = 1000 mm. Therefore, 1 meter contains 1000 millimeters.
In simple words: A meter has 1000 millimeters - multiply 100 cm by 10 mm/cm.

Exam Tip: Chain conversions together - meters to cm, then cm to mm - to find the total count of tiny units.

Question. Can we write 1 mm = 1/1000 m?
Answer: Yes, this is correct. Since 1000 mm = 1 m, dividing both sides by 1000 gives us 1 mm = (1/1000) m. So we can indeed write one millimeter as one one-thousandth of a meter.
In simple words: If 1000 millimeters make 1 meter, then 1 millimeter is 1/1000 of a meter.

Exam Tip: This fraction form shows why we created the thousandths place - it measures millimeters in terms of meters.

Question. How many kilograms is 5 g?
Answer: Since 1 g = (1/1000) kg, then 5 g = 5 × (1/1000) kg = (5/1000) kg = 0.005 kg. So 5 grams equals five one-thousandths of a kilogram.
In simple words: A gram is 1/1000 of a kilogram, so multiply by 5 to get 5 grams in kilograms.

Exam Tip: Use the same method for all unit conversions - set up the base relationship first, then multiply or divide as needed.

Question. How many kilograms is 10 g?
Answer: Since 1 g = (1/1000) kg, then 10 g = 10 × (1/1000) kg = (10/1000) kg. We can simplify this: (10/1000) kg = (1/100) kg = 0.01 kg (or 0.010 kg if we want to show three decimal places). So 10 grams equals one one-hundredth of a kilogram.
In simple words: Ten grams is the same as 1/100 of a kilogram, written as 0.01 kg.

Exam Tip: After converting, simplify the fraction if possible - 10/1000 reduces to 1/100, which is cleaner.

Question. Calculation for 75 paise = 0.75 rupee (continued from previous context).
Answer: Since 100 paise = 1 rupee, we have 75 paise = (75/100) rupee. This breaks down as: (75/100) rupee = (70/100 + 5/100) rupee = (7/10 + 5/100) rupee = 0.75 rupee. So 75 paise equals 0.75 rupees, or 75 one-hundredths of a rupee.
In simple words: Divide paise by 100 to get rupees - 75 divided by 100 is 0.75 rupees.

Exam Tip: Currency conversions follow the same logic as other unit conversions - use the base relationship (100 p = 1 Rs) and divide.

Question. Fill in the blanks below (rupee - paise conversions): - 10 p = Rs 0.10 - 5 p = Rs 0.05 - 36 p = Rs 0.36 - 50 p = Rs 0.50 - 9 p = Rs 0.09 - 250 p = Rs 2.50
Answer: All conversions use the fact that 100 paise = 1 rupee. Divide each paise amount by 100 (move decimal 2 places left):
- 10 p = 10/100 = Rs 0.10
- 5 p = 5/100 = Rs 0.05
- 36 p = 36/100 = Rs 0.36
- 50 p = 50/100 = Rs 0.50
- 9 p = 9/100 = Rs 0.09
- 250 p = 250/100 = Rs 2.50
In simple words: Divide paise by 100 to change to rupees - this moves the decimal point 2 places to the left.

Exam Tip: Currency always uses 2 decimal places - even if the answer is "2.50", never drop the final zero when dealing with money.

Question. Name all the divisions between 1 and 1.1 on the number line.
Answer: Assuming the space between 1 and 1.1 is divided into 10 equal parts representing hundredths (as shown in a magnified view showing 1.04), the divisions would be: 1.01, 1.02, 1.03, 1.04, 1.05, 1.06, 1.07, 1.08, 1.09. These are the nine marked points between 1.00 and 1.10.
In simple words: The gap between 1 and 1.1 contains 10 equal pieces, so we mark them at 1.01, 1.02, and so on through 1.09.

Exam Tip: When zooming in on a number line, each interval gets divided into 10 equal parts - use this to find values at different scales.

Question. Identify and write the decimal numbers against the letters (A, B, C) on the number line shown (between 5 and 5.4).
Answer: The scale shows major markings at 5.1, 5.2, 5.3, and 5.4. The space between each major marking is divided into 10 smaller parts (hundredths). Based on careful reading of point positions:
- Point A appears to be 3 small parts past 5.1, so A represents 5.13
- Point B appears to be 8 small parts past 5.2, so B represents 5.28
- Point C appears to be 5 small parts past 5.3, so C represents 5.35
In simple words: Count the tiny divisions past each major marking to find the exact decimal value at each letter.

Exam Tip: Always count small divisions carefully on magnified number lines - off by one division changes the answer by one hundredth.

