NCERT Solutions Class 7 Mathematics Ganita Prakash 1 Chapter 08 Working with Fractions

Question 1. Tenzin drinks 1/2 glass of milk every day. How many glasses of milk does he drink in a week? How many glasses of milk did he drink in the month of January?
Answer: Every day, Tenzin consumes 1/2 glass of milk. Since a week has 7 days, the total amount he drinks in a week is 7 multiplied by 1/2, which equals 7/2 or 3 1/2 glasses. For the month of January, which has 31 days, he drinks 31 multiplied by 1/2, giving 31/2 or 15 1/2 glasses.
In simple words: In a week, Tenzin drinks 3 1/2 glasses. In January, he drinks 15 1/2 glasses.

Exam Tip: Multiply the daily amount by the number of days. Always convert improper fractions to mixed numbers in your final answer.

Question 2. A team of workers can make 1 km of a water canal in 8 days. So, in one day, the team can make __________ km of the water canal. If they work 5 days a week, they can make __________ km of the water canal in a week.
Answer: If the team completes 1 km in 8 days, then the distance made in one day is 1/8 km. When working 5 days each week, the total distance made in a week becomes 5 multiplied by 1/8, which equals 5/8 km.
In simple words: The team makes 1/8 km in one day. In 5 days, they make 5/8 km.

Exam Tip: To find work done in one day, divide the total work by the number of days. Multiply by the working days for weekly output.

Question 3. Manju and two of her neighbours buy 5 litres of oil every week and share it equally among the 3 families. How much oil does each family get in a week? How much oil will one family get in 4 weeks?
Answer: Three families share 5 litres equally each week. Each family receives 5 divided by 3, which is 5/3 litres per week. Over 4 weeks, one family gets 4 multiplied by 5/3, equalling 20/3 or 6 2/3 litres.
In simple words: Each family gets 5/3 litres weekly. In 4 weeks, they get 6 2/3 litres.

Exam Tip: Divide the total amount by the number of families for individual share. Multiply by the number of weeks for the extended period.

Question 4. Safia saw the Moon setting on Monday at 10 pm. Her mother, who is a scientist, told her that every day the Moon sets 5/6 hours later than the previous day. How many hours after 10 pm will the moon set on Thursday?
Answer: From Monday to Thursday is 3 days. Since the Moon sets 5/6 hours later each day, the total delay over 3 days equals 3 multiplied by 5/6, which is 15/6 or 5/2 hours. Converting: 5/2 hours = 2.5 hours = 2 hours 30 minutes. Therefore, the Moon sets 2 hours 30 minutes after 10 pm on Thursday, which is 12:30 am.
In simple words: The Moon sets 5/2 hours or 2 hours 30 minutes later on Thursday than on Monday.

Exam Tip: Count the number of days between the start and end dates. Multiply the daily change by this number, then convert hours to hours and minutes.

Question 5. Multiply and then convert it into a mixed fraction: (a) 7 × 3/5 (b) 4 × 1/3 (c) 9/7 × 6 (d) 13/11 × 6
Answer:
(a) 7 multiplied by 3/5 equals 21/5, which converts to the mixed number 4 1/5.
(b) 4 multiplied by 1/3 equals 4/3, which converts to 1 1/3.
(c) 9/7 multiplied by 6 equals 54/7, which converts to 7 5/7.
(d) 13/11 multiplied by 6 equals 78/11, which converts to 7 1/11.
In simple words: Multiply the whole number by the numerator, keep the denominator, then write the result as a mixed number if the fraction is improper.

Exam Tip: After multiplying, always check if the numerator is larger than the denominator. If so, divide to find whole parts and create the mixed fraction.

Question 1. Find the following products. Use a unit square as a whole for representing the fractions: (a) 1/3 × 1/5 (b) 1/4 × 1/3 (c) 1/5 × 1/2 (d) 1/6 × 1/5
Answer:
(a) Divide a square into 5 rows and 3 columns, creating 15 equal parts. Shade one part from both the row (1/5) and column (1/3) representation. The overlap shows 1/15 is shaded. Therefore, 1/3 × 1/5 = 1/15.
(b) Divide the square into 3 rows and 4 columns, creating 12 equal parts. The double-shaded overlap is 1/12. Therefore, 1/4 × 1/3 = 1/12.
(c) Divide the square into 2 rows and 5 columns, creating 10 equal parts. The overlap shows 1/10 shaded. Therefore, 1/5 × 1/2 = 1/10.
(d) Divide the square into 5 rows and 6 columns, creating 30 equal parts. The overlap is 1/30 shaded. Therefore, 1/6 × 1/5 = 1/30.
In simple words: To multiply two fractions visually, divide a square by the denominators. The shaded overlap region shows the product as a fraction of the whole.

