Class 7 Mathematics · Part 2 · Chapter 2

2. Operations with Integers

Answers to all Figure it Out and in-text questions on multiplying and dividing integers and integer expressions. CBSE / NCERT Class 7 Ganita Prakash Part 2.

2.1 A Quick Recap of Integers

FIOFigure it Out: Find the two numbers from their sum and difference.

	Sum	Difference	First number	Second number
(a)	27	9	18	9
(b)	4	12	8	−4
(c)	0	10	5	−5
(d)	0	−10	−5	5
(e)	−7	−1	−4	−3
(f)	−7	−13	−10	3

Tip: first number = (sum + difference) ÷ 2, and second number = sum − first number.

?Carrom coin questions

1. The first movement is −4 and the final position is 5. What is the second movement?

−4 + b = 5, so b = 5 − (−4) = 9 (9 units to the right).

2. Movements 1, −2, 3, −4, …, −10. What is the final position?

(1 − 2) + (3 − 4) + … + (9 − 10) = 5 × (−1) = −5.

?From the figures, what can you conclude about the magnitudes and directions of a and b?

1. a is rightward (positive) and b is leftward (negative). b has the bigger magnitude, so P is to the left of 0.

2. a is rightward and b is leftward. a has the bigger magnitude, so P is to the right of 0.

3. a is leftward and b is rightward. They have equal magnitudes, so P = 0.

?Using tokens, argue: (a) 7 − 18 = 7 + (−18) (b) 4 − (−12) = 4 + 12

(a) To remove 18 positives from 7 positives, we add 11 zero pairs and remove 18 positives. 11 negatives are left. Adding 18 negatives to 7 positives also leaves 11 negatives. Both give −11.

(b) To remove 12 negatives from 4 positives, we add 12 zero pairs and remove the 12 negatives. 16 positives are left. Adding 12 positives to 4 positives also gives 16.

2.2 Multiplication of Integers

?Find 4 × (−6) and 9 × (−7). Why do we remove green tokens for (−4) × 2?

4 × (−6) = −24 and 9 × (−7) = −63.

In (−4) × 2, the multiplicand 2 is positive. A negative multiplier means removing it, so we remove 2 positive (green) tokens 4 times.

FIOFigure it Out

1. Using tokens, find:

(a) 3 × (−2) = −6

(b) (−5) × (−2) = 10

(d) (−7) × 3 = −21

2. 123 × 456 = 56088. Find:

(a) (−123) × 456 = −56088

(b) (−123) × (−456) = 56088

3. A simple rule to multiply two integers

Multiply the numbers without signs. If both have the same sign, the product is positive. If the signs are different, the product is negative.

?Take 4 times each token set that represents −2. Do we get the same answer? Check 5 × 4 with different sets.

Yes. Every set gives −8, because zero pairs add nothing. Each set is just −2.

Similarly, any token set for 4 (like 6 green and 2 red) taken 5 times gives 20.

?Can −4 × 2 be found by adding tokens instead of removing them?

Yes. Removing 2 positives is the same as adding 2 negatives. So, add 2 red tokens 4 times: −8.

FIOFigure it Out: Find the products.

(a) 4 × (−3) = −12 (b) (−6) × (−3) = 18 (c) (−5) × (−1) = 5

(d) (−8) × 4 = −32 (e) (−9) × 10 = −90 (f) 10 × (−17) = −170

?Fill in the blanks.

−30 × 12 = −360 and 12 × −30 = −360.

14 × −5 = −70 and −5 × 14 = −70.

The product stays the same when we swap the numbers.

?Mala’s exam (+5 for correct, −2 for wrong, 50 questions): maximum and minimum marks

Maximum: all 50 correct → 50 × 5 = 250.

Minimum: all 50 wrong → 50 × (−2) = −100.

?Elevator part (b) using Method 1: starts 15 m above ground, goes down 3 m per minute for 45 minutes.

Distance moved = 45 × 3 = 135 m downwards. Position = 15 − 135 = −120. It is 120 m below the ground.

?A Magic Grid of Integers: what product do you get? What is special?

Every time, the product is −30,240.

Each number is a row number times a column number: rows 2, −7, 3, 5 and columns 4, −2, 6, −3 (for example, −28 = −7 × 4).

We always pick one number from each row and each column. So, the product is always (2 × −7 × 3 × 5) × (4 × −2 × 6 × −3) = −210 × 144 = −30,240.

The magic is in the way the numbers are made and arranged. To make a new grid, choose 4 row numbers and 4 column numbers and fill each box with their product.

?Summarise the rules for integer division.

Divide the numbers without signs. Same signs give a positive quotient. Different signs give a negative quotient.

FIOFigure it Out

1. Find the values.

(a) 14 × (−15) = −210

(b) −16 × (−5) = 80

(d) (−46) ÷ (−23) = 2

2. Room temperature after 10 hours

32 + 10 × (−5) = 32 − 50 = −18°C.

3. Cement company (profit +8 per white bag, −5 per grey bag)

(a) 3000 × 8 + 5000 × (−5) = 24,000 − 25,000 = −1000. It is a loss of ₹1000.