Question. There is Zero Dilemma! Sonu says that 0.2 can also be written as 0.20, 0.200; Zara thinks that putting zeros on the right side may alter the value of the decimal number. What do you think?
Answer: Sonu is correct. Adding zeros to the right of the last non-zero digit after the decimal point does not change the value of the number. Let's verify: 0.2 represents 2 tenths (2/10). 0.20 represents 2 tenths and 0 hundredths (2/10 + 0/100), which still equals 2/10. 0.200 represents 2 tenths, 0 hundredths, and 0 thousandths (2/10 + 0/100 + 0/1000), which is still 2/10. All three forms are mathematically equal. However, if you add zeros between the decimal point and non-zero digits - like 0.02 or 0.002 - the value does change, because now the digit 2 sits in a different place value.
In simple words: Trailing zeros (zeros at the end after a decimal point) don't change the value - but zeros in the middle do matter.

Exam Tip: 0.2 = 0.20 = 0.200, but 0.2 ≠ 0.02 - leading zeros between the point and the first digit are critical.

Question. Can you tell which of these (0.2, 0.02, 0.002) is the smallest and which is the largest?
Answer: Let's break down what each represents: 0.2 represents 2 tenths (2/10). 0.02 represents 2 hundredths (2/100). 0.002 represents 2 thousandths (2/1000). Comparing using a common denominator: 2/1000 < 2/100 < 2/10, which means 0.002 < 0.02 < 0.2. Therefore, the smallest is 0.002 and the largest is 0.2.
In simple words: The further right the digit is from the decimal point, the smaller its value - so 0.002 is tiny and 0.2 is much bigger.

Exam Tip: Place value determines size in decimals - a digit in the tenths place is worth 10 times a digit in the hundredths place.

Question. Which of these are the same: 4.5, 4.05, 0.405, 4.050, 4.50, 4.005, 04.50?
Answer: To identify which are equal, rewrite each with the same number of decimal places. Using three decimal places as the standard: 4.5 = 4.500; 4.05 = 4.050; 0.405 = 0.405; 4.050 = 4.050; 4.50 = 4.500; 4.005 = 4.005; 04.50 = 4.500 (leading zeros don't matter). Now we can group the equal values:
- Group 1: 4.5, 4.50, 04.50 (all equal 4.500)
- Group 2: 4.05, 4.050 (both equal 4.050)
- Standalone: 0.405 = 0.405 and 4.005 = 4.005 are each unique
In simple words: Trailing zeros don't change value, so 4.5 and 4.50 are the same. But 4.5 and 4.05 are different numbers.

Exam Tip: Always align decimal points and compare place by place - don't rely on how many digits are shown.

Question. Make such number lines (like Figure a) for the decimal numbers: (a) 9.876
Answer: To create a magnified number line for 9.876, follow these steps:

Step 1: Start with a number line showing integers, marking 9 and 10. This shows 9.876 falls between 9 and 10.

Step 2: Magnify the segment between 9 and 10, dividing it into tenths (9.1, 9.2, 9.3, ..., 9.9). Mark 9.8 as the location in this magnified view.

Step 3: Magnify the segment between 9.8 and 9.9, dividing it into hundredths (9.81, 9.82, ..., 9.89). Mark 9.87 as the location in this next magnified view.

Step 4: Magnify the segment between 9.87 and 9.88, dividing it into thousandths (9.871, 9.872, ..., 9.879). Mark the 6th point in this final magnified section, which is exactly 9.876.
In simple words: Zoom in step by step - first to tenths, then to hundredths, then to thousandths - marking where your number falls at each zoom level.

Exam Tip: Magnified number lines show how decimals are organized hierarchically - each zoom level divides the previous section into 10 equal parts.

Question. (b) 0.407
Answer: To create a magnified number line for 0.407:

Step 1: Start with a number line showing integers, marking 0 and 1. This shows 0.407 falls between 0 and 1.

Step 2: Magnify the segment between 0 and 1, dividing it into tenths (0.1, 0.2, ..., 0.9). Mark 0.4 as the location in this magnified view, noting that 0.407 falls in the interval between 0.4 and 0.5.

Step 3: Magnify the segment between 0.4 and 0.5, dividing it into hundredths (0.41, 0.42, ..., 0.49). Mark 0.40 (which is the same as 0.4) as the left boundary. Note that 0.407 falls between 0.40 and 0.41.

Step 4: Magnify the segment between 0.40 and 0.41, dividing it into thousandths (0.401, 0.402, ..., 0.409). Mark the 7th point in this section, which is exactly 0.407.
In simple words: Use the same zoom-in strategy - start at integers, then zoom to tenths, then hundredths, then thousandths - until you reach your target number.

Exam Tip: Whether the number is large or small, the magnification process works the same way - divide into 10 parts at each step.