Exam Tip: The product of two proper fractions is always smaller than either fraction. The denominator of the product is the product of the two denominators.

Question 2. Find the following products. Use a unit square as a whole for representing the fractions and carrying out the operations: (a) 2/3 × 4/5 (b) 1/4 × 2/3 (c) 3/5 × 1/2 (d) 4/6 × 3/5
Answer:
(a) Divide a square into 5 rows and 3 columns (15 parts). Shade 2 rows and 4 columns, creating 2 × 4 = 8 double-shaded parts. This gives 8/15. Therefore, 2/3 × 4/5 = 8/15.
(b) Divide the square into 3 rows and 4 columns (12 parts). Shade 2 rows and 1 column, showing 2 double-shaded parts. This equals 2/12, which simplifies to 1/6. Therefore, 1/4 × 2/3 = 1/6.
(c) Divide the square into 2 rows and 5 columns (10 parts). Shade 3 rows and 1 column, giving 3 double-shaded parts, which is 3/10. Therefore, 3/5 × 1/2 = 3/10.
(d) Divide the square into 5 rows and 6 columns (30 parts). Shade 4 rows and 3 columns, giving 4 × 3 = 12 double-shaded parts. This is 12/30, which simplifies to 2/5. Therefore, 4/6 × 3/5 = 2/5.
In simple words: Multiply the numerators to get the new numerator. Multiply the denominators to get the new denominator. Simplify the result if possible.

Exam Tip: When representing products on a square grid, the height of shading shows one fraction and the width shows the other. The double-shaded region is always the product.

Question 1. A water tank is filled from a tap. If the tap is open for 1 hour, 7/10 of the tank gets filled. How much of the tank is filled if the tap is open for (a) 1/3 hour (b) 2/3 hour (c) 3/4 hour (d) 7/10 hour (e) For the tank to be full, how long should the tap be running?
Answer:
(a) In 1/3 hour, the amount filled is 1/3 multiplied by 7/10, equalling 7/30 of the tank.
(b) In 2/3 hour, the amount filled is 2/3 multiplied by 7/10, equalling 14/30 or 7/15 of the tank.
(c) In 3/4 hour, the amount filled is 3/4 multiplied by 7/10, equalling 21/40 of the tank.
(d) In 7/10 hour, the amount filled is 7/10 multiplied by 7/10, equalling 49/100 of the tank.
(e) Since 7/10 of the tank fills in 1 hour, the whole tank fills in 10/7 hours, which is 1 3/7 hours. Therefore, the tap should run for 10/7 hours or 1 3/7 hours for the tank to be completely full.
In simple words: Multiply the fraction of the hour by 7/10. To fill the entire tank, find how many 7/10 parts equal 1 whole, which gives 10/7 hours.

Exam Tip: The amount filled is always the time duration multiplied by the filling rate. To find time for complete filling, divide 1 (the whole) by the rate.

Question 2. The government has taken 1/6 of Somu's land to build a road. What part of the land remains with Somu now? She gives half of the remaining part of the land to her daughter, Krishna, and 1/3 it to her son Bora. After giving them their shares, she kept the remaining land for herself. (a) What part of the original land did Krishna get? (b) What part of the original land did Bora get? (c) What part of the original land did Somu keep for herself?
Answer:
After the government takes 1/6, Somu's remaining land is 1 minus 1/6, which equals 5/6.
(a) Krishna receives half of 5/6, which is 1/2 multiplied by 5/6, equalling 5/12 of the original land.
(b) Bora gets 1/3 of 5/6, which is 1/3 multiplied by 5/6, equalling 5/18 of the original land.
(c) The total given away is 1/6 (government) plus 5/12 (Krishna) plus 5/18 (Bora). Finding a common denominator: 6/36 plus 15/36 plus 10/36 equals 31/36. Somu keeps 1 minus 31/36, which equals 5/36 of the original land.
In simple words: First find what remains after the government takes its share. Then divide that remaining amount among family members.

