NCERT Solutions Class 7 Mathematics Ganita Prakash 2 Chapter 02 Operations with Integers

Question 1. Let us try to find a few more pairs of numbers from their sums and differences:
(a) Sum = 27, Difference = 9 (b) Sum = 4, Difference = 12 (c) Sum = 0, Difference = 10 (d) Sum = 0, Difference = -10 (e) Sum = -7, Difference = -1 (f) Sum = -7, Difference = -13
Answer:
(a) Sum = 27, Difference = 9
Working through the system: when two numbers add to 27 and their difference is 9, the larger number is 18 and the smaller is 9. We can verify: 18 + 9 = 27 and 18 - 9 = 9.
Hence, the correct pair is (18, 9).

(b) Sum = 4, Difference = 12
Setting up equations: if the two numbers sum to 4 and differ by 12, the larger number is 8 and the smaller is -4. Check: 8 + (-4) = 4 and 8 - (-4) = 12.
Hence, the correct pair is (8, -4).

(c) Sum = 0, Difference = 10
When two numbers sum to zero and have a difference of 10, they are 5 and -5. Verification: 5 + (-5) = 0 and 5 - (-5) = 10.
Hence, the correct pair is (5, -5).

(d) Sum = 0, Difference = -10
For a sum of zero with a difference of -10, the pair is 8 and -8. We can check: -8 + 8 = 0 and -8 - 8 = -16... Let me reconsider. Actually, the pair is (-5, 5) gives sum 0 and difference -10. Check: -5 - 5 = -10. Wait, if difference means first minus second: 8 - (-8) = 16, not -10. Testing: if we want difference = -10, trying -4 and 4: -4 - 4 = -8, no. Trying -5 and 5: -5 - 5 = -10. Yes.
Hence, the correct pair is (-5, 5). Actually from the source table shown, the pair listed is (8, -8). Let me verify the source: the table shows First number 8, Second number -8, giving Sum 0, Difference 16. But the stated Difference is -10, which doesn't match. Checking the visible table on the page: one row has -5, 5, 0, -10. So the correct pair is (-5, 5).
Hence, the correct pair is (-5, 5).

(e) Sum = -7, Difference = -1
Finding two numbers that sum to -7 with a difference of -1: these are -4 and -3. Verify: -4 + (-3) = -7 and -4 - (-3) = -1.
Hence, the correct pair is (-4, -3).

(f) Sum = -7, Difference = -13
When two numbers sum to -7 and differ by -13, working through the equations gives us -10 and 3. Check: -10 + 3 = -7 and -10 - 3 = -13.
Hence, the correct pair is (-10, 3).
In simple words: To find two numbers from their sum and difference, add the sum and difference together and divide by 2 to get the larger number. Then subtract the larger from the sum to find the smaller one.

Exam Tip: Always verify your answer by adding the two numbers to check the sum and subtracting to check the difference. This takes 10 seconds and catches all careless mistakes.

Question 1. Using the token interpretation, find the values of:
(a) 3 × (-2) (b) (-5) × (-2) (c) (-4) × (-1) (d) (-7) × 3
Answer:
(a) 3 × (-2)
We place 3 groups of 2 red tokens (negative tokens) into the box. This gives us 6 red tokens total, which represents -6.
∴ 3 × (-2) = -6

(b) (-5) × (-2)
We need to remove 2 negative tokens from the box, doing this 5 times. Since the box starts empty, we first place 2 zero pairs (one positive and one negative token together) into the box. We then remove 2 negative tokens and repeat this 5 times. Each time we remove the negatives, we are left with 2 positive tokens. After 5 repetitions, we have 10 positive tokens.
∴ (-5) × (-2) = 10

(c) (-4) × (-1)
We remove 1 negative token from the box 4 times. Starting with an empty box, we place 1 zero pair and remove the 1 negative token. Repeating this 4 times means we remove 4 negative tokens and are left with 4 positive tokens.
∴ (-4) × (-1) = 4

(d) (-7) × 3
We remove 3 positive tokens from the box 7 times. Since the box is empty, we place 3 zero pairs and remove 3 positive tokens. Repeating this 7 times, we remove 21 positive tokens and are left with 21 negative tokens.
∴ (-7) × 3 = -21
In simple words: When multiplying, positive times negative gives negative. Negative times negative gives positive. Negative times positive gives negative.

Exam Tip: Remember the token model: red tokens are negative, blue tokens are positive. Removing negatives leaves positives behind, so two negatives multiply to make a positive.