Question. Using similar reasoning find out the decimal numbers in the boxes below (labelled d, e, f, g, h on number lines).
Answer: Based on the magnified number line images:

Number line 1 (between 4.3 and 4.8): The distance is 4.8 - 4.3 = 0.5 units. This interval is divided into 10 equal parts, so each part = 0.5 ÷ 10 = 0.05 units.
- Box 'd' is at the 2nd mark after 4.3: 4.3 + (2 × 0.05) = 4.3 + 0.10 = 4.40
- Box 'e' is at the 7th mark after 4.3: 4.3 + (7 × 0.05) = 4.3 + 0.35 = 4.65

Number line 2 (between 8 and 8.1): The distance is 8.1 - 8 = 0.1 units. This interval is divided into 10 equal parts, so each part = 0.1 ÷ 10 = 0.01 units.
- Box 'f' is at the 1st mark after 8: 8 + (1 × 0.01) = 8.01
- Box 'g' is at the 4th mark after 8: 8 + (4 × 0.01) = 8.04
- Box 'h' is at the 9th mark after 8: 8 + (9 × 0.01) = 8.09
In simple words: Find the interval size, divide by 10, count the marks, and multiply to get the value in each box.

Exam Tip: Always calculate the size of each small division first - this prevents counting errors.

Question 1. Which is larger: 6.456 or 6.465?
Answer: Start by comparing place values from left to right. In the units place, both numbers have 6, so they are the same. In the tenths place, both have 4, still the same. In the hundredths place, the first number has 5 while the second has 6. Since 6 is bigger than 5, we can stop here and conclude that 6.465 is larger than 6.456.
In simple words: Look at each place value from left to right. When you find a place where the digits differ, the number with the bigger digit in that place is the larger number.

Exam Tip: Always compare place values from left to right and stop as soon as you find a difference - there's no need to check the remaining digits.

Question 2. Why can we stop comparing at the hundredths place? Can we be sure that whatever digits are there after this will not affect our conclusion?
Answer: Yes, we can stop comparing once we find a place value where the digits differ. This is because of how the place value system works - each place is 10 times smaller than the one to its left. This means a difference in a higher place value (like hundredths) is always more important than any possible difference in lower place values (like thousandths or ten-thousandths). For example, 0.41 will always be bigger than 0.3999 because the difference in the tenths place (4 versus 3) outweighs any digits that come after it.
In simple words: A bigger difference in a higher place value will always matter more than any difference in lower place values.

Exam Tip: Remember that place value matters more than the number of digits - stop comparing as soon as you find a difference in a particular place value.

Question 3. Which decimal number is greater?
(a) 1.23 or 1.32
(b) 3.81 or 13.800
(c) 1.009 or 1.090
Answer:
(a) Compare the units place: both have 1, so they are the same. Now compare the tenths place: 2 versus 3. Since 3 is bigger than 2, 1.32 is the larger number.
(b) First, look at the whole number parts: 3 versus 13. Since 13 is bigger than 3, 13.800 is the larger number - no need to check the decimal parts.
(c) Start with the units place: both have 1, so they are the same. The tenths place: both have 0, still the same. Next is the hundredths place: 0 versus 9. Since 9 is bigger than 0, 1.090 is the larger number.
In simple words: Compare place values one by one from left to right. The first place where the digits are different tells you which number is bigger.

Exam Tip: For (b), always compare whole number parts first - if they differ, you have your answer right away.

Question 4. Consider 0.9, 1.1, 1.01, 1.11. Identify the decimal number closest to 1.
Answer: First, arrange these numbers in order: 0.9 - 1 - 1.01 - 1.1 - 1.11. Next, calculate the distance from each number to 1. The distance from 1 to 0.9 is 0.1. The distance from 1 to 1.01 is 0.01. The distance from 1 to 1.1 is 0.1. The distance from 1 to 1.11 is 0.11. The smallest distance is 0.01, which means 1.01 is closest to 1.
In simple words: Find how far each number is from 1. The number with the smallest distance is the closest one.

Exam Tip: To find the closest number to a target value, always calculate the distance (the absolute difference) for each option and pick the one with the smallest distance.

Question 5. Which of 0.9, 1.1, 1.01, 1.11 is closest to 1.09?
Answer: Calculate the distance from 1.09 to each of the given numbers. The distance from 1.09 to 0.9 is 0.19. The distance from 1.09 to 1.1 is 0.01. The distance from 1.09 to 1.01 is 0.08. The distance from 1.09 to 1.11 is 0.02. The smallest distance is 0.01, so 1.1 is closest to 1.09.
In simple words: Work out how far 1.09 is from each number. The one with the smallest gap is your answer.

Exam Tip: When calculating distances, use absolute values (ignore minus signs) to ensure you get positive differences.