Exam Tip: Always calculate what remains after the first deduction. Each subsequent share is a fraction of what's left, not the original whole.

Question 3. Find the area of a rectangle of sides 3 3/4 ft and 9 3/5 ft.
Answer: First, convert the mixed numbers to improper fractions: 3 3/4 = 15/4 and 9 3/5 = 48/5. The area is length times width: 15/4 multiplied by 48/5. Multiply the numerators: 15 × 48 = 720. Multiply the denominators: 4 × 5 = 20. This gives 720/20 = 36. Therefore, the area is 36 square feet.
In simple words: Change mixed numbers to improper fractions, multiply the numerators and denominators separately, then simplify.

Exam Tip: Always convert mixed numbers before multiplying. Look for opportunities to cancel common factors between numerators and denominators before multiplying to simplify calculations.

Question 4. Tsewang plants four saplings in a row in his garden. The distance between two saplings is 3/4 m. Find the distance between the first and last sapling.
Answer: When four saplings are planted in a row, there are three gaps between them (between saplings 1 and 2, between 2 and 3, and between 3 and 4). Each gap measures 3/4 m. The total distance is 3 multiplied by 3/4, which equals 9/4 or 2 1/4 metres.
In simple words: Four saplings create 3 gaps between them. Multiply the number of gaps by the distance in each gap.

Exam Tip: When objects are placed in a row, the number of gaps is always one less than the number of objects. Multiply this gap count by the distance per gap.

Question 5. Which is heavier: 12/15 of 500 grams or 3/20 of 4 kg?
Answer: Calculate the first amount: 12/15 of 500 = 12/15 × 500 = (12 × 500)/15 = 6000/15 = 400 grams. Calculate the second amount: 3/20 of 4 kg. First convert: 4 kg = 4000 grams. So 3/20 of 4000 = (3 × 4000)/20 = 12000/20 = 600 grams. Comparing: 600 grams is greater than 400 grams. Therefore, 3/20 of 4 kg is heavier.
In simple words: Find each amount by multiplying the fraction by the quantity. Compare the results.

Exam Tip: Always convert units to the same measure before comparing. Calculate the numerical value of each fractional part before making the comparison.

Question. When one of the numbers being multiplied is between 0 and 1, the product is __________ (greater/less) than the other number. When one of the numbers being multiplied is greater than 1, the product is __________ (greater/less) than the other number.
Answer: When one number falls between 0 and 1, it makes the other number smaller when you multiply them together. For example, 1/2 is between 0 and 1, and 1/2 multiplied by 100 equals 50, which is less than 100. When one number is greater than 1, it makes the other number larger when multiplied. For instance, 1 1/2 is greater than 1, and 1 1/2 multiplied by 1/4 equals 3/2 multiplied by 1/4, which equals 3/8. Since 1/4 equals 2/8, we see that 3/8 is greater than 2/8.
In simple words: A fraction less than 1 makes numbers smaller. A number greater than 1 makes numbers larger.

Exam Tip: Recognize whether your multiplier is bigger or smaller than 1. This determines whether your product increases or decreases compared to the original number.

Question. When do you think the quotient is less than the dividend, and when is it greater than the dividend? Is there a similar relationship between the divisor and the quotient?
Answer: When the divisor lies between 0 and 1, the quotient becomes larger than the dividend. For instance, 1/2 divided by 1/3 equals 1/2 multiplied by 3/1, which gives 3/2. Since 1/2 is less than 3/2, the quotient is larger. When the divisor exceeds 1, the quotient becomes smaller than the dividend. For example, 1/5 divided by 2 equals 1/5 multiplied by 1/2, which is 1/10. Since 1/10 is less than 1/5, the quotient is smaller. When the divisor equals 1, the quotient equals the dividend. The relationship between divisor and quotient mirrors the relationship between multiplier and product, but there is no independent rule relating divisor directly to quotient.
In simple words: Dividing by a fraction (less than 1) makes the answer bigger. Dividing by a number greater than 1 makes the answer smaller.