Question 2. If 123 × 456 = 56088, without calculating, find the value of:
(a) (-123) × 456 (b) (-123) × (-456) (c) (123) × (-456)
Answer:
(a) (-123) × 456
When a negative number is multiplied by a positive number, the result is always negative. The absolute values are multiplied as usual. So we take the product 123 × 456 = 56088 and make it negative.
∴ (-123) × 456 = -56088

(b) (-123) × (-456)
Multiplying two negative numbers together always gives a positive result. We can write this as: (-123) × (-456) = 123 × 456 = 56088
∴ (-123) × (-456) = 56088

(c) (123) × (-456)
A positive number times a negative number results in a negative product. We take the absolute values' product and apply the negative sign.
∴ (123) × (-456) = -56088
In simple words: Different signs give a negative answer. Same signs (both negative or both positive) give a positive answer.

Exam Tip: Before multiplying, count how many negative numbers you have. If it is an even count, the answer is positive. If it is odd, the answer is negative.

Question 3. Try to frame a simple rule to multiply two integers.
Answer: To multiply any two integers, follow these steps:
First, multiply the absolute values (the numbers without their signs) together.
Then, look at the signs of both numbers. If they are different - one positive and one negative - the final answer is negative. If both numbers have the same sign - either both positive or both negative - the final answer is positive.
In simple words: Multiply the numbers normally. If the signs are different, make it negative. If the signs match, make it positive.

Exam Tip: This rule works every single time. Write it down and refer to it whenever you multiply integers. It removes all guessing.

Question 1. Find the following products.
(a) 4 × (-3) (b) (-6) × (-3) (c) (-5) × (-1) (d) (-8) × 4 (e) (-9) × 10 (f) 10 × (-17)
Answer:
(a) Here, 4 × (-3)
The first factor is positive and the second is negative, so the product must be negative. Multiply the absolute values: 4 × 3 = 12. Apply the negative sign.
∴ 4 × (-3) = -12

(b) Here, (-6) × (-3)
Both factors are negative, so the product is positive. Multiply the absolute values: 6 × 3 = 18.
∴ (-6) × (-3) = 18

(c) Here, (-5) × (-1)
Both factors are negative, making the product positive. Multiply: 5 × 1 = 5.
∴ (-5) × (-1) = 5

(d) Here, (-8) × 4
The first factor is negative and the second is positive, so the product is negative. Multiply the absolute values: 8 × 4 = 32. Apply the negative sign.
∴ (-8) × 4 = -32

(e) Here, (-9) × 10
The first is negative and the second is positive, making the product negative. Multiply: 9 × 10 = 90. Apply the negative sign.
∴ (-9) × 10 = -90

(f) Here, 10 × (-17)
The first is positive and the second is negative, so the product is negative. Multiply the absolute values: 10 × 17 = 170. Apply the negative sign.
∴ 10 × (-17) = -170
In simple words: If one number is negative and the other is positive, the answer is negative. If both are negative, the answer is positive.

Exam Tip: Always check the signs first before doing any multiplication. This prevents sign errors, which are the most common mistakes in integer multiplication.

Question 1. Find the values of:
(a) 14 × (-15) (b) -16 × (-5) (c) 36 ÷ (-18) (d) (-46) ÷ (-23)
Answer:
(a) For 14 × (-15)
We have a positive number times a negative number. Multiply the absolute values: 14 × 15 = 210. Since the signs are different, the result is negative.
∴ 14 × (-15) = -210

(b) For -16 × (-5)
We multiply two negative numbers together. Multiply the absolute values: 16 × 5 = 80. Since both numbers are negative, the result is positive.
∴ -16 × (-5) = 80

(c) For 36 ÷ (-18)
We divide a positive number by a negative number. Divide the absolute values: 36 ÷ 18 = 2. Since the signs differ, the result is negative.
∴ 36 ÷ (-18) = -2

(d) For (-46) ÷ (-23)
We divide two negative numbers. Divide the absolute values: 46 ÷ 23 = 2. Since both are negative, the result is positive.
∴ (-46) ÷ (-23) = 2
In simple words: For both multiplication and division, different signs give a negative answer, and the same signs give a positive answer.

Exam Tip: The sign rules for division are identical to those for multiplication. Master one, and you master both.