Question 6. Which among 3.56, 3.65, 3.099 is closest to 4?
Answer: Find the distance from each number to 4. The distance from 4 to 3.56 is 0.44. The distance from 4 to 3.65 is 0.35. The distance from 4 to 3.099 is 0.901. The smallest distance is 0.35, which means 3.65 is closest to 4.
In simple words: Subtract each number from 4 to find the gap. The number with the smallest gap is the closest.

Exam Tip: Don't be fooled by numbers with more decimal places - focus on which one has the smallest difference from the target value.

Question 7. Which among 0.8, 0.69, 1.08 is closest to 1?
Answer: Find the distance from each number to 1. The distance from 1 to 0.8 is 0.20. The distance from 1 to 0.69 is 0.31. The distance from 1 to 1.08 is 0.08. The smallest distance is 0.08, so 1.08 is closest to 1.
In simple words: Calculate how far away each number is from 1. Pick the one that's nearest.

Exam Tip: Numbers that are bigger than the target (like 1.08 being bigger than 1) can still be closest if the difference is small.

Question 8. Using the digits 4, 1, 8, 2, and 5 exactly once, make a decimal number as close as possible to 25.
Answer: One approach is to use 2 and 5 for the whole number part to get 25. Then arrange the remaining digits (1, 4, 8) in ascending order after the decimal point to keep the fractional part as small as possible. This gives 25.148, with a distance of 0.148 from 25. Alternatively, you could use 2 and 4 for the whole number part to get 24, and arrange the remaining digits (1, 5, 8) in descending order to make the fractional part as large as possible, giving 24.851 with a distance of 0.149 from 25. Comparing these, 25.148 is closer since 0.148 is smaller than 0.149.
In simple words: Try different ways to use the digits. Calculate how far each attempt is from 25 and pick the closest one.

Exam Tip: When constructing a number close to a target, try different arrangements of the whole and fractional parts, then measure the distance for each to find the best one.

Question 9. Priya requires 2.7 m of cloth for her skirt, and Shylaja requires 3.5 m for her kurti. What is the total quantity of cloth needed?
Answer: To find the total, add 2.7 m and 3.5 m using decimal addition. Line up the decimal points and add column by column from right to left. The tenths place gives 7 + 5 = 12 tenths, which means 1 whole and 2 tenths carried over. The units place gives 2 + 3 + 1 (carried) = 6. So the total is 6.2 m.
In simple words: Line up the decimal points and add the numbers just like you would add whole numbers.

Exam Tip: Always line up decimal points before adding - this keeps place values aligned and prevents mistakes.

Question 10. How much longer is Shylaja's cloth (3.5 m) compared to Priya's (2.7 m)?
Answer: To find the difference, subtract 2.7 m from 3.5 m using decimal subtraction. Line up the decimal points. In the tenths place, we need to subtract 7 from 5, which is not possible, so we borrow 1 from the units place. This turns 5 tenths into 15 tenths. Now 15 - 7 = 8 tenths. In the units place, 3 - 1 (borrowed) - 2 = 0. So the difference is 0.8 m.
In simple words: Subtract the smaller number from the larger one, borrowing when needed, just like subtracting whole numbers.

Exam Tip: When a digit in the minuend is smaller than the digit in the subtrahend, borrow from the next place value.

Question 11. Fill in the blanks: convert grams to kilograms.
Answer:
465 g = 0.465 kg (since 465 - 1000 = 0.465)
68 g = 0.068 kg (since 68 - 1000 = 0.068)
1560 g = 1.560 kg or 1.56 kg (since 1560 - 1000 = 1.560)
704 g = 0.704 kg (since 704 - 1000 = 0.704)
560 g = 0.56 kg (since 0.56 × 1000 = 560)
2500 g = 2.5 kg (since 2.5 × 1000 = 2500)
In simple words: To change grams to kilograms, divide by 1000, which moves the decimal point three places to the left.

Exam Tip: Remember that 1 kg = 1000 g. When converting, always divide or multiply accordingly.

Question 12. What is 1 milligram in terms of grams?
Answer: One gram equals 1000 milligrams. Therefore, 1 milligram is 1 - 1000 of a gram, which is 0.001 g.
In simple words: One milligram is a tiny fraction of a gram - divide 1 gram by 1000 to get a milligram.

Exam Tip: Keep in mind the conversion hierarchy: 1 g = 1000 mg and 1 kg = 1000 g. Use these to convert between units.

Question 13. What is the relationship between paise and rupees?
Answer: There are 100 paise in 1 rupee. This means 1 paisa equals 1 - 100 of a rupee, or 0.01 rupee. Historically, paisa coins were used as smaller denominations in Indian currency.
In simple words: One rupee is made up of 100 paise, so each paisa is worth 0.01 rupees.

Exam Tip: When converting paise to rupees, divide the number of paise by 100.