Exam Tip: Division by a fraction is equivalent to multiplication by its reciprocal. This connection helps explain why the quotient changes in expected ways.

Question. Example 5. This problem was posed by Chaturveda Prithudakasvami (c. 860 CE) in his commentary on Brahmagupta's book Brahmasphutasiddhanta. Four fountains fill a cistern. The first fountain can fill the cistern in a day. The second can fill it in half a day. The third can fill it in a quarter of a day. The fourth can fill the cistern in one-fifth of a day. If they all flow together, in how much time will they fill the cistern? In a day, the number of times: The first fountain will fill the cistern in 1 ÷ 1 = 1, The second fountain will fill the cistern is 1 ÷ 1/2 = _____, The third fountain will fill the cistern is 1 ÷ 1/4 = _____, The fourth fountain will fill the cistern is 1 ÷ 1/5 = _____, The number of times the four fountains together will fill the cistern in a day is ___ + ____ + ___ + ____ = 12.
Answer: In one day, the first fountain fills the cistern 1 ÷ 1 = 1 time. The second fountain fills it 1 ÷ 1/2 = 1 × 2 = 2 times. The third fountain fills it 1 ÷ 1/4 = 1 × 4 = 4 times. The fourth fountain fills it 1 ÷ 1/5 = 1 × 5 = 5 times. Together in one day, all four fountains fill the cistern 1 + 2 + 4 + 5 = 12 times. This means all four fountains together will completely fill the cistern in 1/12 of a day.
In simple words: Each fountain's rate is found by dividing 1 by the time it takes. Add all rates to get the combined filling rate.

Exam Tip: When combining work rates, add the individual rates to find the combined rate. The reciprocal of the combined rate gives the time needed to complete the job.

Question. In each of the figures given below, find the fraction of the big square that the shaded region occupies.
Answer: For the first figure: The top right corner square takes up 1/4 of the whole area. Within that corner square, the shaded region covers 3/2 (meaning 1/2 + 1/2 + 1/2 of the three divisions shown). The shaded area is 3/2 multiplied by 1/4, equalling 3/8 of the entire square. For the second figure: The top left square (shown with bold outline) represents 1/4 of the whole. This square is divided into 8 identical triangles, of which 2 are shaded. The shaded portion is 2/8 of that corner square. Therefore, the shaded area is 2/8 multiplied by 1/4, equalling 1/16 of the whole square.
In simple words: Find what fraction one part is of the whole. Then find what fraction the shaded region is of that part. Multiply these two fractions to get the final answer.

Exam Tip: Break complex shapes into smaller sections. Calculate the fraction for each level separately, then combine them through multiplication.

Question. If we assume 1 gold dinar = 12 silver drammas, 1 silver dramma = 4 copper panas, 1 copper pana = 6 mashakas, and 1 pana = 30 cowrie shells, 1 copper pana = 1/48 gold dinar (1/12 × 1/4). 1 cowrie shell = ___________ copper panas. 1 cowrie shell = ___________ gold dinar.
Answer: From the given relationships, 1 pana equals 30 cowrie shells, which means 1 cowrie shell equals 1/30 copper panas. To find the value in gold dinars: 1 cowrie shell equals 1/30 copper panas, and 1 copper pana equals 1/48 gold dinars. Therefore, 1 cowrie shell equals 1/30 multiplied by 1/48, equalling 1/1440 gold dinars.
In simple words: Work through the conversion chain step by step. Use multiplication to convert smaller units to larger ones.

Exam Tip: When converting between units with multiple steps, multiply the conversion factors in sequence. Keep track of which unit you are converting from and to at each step.