Question 2. A freezing process requires that the room temperature be lowered from 32°C at the rate of 5°C every hour. What will be the room temperature 10 hours after the process begins?
Answer: The temperature drops by 5°C each hour. Over 10 hours, the total drop is 5°C/hour × 10 hours = 50°C. The starting temperature is 32°C. To find the final temperature, we subtract the total drop from the starting value: 32°C - 50°C = -18°C. The negative sign shows the temperature has fallen below zero.
∴ Final temperature = -18°C
In simple words: The room cools down by 50 degrees in total. Starting at 32 degrees above zero and cooling 50 degrees takes us 18 degrees below zero.

Exam Tip: Write the temperature drop as a negative number in your calculation. This automatically gives you the correct sign in the final answer without extra thinking.

Question 3. A cement company earns a profit of Rs 8 per bag of white cement sold and a loss of Rs 5 per bag of grey cement sold.
Answer:
(a) The company gets Rs 8 profit per white cement bag and loses Rs 5 per grey cement bag. From white cement: 3000 bags × Rs 8/bag = Rs 24,000 profit. From grey cement: 5000 bags × Rs 5/bag = Rs 25,000 loss. The total loss is Rs 25,000 - Rs 24,000 = Rs 1,000.
∴ The company has a loss of Rs 1,000.

(b) Let x = the number of white cement bags to sell. For neither profit nor loss, the total must be zero. Profit from white cement minus loss from grey cement equals zero: x × 8 - 6400 × 5 = 0. Simplifying: 8x - 32,000 = 0, so 8x = 32,000. Dividing: x = 4,000.
∴ The company must sell 4,000 bags of white cement to break even.
In simple words: Profit and loss are opposite. Add them together to get the overall result. When they are equal, the total is zero - that is break-even.

Exam Tip: Always represent profit as positive and loss as negative when setting up equations. This keeps your calculation clear and prevents sign errors.

Question 4. Replace the blank with an integer to make a true statement.
(a) (-3) × ________ = 27 (b) 5 × ________ = (-35) (c) ________ × (-8) = (-56) (d) ________ × (-12) = 132 (e) ________ ÷ (-8) = 7 (f) ________ ÷ 12 = -11
Answer:
(a) Let x be the missing number. We have -3 × x = 27. Divide both sides by -3: x = 27 ÷ (-3) = -9.
∴ x = -9

(b) Let x be the missing number. We have 5 × x = -35. Divide both sides by 5: x = -35 ÷ 5 = -7.
∴ x = -7

(c) Let x be the missing number. We have x × (-8) = -56. Divide both sides by -8: x = -56 ÷ (-8) = 7.
∴ x = 7

(d) Let x be the missing number. We have x × (-12) = 132. Divide both sides by -12: x = 132 ÷ (-12) = -11.
∴ x = -11

(e) Let x be the missing number. We have x ÷ (-8) = 7. Multiply both sides by -8: x = 7 × (-8) = -56.
∴ x = -56

(f) Let x be the missing number. We have x ÷ 12 = -11. Multiply both sides by 12: x = -11 × 12 = -132.
∴ x = -132
In simple words: To find the missing number, undo the operation. If you see multiplication, divide. If you see division, multiply. Remember the sign rules when you divide or multiply.

Exam Tip: Always verify your answer by plugging it back into the original equation. This catches mistakes instantly.

Question 1. Find the values of the following expressions:
(a) (-5) × (18 + (-3)) (b) (-7) × 4 × (-1) (c) (-2) × (-1) × (-5) × (-3)
Answer:
(a) (-5) × (18 + (-3))
First, evaluate inside the brackets: 18 + (-3) = 15. Now multiply: (-5) × 15 = -75. Alternatively, using the distributive property: -5 × 18 + (-5) × (-3) = -90 + 15 = -75.
∴ (-5) × (18 + (-3)) = -75

(b) (-7) × 4 × (-1)
Multiply from left to right. First: (-7) × 4 = -28. Then: (-28) × (-1) = 28.
∴ (-7) × 4 × (-1) = 28

(c) (-2) × (-1) × (-5) × (-3)
Group the factors into pairs and multiply each pair. First pair: (-2) × (-1) = 2. Second pair: (-5) × (-3) = 15. Now multiply these results: 2 × 15 = 30.
∴ (-2) × (-1) × (-5) × (-3) = 30
In simple words: Work step by step. Use brackets to group your work. Remember that two negatives always give a positive, so counting negatives in groups of 2 helps.

Exam Tip: When multiplying many numbers, group them strategically. Pairing negatives together simplifies your work and reduces errors.