Question 14. Convert 75 paise to rupees.
Answer: To convert paise to rupees, divide by 100. So 75 paise equals 75 - 100 rupees. This can be written as (70 - 100 + 5 - 100) rupees, or (7 - 10 + 5 - 100) rupees, which equals 0.75 rupees.
In simple words: Move the decimal point two places to the left: 75 paise becomes 0.75 rupees.

Exam Tip: To change paise to rupees quickly, drop the last two digits and place a decimal point there.

Question 15. Write the detailed place value computation for 84.691 - 77.345 and its compact form.
Answer: In compact form, line up the decimal points and subtract:
84.691
- 77.345
= 7.346
In the detailed place value approach, break down each number by place value: (80 + 4 + 6/10 + 9/100 + 1/1000) - (70 + 7 + 3/10 + 4/100 + 5/1000). Starting from the right: thousandths place, 1/1000 - 5/1000 requires borrowing 1/100 (which becomes 10/1000), so 11/1000 - 5/1000 = 6/1000. The hundredths place becomes 8/100 - 4/100 = 4/100. The tenths place is 6/10 - 3/10 = 3/10. For the units place, 4 - 7 requires borrowing 10 from the tens, so 14 - 7 = 7. The tens place becomes 70 - 70 = 0. The result is 7.346.
In simple words: Subtract place by place from right to left, borrowing when a digit in the top number is smaller than the one below it.

Exam Tip: Use borrowing strategically - if you can't subtract in a particular place, borrow from the next place to the left.

Question 16. Find the sums: (a) 5.3 + 2.6 (b) 18 + 8.8 (c) 2.15 + 5.26 (d) 9.01 + 9.10 (e) 29.19 + 9.91 (f) 6.236 + 0.487 (g) 0.75 + 0.03
Answer:
(a) 5.3 + 2.6 = 7.9
(b) 18 + 8.8 = 18.0 + 8.8 = 26.8
(c) 2.15 + 5.26 = 7.41
(d) 9.01 + 9.10 = 18.11
(e) 29.19 + 9.91 = 39.10 (or 39.1)
(f) 6.236 + 0.487 = 6.723
(g) 0.75 + 0.03 = 0.78
In simple words: Line up the decimal points, add column by column, and place the decimal point in the answer directly below the other decimal points.

Exam Tip: When one number is a whole number, write it with a decimal point to avoid confusion - for example, write 18 as 18.0.

Question 17. Find the differences: (a) 5.6 - 2.3 (b) 18 - 8.8 (c) 10.4 - 4.5 (d) 17 - 16.198 (e) 17 - 0.05 (f) 34.505 - 18.1 (g) 9.9 - 9.09 (h) 6.236 - 0.487
Answer:
(a) 5.6 - 2.3 = 3.3
(b) 18 - 8.8 = 18.0 - 8.8 = 9.2
(c) 10.4 - 4.5 = 5.9
(d) 17 - 16.198 = 17.000 - 16.198 = 0.802
(e) 17 - 0.05 = 17.00 - 0.05 = 16.95
(f) 34.505 - 18.1 = 34.505 - 18.100 = 16.405
(g) 9.9 - 9.09 = 9.90 - 9.09 = 0.81
(h) 6.236 - 0.487 = 5.749
In simple words: Write both numbers with the same number of decimal places, line up the decimal points, and subtract as you would with whole numbers.

Exam Tip: Adding trailing zeros after the decimal point doesn't change the value - it just makes subtraction easier by keeping place values aligned.

Question 18. Identify the change and write the next 3 terms for each sequence: (a) 4.4, 4.45, 4.5, ... (b) 25.75, 26.25, 26.75, ... (c) 10.56, 10.67, 10.78, ... (d) 13.5, 16, 18.5, ... (e) 8.5, 9.4, 10.3, ... (f) 5, 4.95, 4.90, ... (g) 12.45, 11.95, 11.45, ... (h) 36.5, 33, 29.5, ...
Answer:
(a) Change: +0.05. Next 3 terms: 4.55, 4.60, 4.65
(b) Change: +0.50 (or +0.5). Next 3 terms: 27.25, 27.75, 28.25
(c) Change: +0.11. Next 3 terms: 10.89, 11.00, 11.11
(d) Change: +2.5. Next 3 terms: 21.0, 23.5, 26.0
(e) Change: +0.9. Next 3 terms: 11.2, 12.1, 13.0
(f) Change: -0.05. Next 3 terms: 4.85, 4.80, 4.75
(g) Change: -0.50 (or -0.5). Next 3 terms: 10.95, 10.45, 9.95
(h) Change: -3.5. Next 3 terms: 26.0, 22.5, 19.0
In simple words: Find what you add or subtract to get from one term to the next. Keep doing the same operation to find the remaining terms.