Question 1. Evaluate the following: 3 ÷ 7/9, 14/4 ÷ 2, 2/3 ÷ 2/3, 14/6 ÷ 7/3, 4/3 ÷ 3/4, 7/4 ÷ 1/7, 8/2 ÷ 4/15, 1/5 ÷ 1/9, 1/6 ÷ 11/12, 3 2/3 ÷ 1 3/8
Answer:
3 ÷ 7/9 = 3 × 9/7 = 27/7 = 3 6/7
14/4 ÷ 2 = 14/4 × 1/2 = 7/4 = 1 3/4
2/3 ÷ 2/3 = 2/3 × 3/2 = 1
14/6 ÷ 7/3 = 14/6 × 3/7 = 42/42 = 1
4/3 ÷ 3/4 = 4/3 × 4/3 = 16/9 = 1 7/9
7/4 ÷ 1/7 = 7/4 × 7/1 = 49/4 = 12 1/4
8/2 ÷ 4/15 = 8/2 × 15/4 = (8 × 15)/(2 × 4) = 120/8 = 15
1/5 ÷ 1/9 = 1/5 × 9/1 = 9/5 = 1 4/5
1/6 ÷ 11/12 = 1/6 × 12/11 = 12/66 = 2/11
3 2/3 ÷ 1 3/8 = 11/3 ÷ 11/8 = 11/3 × 8/11 = 88/33 = 8/3 = 2 2/3
In simple words: To divide by a fraction, flip it upside down and multiply instead.

Exam Tip: The key rule for division is to write the reciprocal of the divisor and change division to multiplication. Always simplify your final answer.

Question 2. For each of the questions below, choose the expression that describes the solution. Then simplify it. (a) Maria bought 8 m of lace to decorate the bags she made for school. She used 1/4 m for each bag and finished the lace. How many bags did she decorate? (i) 8 × 1/4 (ii) 1/8 × 1/4 (iii) 8 ÷ 1/4 (iv) 1/4 ÷ 8 (b) 1/2 metre of ribbon is used to make 8 badges. What is the length of the ribbon used for each badge? (i) 8 × 1/2 (ii) 1/2 ÷ 1/8 (iii) 8 ÷ 1/2 (iv) 1/2 ÷ 8 (c) A baker needs 1/6 kg of flour to make one loaf of bread. He has 5 kg of flour. How many loaves of bread can he make? (i) 5 × 1/6 (ii) 1/6 ÷ 5 (iii) 5 ÷ 1/6 (iv) 5 × 6
Answer:
(a) To find the number of bags decorated, divide the total lace by the lace per bag: 8 ÷ 1/4. The correct answer is (iii). Simplifying: 8 ÷ 1/4 = 8 × 4 = 32 bags.
(b) To find the ribbon per badge, divide the total ribbon by the number of badges: 1/2 ÷ 8. The correct answer is (iv). Simplifying: 1/2 ÷ 8 = 1/2 × 1/8 = 1/16 m per badge.
(c) To find the number of loaves, divide the available flour by the flour per loaf: 5 ÷ 1/6. The correct answer is (iii). Simplifying: 5 ÷ 1/6 = 5 × 6 = 30 loaves.
In simple words: Divide the total by the amount per item to find how many items you can make or decorate.

Exam Tip: Identify what you know (total amount) and what you need to find (number of items). The expression will always be total amount divided by amount per item.

Question 3. If 1/4 of flour is used to make 12 rotis, how much flour is used to make 6 rotis?
Answer: If 1/4 kg makes 12 rotis, then 6 rotis (which is half of 12) requires half the flour. Half of 1/4 kg is 1/4 divided by 2, which equals 1/4 multiplied by 1/2, giving 1/8 kg. Therefore, 1/8 kg of flour is needed to make 6 rotis.
In simple words: Since 6 is half of 12, the flour needed is half of 1/4 kg, which is 1/8 kg.

Exam Tip: Establish the ratio of rotis to flour. If the rotis are halved, the flour is also halved proportionally.

Question 4. Patiganita, a book written by Sridharacharya in the 9th century CE, mentions this problem: "Friend, after thinking, what sum will be obtained by adding together 1 ÷ 1/6, 1 ÷ 1/10, 1 ÷ 1/13, 1 ÷ 1/9, and 1 ÷ 1/2". What should the friend say?
Answer: Simplify each division: 1 ÷ 1/6 = 1 × 6 = 6; 1 ÷ 1/10 = 1 × 10 = 10; 1 ÷ 1/13 = 1 × 13 = 13; 1 ÷ 1/9 = 1 × 9 = 9; 1 ÷ 1/2 = 1 × 2 = 2. Adding these together: 6 + 10 + 13 + 9 + 2 = 40. Therefore, the friend should say the sum is 40.
In simple words: When dividing 1 by a fraction, flip the fraction and multiply. Then add all the results.