Question 2. Find the values of the following expressions:
(a) (-27) ÷ 9 (b) 84 ÷ (-4) (c) (-56) ÷ (-2)
Answer:
(a) (-27) ÷ 9
Think: what number times 9 gives -27? Since 9 × (-3) = -27, the answer is -3.
∴ (-27) ÷ 9 = -3

(b) 84 ÷ (-4)
Think: what number times -4 gives 84? Since (-4) × (-21) = 84, the answer is -21.
∴ 84 ÷ (-4) = -21

(c) (-56) ÷ (-2)
Think: what number times -2 gives -56? Since (-2) × 28 = -56, the answer is 28.
∴ (-56) ÷ (-2) = 28
In simple words: Division is the opposite of multiplication. If you are stuck, ask yourself: "What number times the divisor gives the dividend?" Use the same sign rules as multiplication.

Exam Tip: Turning division into a multiplication question makes it easier to think about. This method rarely fails.

Question 3. Find the integer whose product with (-1) is:
(a) 27 (b) -31 (c) -1 (d) 1 (e) 0
Answer:
(a) We need to find a number that when multiplied by -1 gives 27. Since (-1) × (-27) = 27, the number is -27.
∴ -27 is the required integer.

(b) We need (-1) × (?) = -31. Since (-1) × 31 = -31, the number is 31.
∴ 31 is the required integer.

(c) We need (-1) × (?) = -1. Since (-1) × 1 = -1, the number is 1.
∴ 1 is the required integer.

(d) We need (-1) × (?) = 1. Since (-1) × (-1) = 1, the number is -1.
∴ (-1) is the required integer.

(e) We need (-1) × (?) = 0. Since (-1) × 0 = 0, the number is 0.
∴ 0 is the required integer.
In simple words: Multiplying by -1 flips the sign of a number. Positive becomes negative, and negative becomes positive. Zero stays zero.

Exam Tip: Multiplying by -1 is like asking "what is the opposite of this number?" Use this shortcut to solve these problems instantly.

Question 4. If 47 - 56 + 14 - 8 + 2 - 8 + 5 = -4, then find the value of -47 + 56 - 14 + 8 - 2 + 8 - 5 without calculating the full expression.
Answer: Notice that the second expression is exactly the negative of the first expression - each term has the opposite sign. When we take the negative of an entire expression, we multiply every term by -1. So: -47 + 56 - 14 + 8 - 2 + 8 - 5 = -1 × (47 - 56 + 14 - 8 + 2 - 8 + 5) = -1 × (-4) = 4.
∴ The value is 4.
In simple words: If you know what an expression equals, and the second expression is its exact opposite, just multiply the first answer by -1 to get the second answer.

Exam Tip: Always look for patterns and relationships between expressions. This can save you from doing long calculations and is much faster than computing from scratch.

Question 5. Do you remember the Collatz Conjecture from last year? Try a modified version with integers. The rule is - start with any number; if the number is even, take half of it; if the number is odd, multiply it by -3 and add 1; repeat. Try this with different starting numbers: (-21), (-6), and so on. Describe the patterns you observe.
Answer:
(a) For the sequence starting with -7 (which is odd):
-7 (odd) → [(-7) × (-3)] + 1 = 21 + 1 = 22 (even)
22 (even) → 22 ÷ 2 = 11 (odd)
11 (odd) → [11 × (-3)] + 1 = -33 + 1 = -32 (even)
-32 (even) → (-32) ÷ 2 = -16 (even)
-16 (even) → (-16) ÷ 2 = -8 (even)
-8 (even) → (-8) ÷ 2 = -4 (even)
-4 (even) → (-4) ÷ 2 = -2 (even)
-2 (even) → (-2) ÷ 2 = -1 (odd)
-1 (odd) → [(-1) × (-3)] + 1 = 3 + 1 = 4 (even)
4 (even) → 4 ÷ 2 = 2 (even)
2 (even) → 2 ÷ 2 = 1 (odd)
The sequence is: -7, 22, 11, -32, -16, -8, -4, -2, -1, 4, 2, 1, and then it would repeat the pattern.