Exam Tip: Always check your pattern by verifying that the change is consistent between every pair of consecutive terms.

Question 19. Verify the claim: the sum of two decimal numbers lies between the sum of their whole parts and that sum plus 2. Will this work for any two decimal numbers?
Answer: For the numbers 25.936 and 8.202, the sum is 34.138. The sum of whole parts is 25 + 8 = 33. We check: is 33 < 34.138 < 35? Yes. Now let's prove this works for any two decimal numbers A.a and B.b (where A and B are whole parts and a and b are fractional parts between 0 and 1). The sum is A.a + B.b = (A + B) + (a + b). Since a and b are at least 0, the sum is at least A + B. Since a and b are each less than 1, their sum is less than 2, so the total sum is less than (A + B) + 2. Therefore, the claim holds: (A + B) < Sum < (A + B) + 2. This works regardless of how many decimal places the numbers have.
In simple words: When you add two decimals, the answer is always bigger than the whole parts added together, but smaller than that sum plus 2.

Exam Tip: Use place value reasoning to justify inequality claims - breaking numbers into whole and fractional parts makes the logic clear.

Question 20. Come up with a way to narrow down the range of whole numbers within which the difference of two decimal numbers will lie.
Answer: Let the numbers be A.a and B.b, where A and B are whole parts and a and b are fractional parts (0 ≤ a < 1, 0 ≤ b < 1). The reasonable range for the difference is (A - B - 1) < Difference < (A - B + 1). For example, with 25.936 and 8.202, we get A = 25, B = 8, so A - B - 1 = 16 and A - B + 1 = 18. The actual difference is 17.734, which lies between 16 and 18, confirming the formula works.
In simple words: If you subtract the whole parts, the real answer won't be more than 1 away from that result - it will be within 1 unit in either direction.

Exam Tip: This estimation technique helps you check if your answer is reasonable before calculating the exact difference.

Question 21. Where else can we see such 'non-decimals' with a decimal-like notation?
Answer: Several contexts use notation that looks like decimals but doesn't follow the base-10 fractional system. In time, 2.5 hours means 2 hours and 30 minutes (half an hour), not 2 hours and 50 minutes. For feet and inches, 2.5 ft sometimes means 2 feet and 6 inches (half a foot), though the standard notation is 2' 6". For angles and coordinates, degrees, minutes, and seconds are sometimes written in pseudo-decimal forms in specific contexts, though the standard is to use symbols (°, ', ") to show they are base-60, not base-10 units.
In simple words: A decimal point in time or measurement sometimes means something different than in math - 2.5 hours is two and a half hours, not two hours and fifty hundredths.

Exam Tip: Always check the context when you see a decimal-looking number - in real-world situations, the fractional part might use a different base.

Question 22. Convert the following fractions into decimals: (a) 5/100 (b) 16/1000 (c) 12/10 (d) 254/1000
Answer:
(a) 5/100 = 0.05
(b) 16/1000 = 0.016
(c) 12/10 = 1.2
(d) 254/1000 = 0.254
In simple words: To change a fraction with a power of 10 in the denominator to a decimal, count the zeros in the denominator - that's how many decimal places you need.

Exam Tip: The number of zeros in 10, 100, 1000 tells you exactly where to place the decimal point in your answer.

Question 23. Convert the following decimals into a sum of tenths, hundredths and thousandths: (a) 0.34 (b) 1.02 (c) 0.8 (d) 0.362
Answer:
(a) 0.34 = 3 × 1/10 + 4 × 1/100
(b) 1.02 = 1 × 1 + 0 × 1/10 + 2 × 1/100
(c) 0.8 = 8 × 1/10
(d) 0.362 = 3 × 1/10 + 6 × 1/100 + 2 × 1/1000
In simple words: Write out what each digit represents - the first digit after the decimal is tenths, the second is hundredths, the third is thousandths.

Exam Tip: This expanded form shows you understand place value - each digit is multiplied by its place value fraction.

Question 24. What decimal number does each letter represent in the number line? (Scale from 6.4 to 6.6 shown, divided into hundredths)
Answer: Each major mark on the scale represents 0.1 (6.4, 6.5, 6.6). Each small mark represents 0.01. The letter 'a' is 8 marks after 6.4, so its value is 6.48. The letter 'b' is 9 marks after 6.5, so its value is 6.59. The letter 'c' is 2 marks after 6.4, so its value is 6.42.
In simple words: Count the small marks from a known point and add that count (in hundredths) to find the value.

Exam Tip: On a number line, always identify the scale first - how much does each mark represent - then count carefully from a reference point.