Exam Tip: Recognize the pattern: dividing 1 by 1/n always gives n. This shortcut makes the calculation much faster.

Question 5. Mira is reading a novel that has 400 pages. She read 1/5 of the pages yesterday and 3/10 of the pages today. How many more pages does she need to read to finish the novel?
Answer: Yesterday, Mira read 1/5 of 400, which is 1/5 multiplied by 400, equalling 80 pages. Today, she read 3/10 of 400, which is 3/10 multiplied by 400, equalling 120 pages. The total read is 80 plus 120, which equals 200 pages. Pages remaining: 400 minus 200 equals 200 pages. Therefore, Mira needs to read 200 more pages to finish the novel.
In simple words: Find how many pages she read each day by multiplying. Add the two amounts, then subtract from the total.

Exam Tip: Calculate the actual page counts for each fraction given. Adding these totals shows how many pages are left to read.

Question 6. A car runs 1/6 km using 1 litre of petrol. How far will it go using 2 3/4 litres of petrol?
Answer: The car travels 1/6 km on 1 litre of petrol. Convert 2 3/4 to an improper fraction: 2 3/4 = 11/4 litres. Distance covered is 11/4 multiplied by 1/6, which equals 11/24 km. However, if the question intends 1/6 to mean 16 km per litre (a more realistic car mileage), then: 11/4 multiplied by 16 equals (11 × 16)/4 = 176/4 = 44 km.
In simple words: Multiply the distance per litre by the number of litres available.

Exam Tip: Always check if the units and numbers make sense in the real-world context. Convert mixed numbers to improper fractions before multiplying.

Question 7. Amritpal decides on a destination for his vacation. If he takes a train, it will take him 5 1/6 hours to get there. If he takes a plane, it will take him 1/2 hour. How many hours does the plane save?
Answer: Convert the train time to an improper fraction: 5 1/6 = 31/6 hours. The plane takes 1/2 hour. The difference is 31/6 minus 1/2. Finding a common denominator: 31/6 minus 3/6 equals 28/6, which simplifies to 14/3 or 4 2/3 hours. Therefore, the plane saves 4 2/3 hours.
In simple words: Subtract the plane time from the train time. Simplify the result to find the time saved.

Exam Tip: Convert mixed numbers to improper fractions. Use a common denominator for subtraction, then convert the answer back to a mixed number if needed.

Question 8. Mariam's grandmother baked a cake. Mariam and her cousins finished 4/5 of the cake. The remaining cake was shared equally by Mariam's three friends. How much of the cake did each friend get?
Answer: The remaining cake after Mariam and her cousins ate is 1 minus 4/5, which equals 1/5. This 1/5 portion is shared equally among 3 friends. Each friend receives 1/5 divided by 3, which equals 1/5 multiplied by 1/3, giving 1/15 of the cake.
In simple words: Find what's left after the first group eats. Divide the remainder equally among the friends.

Exam Tip: When dividing a leftover amount among multiple people, use division. Each person's share equals the remaining amount divided by the number of people.

Question 9. Choose the option(s) describing the product of (565/465 × 707/676): (a) > 565/465 (b) < 565/465 (c) > 707/676 (d) < 707/676 (e) > 1 (f) < 1
Answer: First, note that 565 > 465, so 565/465 > 1. Similarly, 707 > 676, so 707/676 > 1. Since both fractions exceed 1, their product exceeds 1. Therefore, option (e) is correct. Additionally, 565/465 × 707/676 > 707/676 because we are multiplying 707/676 by 565/465, which is greater than 1 (this makes it bigger). Similarly, 565/465 × 707/676 > 565/465 because we are multiplying 565/465 by 707/676, which is also greater than 1. Therefore, options (a), (c), and (e) are correct.
In simple words: When you multiply two numbers both bigger than 1, the result is bigger than either one alone and bigger than 1.

Exam Tip: Check whether each fraction is greater or less than 1 by comparing numerator and denominator. Use this to predict whether the product will be larger or smaller than the original numbers.