(b)(i) For -21 (which is odd):
-21 (odd) → [(-21) × (-3)] + 1 = 63 + 1 = 64 (even)
64 (even) → 64 ÷ 2 = 32 (even)
32 (even) → 32 ÷ 2 = 16 (even)
16 (even) → 16 ÷ 2 = 8 (even)
8 (even) → 8 ÷ 2 = 4 (even)
4 (even) → 4 ÷ 2 = 2 (even)
2 (even) → 2 ÷ 2 = 1 (odd)
1 (odd) → [1 × (-3)] + 1 = -3 + 1 = -2 (even)
-2 (even) → (-2) ÷ 2 = -1 (odd)
-1 (odd) → [(-1) × (-3)] + 1 = 3 + 1 = 4 (even)
4 (even) → 4 ÷ 2 = 2 (even)
2 (even) → 2 ÷ 2 = 1 (odd)
The sequence for -21 is: -21, 64, 32, 16, 8, 4, 2, 1, -2, -1, 4, 2, 1, -2, ... (enters a repeating cycle).

(ii) For -6 (which is even):
-6 (even) → (-6) ÷ 2 = -3 (odd)
-3 (odd) → [(-3) × (-3)] + 1 = 9 + 1 = 10 (even)
10 (even) → 10 ÷ 2 = 5 (odd)
5 (odd) → [5 × (-3)] + 1 = -15 + 1 = -14 (even)
-14 (even) → (-14) ÷ 2 = -7 (odd)
-7 (odd) → [(-7) × (-3)] + 1 = 21 + 1 = 22 (even)
22 (even) → 22 ÷ 2 = 11 (odd)
11 (odd) → [11 × (-3)] + 1 = -33 + 1 = -32 (even)
-32 (even) → (-32) ÷ 2 = -16 (even)
-16 (even) → (-16) ÷ 2 = -8 (even)
-8 (even) → (-8) ÷ 2 = -4 (even)
-4 (even) → (-4) ÷ 2 = -2 (even)
-2 (even) → (-2) ÷ 2 = -1 (odd)
-1 (odd) → [(-1) × (-3)] + 1 = 4 (even)
4 (even) → 4 ÷ 2 = 2 (even)
2 (even) → 2 ÷ 2 = 1 (odd)
The sequence for -6 is: -6, -3, 10, 5, -14, -7, 22, 11, -32, -16, -8, -4, -2, -1, 4, 2, 1, -2, ... (repeats the cycle -2, -1, 4, 2, 1).

Observation: No matter which starting number we choose, the sequence eventually enters a repeating loop. Specifically, all sequences eventually reach and repeat the cycle: -2, -1, 4, 2, 1, -2, -1, 4, 2, 1, ... This pattern suggests that all integers eventually get drawn into this single repeating cycle.
In simple words: With any starting number, keep following the rules. Eventually you will get stuck in a repeating pattern that goes -2, -1, 4, 2, 1, and then back to -2. All numbers end up here.

Exam Tip: When working with sequences, track odd and even numbers separately. The rule for odd numbers and the rule for even numbers help you predict what comes next.

Question 6. In a test, (+4) marks are given for every correct answer and (-2) marks are given for every incorrect answer.
Answer:
(a) Anita scored 15 correct answers, earning 15 × 4 = 60 marks. Her total score is 40 marks. So she lost 60 - 40 = 20 marks to incorrect answers. Since each incorrect answer costs 2 marks, the number of incorrect answers is 20 ÷ 2 = 10. The total number of questions is the sum of correct and incorrect answers: 15 + 10 = 25.
∴ Anita had 10 incorrect answers out of 25 total questions.

(b) Anil scored 5 correct answers, earning 5 × 4 = 20 marks. His total score is -10 marks. This means he lost 20 - (-10) = 30 marks to incorrect answers. Since each incorrect answer costs 2 marks, he had 30 ÷ 2 = 15 incorrect answers. The test has 25 questions total (from part a). Anil answered 5 + 15 = 20 questions, so he left 25 - 20 = 5 questions unanswered.
∴ Anil had 15 incorrect answers and left 5 questions unanswered. Yes, he did leave some questions unanswered.
In simple words: Correct answers add points. Incorrect answers subtract points. Find how many points were lost, then divide by the penalty per wrong answer to find how many were wrong.

Exam Tip: Always find the total number of questions by adding correct, incorrect, and unanswered. This should match the test total and confirms your work.

Question 7. Pick the pattern - find the operations done by the machine shown below:
Answer: By testing each row, the pattern becomes clear. The machine performs this operation: Take the first number, subtract the product of the second and third numbers. In symbol form: (First Number) - (Second Number × Third Number).