Question 25. Arrange the following quantities in descending order: (a) 11.01, 1.011, 1.101, 11.10, 1.01 (b) 2.567, 2.675, 2.768, 2.499, 2.698 (c) 4.678 g, 4.595 g, 4.600 g, 4.656 g, 4.666 g (d) 33.13 m, 33.31 m, 33.133 m, 33.331 m, 33.313 m
Answer:
(a) 11.10, 11.01, 1.101, 1.011, 1.01
(b) 2.768, 2.698, 2.675, 2.567, 2.499
(c) 4.678 g, 4.666 g, 4.656 g, 4.600 g, 4.595 g
(d) 33.331 m, 33.313 m, 33.31 m, 33.133 m, 33.13 m
In simple words: Compare place values from left to right. Start with the whole number part, then the tenths, hundredths, and so on to arrange them from biggest to smallest.

Exam Tip: When sorting decimals, always check the whole number part first - this usually separates the larger from the smaller numbers right away.

Question 26. Using the digits 1, 4, 0, 8, and 6, make the decimal number closest to 30.
Answer: Consider different possibilities: numbers like 18.xxx, 40.xxx, and 16.xxx can all be formed. For 18.640, the distance from 30 is 30 - 18.640 = 11.360. For 40.168, the distance from 30 is 40.168 - 30 = 10.168. Since 10.168 is smaller than 11.360, the number 40.168 is closest to 30.
In simple words: Try a few different arrangements, calculate how far each is from 30, and pick the one with the smallest distance.

Exam Tip: When you need to construct a number close to a target, think about which whole number part gives you options nearest to that target.

Question 27. Using the digits 1, 4, 0, 8, and 6, make the smallest possible decimal number between 100 and 1000.
Answer: For a number to fall between 100 and 1000, it must have 3 digits before the decimal point. To make it as small as possible, arrange the digits in ascending order: start with 1 (the smallest non-zero digit for the hundreds place), then 0, then 4. Put the remaining digits (6 and 8) after the decimal point, arranged in ascending order as well. The smallest number is 104.68.
In simple words: Use the smallest available digits in the biggest place values first to minimize the overall number.

Exam Tip: When constructing a small number, prioritize placing smaller digits in higher place values.

Question 28. Will a decimal number with more digits be greater than a decimal number with fewer digits?
Answer: Not necessarily. Comparison depends entirely on the place value of the digits, not on how many digits the number has. For example, 0.5 (which has one digit after the decimal) is greater than 0.123 (which has three digits after the decimal) because 5 tenths is bigger than 1 tenth. However, when comparing whole number parts, 12 (which has two digits) is greater than 9 (which has one digit). The total number of digits is not the deciding factor - place value is.
In simple words: More digits doesn't mean a bigger number. What matters is which place values the digits occupy.

Exam Tip: Always compare place values, not digit count, when determining which decimal is larger.

Question 29. Mahi purchases 0.25 kg beans, 0.3 kg carrots, 0.5 kg potatoes, 0.2 kg capsicums and 0.05 kg ginger. Calculate the total weight.
Answer: Add all the quantities: 0.25 + 0.3 + 0.5 + 0.2 + 0.05 kg. To make the addition easier, write each with the same number of decimal places: 0.25 + 0.30 + 0.50 + 0.20 + 0.05 = 1.30 kg. The total weight is 1.3 kg.
In simple words: Line up all the decimal points and add column by column from right to left.

Exam Tip: When adding several decimals, write them all with the same number of decimal places to prevent alignment errors.

Question 30. Pinto supplies 3.79 L, 4.2 L and 4.25 L of milk in the first three days. In 6 days, he supplies 25 litres of milk. Find the total quantity supplied in the last three days.
Answer: First, find the total supply in the first 3 days: 3.79 + 4.20 + 4.25 = 12.24 L. The total supply for all 6 days is 25 L. To find the supply in the last 3 days, subtract: 25.00 - 12.24 = 12.76 L. So Pinto supplied 12.76 L in the last three days.
In simple words: Add up the first three days' supply, then subtract from the total to find what was supplied in the remaining three days.

Exam Tip: When dealing with multi-day totals, always add the known parts first, then subtract from the overall total to find the unknown part.

Question 31. Tinku weighed 35.75 kg in January and 34.50 kg in February. Has he gained or lost weight? How much is the change?
Answer: Compare the two weights: January weight is 35.75 kg, February weight is 34.50 kg. Since 34.50 < 35.75, Tinku has lost weight. The change is calculated as: 35.75 - 34.50 = 1.25 kg. Therefore, Tinku lost 1.25 kg.
In simple words: Subtract the smaller weight from the larger one to find how much the weight changed.

Exam Tip: Always state clearly whether it's a gain or loss, and include the quantity of change in your answer.