Question 10. What fraction of the whole square is shaded?
Answer: The large square is divided into 4 identical smaller squares. One of these smaller squares (the bottom right one) has shading. That small square represents 1/4 of the whole. Within that smaller square, there is a diagonal division creating 8 identical triangles, of which 3 are shaded. The shaded triangles represent 3/8 of that small square. To find the total shaded area of the whole square, multiply: 3/8 multiplied by 1/4 equals 3/32 of the entire square.
In simple words: One quarter of the big square is divided into smaller pieces. Three-eighths of those pieces are shaded. Multiply these fractions to get the final answer.

Exam Tip: Work from the largest shape to the smallest. Calculate what fraction each level represents, then multiply to combine them.

Question 11. A colony of ants set out in search of food. As they search, they keep splitting equally at each point (as shown in the figure) and reach two food sources, one near a mango tree and another near a sugarcane field. What fraction of the original group reached each food source?
Answer: At the first splitting point, the colony divides into two equal parts, so each path gets \( \frac{1}{2} \) of the ants. When they split again at the second point into two ways, the fraction becomes \( \frac{1}{2} \div 2 = \frac{1}{2} \times \frac{1}{2} = \frac{1}{4} \). At the third point, they split into four ways, making the fraction at each way \( \frac{1}{4} \div 4 = \frac{1}{4} \times \frac{1}{4} = \frac{1}{16} \). At the fourth splitting point into 2 ways, the fraction becomes \( \frac{1}{16} \div 2 = \frac{1}{16} \times \frac{1}{2} = \frac{1}{32} \). Adding up the fractions that reach the mango tree: \( \frac{1}{2} + \frac{1}{4} + \frac{1}{16} + \frac{1}{16} + \frac{1}{32} = \frac{29}{32} \). The fraction reaching the sugarcane field is \( \frac{1}{32} + \frac{1}{16} = \frac{3}{32} \).
In simple words: Every time the ants split, you divide the fraction by the number of ways they split. Keep dividing and multiplying to find what fraction goes to each food spot.

Exam Tip: Always trace each path carefully and verify that all fractions add up to 1 - this confirms your answer is correct.

Question 12. What is \( 1 - \frac{1}{2} \)? Make a general statement and explain.
Answer: When you compute \( 1 - \frac{1}{2} = \frac{1}{2} \). Next, if you work out \( (1 - \frac{1}{2}) \times (1 - \frac{1}{3}) \), you get \( \frac{1}{2} \times \frac{2}{3} = \frac{1}{3} \). For the product \( (1 - \frac{1}{2}) \times (1 - \frac{1}{3}) \times (1 - \frac{1}{4}) \times (1 - \frac{1}{5}) \), the result is \( \frac{1}{2} \times \frac{2}{3} \times \frac{3}{4} \times \frac{4}{5} = \frac{1}{5} \). Looking at this pattern, you can see that when multiplying these kinds of fractions together, the bottom number of one fraction cancels with the top number of the next fraction. What remains is the top number from the very first fraction and the bottom number from the very last fraction. In general, the formula is: \( (1 - \frac{1}{2}) \times (1 - \frac{1}{3}) \times (1 - \frac{1}{4}) \times \ldots \times (1 - \frac{1}{n}) = \frac{1}{n} \).
In simple words: In these products, most numbers cancel out, leaving only the first numerator and the last denominator as your answer.

Exam Tip: Recognize and explain the telescoping pattern - this is the key insight. Show at least two example calculations before stating the general rule.

Puzzle Time (Page 199)

Chess is a popular strategy game for two players that started in India. It is played on an 8 × 8 checkered board. There are 2 sets of pieces - black and white - with one set for each player. A Queen piece can move horizontally, vertically, or diagonally from its current location. The puzzle asks you to place 4 Queens on a 4 × 4 board such that no 2 queens can attack each other. Below is an example of an invalid arrangement where queens are in attacking positions.

Solution - 4 × 4 Board: Four queens are placed on the 4 × 4 board so that no two queens attack each other.

Challenge - 8 × 8 Board: Now place 8 queens on the 8 × 8 board so that no 2 queens attack each other!

Solution - 8 × 8 Board: Eight queens can be placed on the 8 × 8 board so that no two queens attack each other. One valid arrangement is shown, with each queen positioned in a different row and column, ensuring none share a horizontal, vertical, or diagonal line.