Let me verify this with each row:
Row I: 4 - (8 × (-3)) = 4 - (-24) = 4 + 24 = 28 ✓
Row II: 6 - (9 × 6) = 6 - 54 = -48 ✓
Row III: 2 - (3 × (-2)) = 2 - (-6) = 2 + 6 = 8 ✓
Row IV: (-9) - (5 × (-8)) = (-9) - (-40) = (-9) + 40 = 31 ✓
Row V: 7 - ((-4) × (-6)) = 7 - 24 = -17 ✓
Row VI: (-16) - ((-6) × (-9)) = (-16) - 54 = -70 ✓

∴ The missing operation for Row VI is -70.
In simple words: Always multiply the second and third numbers first. Then subtract that product from the first number. Subtract means change the sign inside the brackets before combining.

Exam Tip: When finding patterns with operations, test your pattern against every row. If it works for all rows, you have found the correct pattern.

Question 8. Imagine you're in a place where the temperature drops by 5°C each hour. If the temperature is currently at 8°C, write an expression that denotes the temperature after 4 hours.
Answer: The starting temperature is 8°C. The temperature falls by 5°C every hour. After 4 hours, the total drop is 5°C/hour × 4 hours = 20°C. To find the temperature after 4 hours, we subtract this drop from the current temperature: 8 - 20. This can also be written as 8 - (5 × 4).
∴ The required expression is 8 - (5 × 4) or equivalently 8 - 5 × 4.
In simple words: The temperature starts at 8. Each hour it drops 5 degrees. In 4 hours, it drops a total of 5 times 4. So the final temperature is 8 minus that drop.

Exam Tip: Always identify the starting value, the rate of change, and the time. The expression should reflect: starting value - (rate × time) for decreasing quantities.

Question 9. Find 3 consecutive numbers with a product of (a) -6, (b) 120.
Answer:
(a) Let the three consecutive numbers be n - 1, n, and n + 1. Their product is (n - 1) × n × (n + 1) = -6. We need the product to be negative, so an odd number of the three must be negative. Testing -3, -2, -1: (-3) × (-2) × (-1) = 6 × (-1) = -6. This works!
∴ The consecutive integers are -3, -2, -1.

(b) Let the three consecutive numbers be n - 1, n, and n + 1. Their product is (n - 1) × n × (n + 1) = 120. The cube root of 120 is approximately 4.93, so we expect n to be close to 5. Testing n = 5: (5 - 1) × 5 × (5 + 1) = 4 × 5 × 6 = 120. This works perfectly!
∴ The consecutive integers are 4, 5, and 6.
In simple words: For negative products, count how many negatives you have - it must be odd (1 or 3). For finding the middle number, take the cube root of the target product and round to a nearby whole number, then test it.

Exam Tip: Once you have a candidate answer, always verify by multiplying all three numbers together. This takes 10 seconds and prevents mistakes.

Question 10. An alien society uses a peculiar currency called 'pibs' with just two denominations of coins - a +13 pibs coin and a -9 pibs coin. You have several of these coins. Is it possible to purchase an item that costs +85 pibs? Yes, we can use 10 coins of +13 pibs and 5 coins of -9 pibs to make a total of +85. Using the two denominations, try to get the following totals:
(a) +20 (b) +40 (c) -50 (d) +8 (e) +10 (f) -2 (g) +1
(h) Is it possible to purchase an item that costs 1568 pibs?
Answer: This problem requires solving equations of the form 13x - 9y = total, where x and y are non-negative integers representing the number of +13 pibs coins and -9 pibs coins respectively.

(a) +20 pibs
We need 13x - 9y = 20. Testing values: if x = 5, then 13(5) - 9y = 20 gives 65 - 9y = 20, so 9y = 45 and y = 5. This works: 5 coins of +13 and 5 coins of -9 gives 65 - 45 = 20.
∴ Use 5 coins of +13 and 5 coins of -9.

(b) +40 pibs
We need 13x - 9y = 40. If we use 10 coins of +13 and 10 coins of -9: 10(13) - 10(9) = 130 - 90 = 40.
∴ Use 10 coins of +13 and 10 coins of -9.

(c) -50 pibs
We need 13x - 9y = -50. If we use 10 coins of +13 and 20 coins of -9: 10(13) - 20(9) = 130 - 180 = -50.
∴ Use 10 coins of +13 and 20 coins of -9.

(d) +8 pibs
We need 13x - 9y = 8. If we use 2 coins of +13 and 2 coins of -9: 2(13) - 2(9) = 26 - 18 = 8.
∴ Use 2 coins of +13 and 2 coins of -9.

(e) +10 pibs
We need 13x - 9y = 10. If we use 7 coins of +13 and 9 coins of -9: 7(13) - 9(9) = 91 - 81 = 10.
∴ Use 7 coins of +13 and 9 coins of -9.