Question 32. Extend the pattern: 5.5, 6.4, 6.39, 7.29, 7.28, 6.18, 6.17, ...
Answer: Looking at the pattern, the changes between consecutive terms are: +0.9, -0.01, +0.9, -0.01, -1.1, -0.01. This is not a simple constant rule. The pattern appears to involve alternating operations or possibly contains errors. Without a clear, consistent rule, reliably extending this pattern is not straightforward.
In simple words: This pattern doesn't follow a simple rule, so it's hard to predict the next terms with confidence.

Exam Tip: When a pattern isn't immediately clear, check if there's an alternating pattern, if numbers repeat, or if the rule changes at different points.

Question 33. How many millimeters make 1 kilometer?
Answer: Start with the conversions: 1 km = 1000 m and 1 m = 1000 mm. To find how many millimeters are in 1 kilometer, multiply: 1000 m × 1000 mm/m = 1,000,000 mm. One million millimeters equal 1 kilometer.
In simple words: Multiply the conversion factors: 1000 times 1000 equals 1 million.

Exam Tip: When converting between units, multiply the conversion factors step by step, or write out all the factors and multiply in one go.

Question 34. Indian Railways insurance costs 45 paise per passenger. If 1 lakh people opt for insurance in a day, what is the total insurance fee paid?
Answer: 1 lakh equals 100,000 people. The total fee is calculated as: 100,000 people × 45 paise per person = 4,500,000 paise. To convert to rupees, divide by 100 (since 100 paise = 1 rupee): 4,500,000 ÷ 100 = Rs 45,000. The total insurance fee paid is Rs 45,000.
In simple words: Multiply the number of people by the cost per person to get the total in paise, then change paise to rupees by dividing by 100.

Exam Tip: Always convert currency units at the end if the problem asks for the answer in a specific denomination.

Question 35. Which is greater? (a) 10/1000 or 1/10 (b) One-hundredth or 90 thousandths (c) One-thousandth or 90 hundredths
Answer:
(a) Convert to decimals: 10/1000 = 0.01 and 1/10 = 0.1. Since 0.1 > 0.01, 1/10 is greater.
(b) Convert to decimals: one-hundredth = 1/100 = 0.01 and 90 thousandths = 90/1000 = 0.090. Since 0.090 > 0.010, 90 thousandths is greater.
(c) Convert to decimals: one-thousandth = 1/1000 = 0.001 and 90 hundredths = 90/100 = 0.90. Since 0.90 > 0.001, 90 hundredths is greater.
In simple words: Turn word fractions into decimals, then compare by looking at the place values.

Exam Tip: When comparing fractions and their decimal forms, always convert to the same form (decimal) to make comparison easier.

Question 36. Write the decimal forms of the quantities mentioned: (a) 87 ones, 5 tenths and 60 hundredths (b) 12 tens and 12 tenths (c) 10 tens, 10 ones, 10 tenths, and 10 hundredths (d) 25 tens, 25 ones, 25 tenths, and 25 hundredths
Answer:
(a) 87 ones + 5 tenths + 60 hundredths = 87 + 0.5 + 0.60 = 88.10
(b) 12 tens + 12 tenths = (12 × 10) + (12 × 1/10) = 120 + 1.2 = 121.2
(c) 10 tens + 10 ones + 10 tenths + 10 hundredths = 100 + 10 + 1 + 0.10 = 111.10 (or 111.1)
(d) 25 tens + 25 ones + 25 tenths + 25 hundredths = 250 + 25 + 2.5 + 0.25 = 277.75
In simple words: Multiply each quantity by its place value, then add all the results together.

Exam Tip: Break the number down by place value, calculate each part, and then add them all to get the final decimal.

Question 37. Using each digit 0-9 not more than once, fill the boxes so that the sum is closest to 10.5: [ . ] + [ . ]
Answer: To get a sum closest to 10.5 using unique digits from 0-9, try forming numbers that add up to exactly 10.5 or very close to it. One possibility is 6.92 + 3.58 = 10.50. This uses the digits 6, 9, 2, 3, 5, 8 - all unique and from the available set. The sum is exactly 10.5.
In simple words: Create two decimal numbers using different digits, add them, and check if their sum is very close to 10.5.

Exam Tip: When constructing numbers for a specific sum, work backwards: think about what digits in the tenths and units places would give you the target sum.

Question 38. Write the following fractions in decimal form: (a) 1/2 (b) 3/2 (c) 1/4 (d) 3/4 (e) 1/5 (f) 3/5
Answer:
(a) 1/2 = 0.5
(b) 3/2 = 1.5
(c) 1/4 = 0.25
(d) 3/4 = 0.75
(e) 1/5 = 0.2
(f) 3/5 = 0.6
In simple words: Divide the top number by the bottom number to change a fraction into a decimal.

Exam Tip: These are common fraction-to-decimal conversions worth memorizing, as they appear frequently in mathematics problems.