(f) -2 pibs
We need 13x - 9y = -2. If we use 13 coins of +13 and 19 coins of -9: 13(13) - 19(9) = 169 - 171 = -2.
∴ Use 13 coins of +13 and 19 coins of -9.

(g) +1 pibs
We need 13x - 9y = 1. If we use 7 coins of +13 and 10 coins of -9: 7(13) - 10(9) = 91 - 90 = 1.
∴ Use 7 coins of +13 and 10 coins of -9.

(h) 1568 pibs
We need 13x - 9y = 1568. If we use 122 coins of +13 and 2 coins of -9: 122(13) - 2(9) = 1586 - 18 = 1568. Yes, it is possible!
∴ It is possible to make 1568 pibs. Use 122 coins of +13 and 2 coins of -9.
In simple words: Try different combinations of the two coins until you find a pair that adds up to the target amount. Start with equal numbers of each coin and adjust from there.

Exam Tip: These are linear Diophantine equations. Start by trying simple combinations like equal numbers of each coin, then adjust the numbers systematically until you hit the target.

Question 11. Find the values of:
(a) (32 × (-18)) ÷ ((-36)) (b) (32) ÷ ((-36) × (-18)) (c) (25 × (-12)) ÷ ((45) × (-27)) (d) (280 × (-7)) ÷ ((-8) × (-35))
Answer:
(a) First, calculate the numerator: 32 × (-18) = -576. Then divide by the denominator: (-576) ÷ (-36) = 16 (since both are negative, the result is positive).
∴ 16

(b) First, calculate the denominator: (-36) × (-18) = 648 (negative times negative is positive). Then divide: 32 ÷ 648 = 4/81 (after simplifying by dividing both by 8).
∴ 4/81

(c) First, calculate the numerator: 25 × (-12) = -300. Next, calculate the denominator: 45 × (-27) = -1215. Then divide: (-300) ÷ (-1215) = 300/1215 = 20/81 (after simplifying by dividing both by 15).
∴ 20/81

(d) First, calculate the numerator: 280 × (-7) = -1960. Next, calculate the denominator: (-8) × (-35) = 280. Then divide: (-1960) ÷ 280 = -7.
∴ -7
In simple words: Always work inside the brackets first to find the numerator and denominator separately. Then divide. Remember that negative divided by negative is positive.

Exam Tip: Break complex division problems into two steps: first compute the top, then the bottom, then divide. This organized approach prevents errors.

Question 12. Arrange the expressions given below in increasing order:
(a) (-348) + (-1064) (b) (-348) - (-1064) (c) 348 - (-1064) (d) (-348) × (-1064) (e) 348 × (-1064) (f) 348 × 964
Answer:
Let me evaluate each expression:

(a) (-348) + (-1064): Adding two negatives gives a negative. Result = -1412.

(b) (-348) - (-1064): Subtracting a negative is the same as adding a positive. Result = -348 + 1064 = 716.

(c) 348 - (-1064): Subtracting a negative means adding a positive. Result = 348 + 1064 = 1412.

(d) (-348) × (-1064): Multiplying two negatives gives a positive. Result = 348 × 1064 = 370,272.

(e) 348 × (-1064): Multiplying a positive and a negative gives a negative. Result = -370,272.

(f) 348 × 964: Multiplying two positives gives a positive. Result = 335,472.

Now arranging from smallest to largest: -370,272 < -1,412 < 716 < 1,412 < 335,472 < 370,272.

∴ The increasing order is: (e) < (a) < (b) < (c) < (f) < (d).
In simple words: First, calculate what each expression equals. Then line them up from the smallest negative number all the way to the largest positive number.

Exam Tip: Always compute each value first. Only then arrange them. Trying to arrange without computing leads to careless mistakes.

Question 13. Given that (-548) × 972 = -532656, write the values of:
(a) (-547) × 972 (b) (-548) × 971 (c) (-547) × 971
Answer:
(a) For (-547) × 972, we can rewrite -547 as (-548 + 1). So: (-547) × 972 = [(-548) + 1] × 972 = (-548) × 972 + 1 × 972 = -532,656 + 972 = -531,684.
∴ (-547) × 972 = -531,684

(b) For (-548) × 971, we can rewrite 971 as (972 - 1). So: (-548) × 971 = (-548) × (972 - 1) = (-548) × 972 - (-548) × 1 = -532,656 - (-548) = -532,656 + 548 = -532,108.
∴ (-548) × 971 = -532,108

(c) For (-547) × 971, we rewrite it as [(-548) + 1] × (972 - 1). Expanding: (-548) × 972 + 1 × 972 + (-548) × (-1) + 1 × (-1) = -532,656 + 972 + 548 - 1 = -531,137.
∴ (-547) × 971 = -531,137
In simple words: Break down the numbers by adding or subtracting 1. Then use the distributive property to express your answer in terms of the known product.

Exam Tip: This technique of decomposing numbers near the given ones makes use of the distributive property. It is much faster than multiplying from scratch.

Question 14. Given that 207 × (-33 + 7) = -5382, write the value of -207 × (33 - 7) = ________
Answer: First, simplify what we know: 207 × (-33 + 7) = 207 × (-26) = -5382. Now for the expression we need: -207 × (33 - 7) = -207 × 26. We can rewrite this as -1 × 207 × 26 = -1 × [207 × 26]. Since 207 × 26 is the same magnitude as 207 × (-26) but with opposite sign, we have 207 × 26 = 5382. Therefore: -207 × 26 = -5382.
∴ -207 × (33 - 7) = -5382
In simple words: The second expression simplifies to the same operation as the first, just written differently. So the answer is the same.

Exam Tip: Always simplify expressions inside brackets first. Then look for patterns that link the given information to the new question.

Question 15. Use the numbers 3, -2, 5, -6 exactly once and the operations '+', '-', and '×' exactly once and brackets as necessary to write an expression such that-
(a) The result is the maximum possible
(b) The result is the minimum possible
Answer:
(a) To get the maximum possible result, we want to multiply by -6 (a large negative) after creating a small negative number inside brackets. Consider: [(-2) - (3 + 5)] × (-6) = [(-2) - 8] × (-6) = (-10) × (-6) = 60. Checking: we used -2, 3, 5, -6, with operations -, +, ×. All numbers and operations used exactly once.
∴ Maximum possible value = 60

(b) To get the minimum possible result, we want to add all the numbers (creating a large positive) and then multiply by -6 (the negative). Consider: [3 - (-2) + 5] × (-6) = [3 + 2 + 5] × (-6) = 10 × (-6) = -60. Checking: we used 3, -2, 5, -6, with operations -, +, ×. All numbers and operations used exactly once.
∴ Minimum possible value = -60
In simple words: To maximize, create a small negative and multiply by a large negative. To minimize, create a large positive and multiply by a negative.

Exam Tip: Always think strategically about which operations will give you the extremes. Negative times negative is large and positive. Negative times positive is large and negative.

Question 16. Fill in the blanks in at least 5 different ways with integers:
(a) _______ + _______ × _______ = -36
(b) (_______ - _______) × _______ = 12
(c) (_______ - (_______ - _______)) = -1
Answer:
(a) The equation is: a + b × c = -36. We need to find five different sets of values.
(i) 0 + (-6) × 6 = 0 - 36 = -36 ✓
(ii) 4 + (-5) × 8 = 4 - 40 = -36 ✓
(iii) -4 + (-8) × 4 = -4 - 32 = -36 ✓
(iv) 12 + (-8) × 6 = 12 - 48 = -36 ✓
(v) -3 + 3 × (-11) = -3 - 33 = -36 ✓

(b) The equation is: (a - b) × c = 12. We need five different sets.
(i) (13 - 1) × 1 = 12 × 1 = 12 ✓
(ii) (10 - 4) × 2 = 6 × 2 = 12 ✓
(iii) (1 - 3) × (-6) = (-2) × (-6) = 12 ✓
(iv) (14 - 10) × 3 = 4 × 3 = 12 ✓
(v) (16 - 13) × 4 = 3 × 4 = 12 ✓

(c) The equation is: (a - (b - c)) = -1. We need five different sets.
(i) (5 - (10 - 4)) = 5 - 6 = -1 ✓
(ii) (0 - (3 - 2)) = 0 - 1 = -1 ✓
(iii) (-1 - (1 - 1)) = -1 - 0 = -1 ✓
(iv) (-5 - (0 - 4)) = -5 - (-4) = -5 + 4 = -1 ✓
(v) (-10 - (-5 - 4)) = -10 - (-9) = -10 + 9 = -1 ✓
In simple words: There are many ways to fill in blanks to get the same answer. Test each one by substituting the numbers and working through the operations in order.

Exam Tip: When filling in blanks, work backwards. Decide what you want the result to be, then choose numbers and operations that will create it. Always verify your work by substituting back.