(b) Loss on grey bags = 6400 × 5 = ₹32,000. White bags needed = 32,000 ÷ 8 = 4000 bags.

4. Fill in the blanks.

(a) (−3) × (−9) = 27

(b) 5 × (−7) = −35

(d) (−11) × (−12) = 132

(e) (−56) ÷ (−8) = 7

(f) (−132) ÷ 12 = −11

?Expressions using integers

5 × −3 × 4 in any order: (5 × −3) × 4 = −60, 5 × (−3 × 4) = −60, (5 × 4) × −3 = −60.

25 × −6 × 12: (25 × −6) × 12 = −1800, 25 × (−6 × 12) = −1800, (25 × 12) × −6 = −1800. The product is always −1800.

Sign rule: if the number of negative factors is even, the product is positive. If it is odd, the product is negative. (So, five −1s multiply to −1.)

Distributive property: (−2) × (4 + (−3)) = (−2) × 1 = −2, and (−2) × 4 + (−2) × (−3) = −8 + 6 = −2. They are equal.

?Visually show −4 × (2 + (−3)).

2 + (−3) is 2 green and 3 red tokens. Its inverse is 2 red and 3 green tokens. Put this inverse in the bag 4 times.

We get 8 red and 12 green tokens, which is −8 + 12 = 4. This equals −4 × 2 + (−4) × (−3) = −8 + 12 = 4.

?Pick the Pattern

Machine 1 (a + b − c): (−10) + (−12) − (−9) = −13.

Machine 2: the rule is −(a × b) − c. Check: −(4 × 8) − (−3) = −32 + 3 = −29. Last row: −((−10) × (−12)) − (−9) = −120 + 9 = −111.

FIOFigure it Out

1. Find the values.

(a) (−5) × 15 = −75

(b) (−7) × 4 × (−1) = 28

2. Find the values.

(a) −3

(b) −21

3. Find the integer whose product with (−1) is:

(a) −27 (b) 31 (c) 1 (d) −1 (e) 0

4. 47 − 56 + 14 − 8 + 2 − 8 + 5 = −4. Find −47 + 56 − 14 + 8 − 2 + 8 − 5.

Every term has changed sign, so the new expression is (−1) × (−4) = 4.

5. Modified Collatz (even → half; odd → × (−3) + 1)

Starting with −7: −7, 22, 11, −32, −16, −8, −4, −2, −1, 4, 2, 1, −2, … (11 × (−3) + 1 = −32).

Starting with −21: −21, 64, 32, 16, 8, 4, 2, 1, −2, −1, 4, 2, 1, …

Starting with −6: −6, −3, 10, 5, −14, −7, 22, 11, −32, −16, −8, −4, −2, −1, 4, 2, 1, …

Pattern: every sequence reaches the loop 1, −2, −1, 4, 2, 1 and keeps repeating it.

6. Test with +4 for correct and −2 for incorrect

(a) 15 correct give 60 marks. 40 − 60 = −20 comes from wrong answers: 20 ÷ 2 = 10 incorrect. Questions in the test = 15 + 10 = 25.

(b) 5 correct give 20 marks. −10 − 20 = −30 comes from wrong answers: 15 incorrect. If the test has 25 questions, he left 5 unanswered.

7. Find the operations done by the machine.

The rule is a − b × c. Check: 4 − 8 × (−3) = 4 + 24 = 28. Last row: −16 − (−6) × (−9) = −16 − 54 = −70.

8. Temperature after 4 hours

8 + 4 × (−5) = 8 − 20 = −12°C

9. Three consecutive numbers with product (a) −6 (b) 120

(a) −3, −2, −1

(b) 4, 5, 6

10. Coins of +13 and −9 pibs

(a) +20 = 5 × 13 + 5 × (−9)

(b) +40 = 10 × 13 + 10 × (−9)

(d) +8 = 2 × 13 + 2 × (−9)

(e) +10 = 7 × 13 + 9 × (−9)

(f) −2 = 4 × 13 + 6 × (−9)

(g) +1 = 7 × 13 + 10 × (−9)

(h) Yes. 1568 = 122 × 13 + 2 × (−9) = 1586 − 18. (Since +1 can be made, any total can be made.)

11. Find the values.

(a) (−576) ÷ (−36) = 16

(b) 32 ÷ 648 = 481

(d) (−1960) ÷ 280 = −7

12. Arrange in increasing order.

Values: (a) −1412, (b) 716, (c) 1412, (d) 3,70,272, (e) −3,70,272, (f) 3,35,472.

(e) < (a) < (b) < (c) < (f) < (d)

13. Given (−548) × 972 = −532656

(a) (−547) × 972 = −532656 + 972 = −531684

(b) (−548) × 971 = −532656 + 548 = −532108

14. −207 × (33 − 7)

−207 × (33 − 7) = 207 × (−33 + 7) = −5382

15. Using 3, −2, 5, −6 and +, −, × once each

(a) Maximum: (−6) × ((−2) − (5 + 3)) = (−6) × (−10) = 60

(b) Minimum: ((3 + 5) − (−2)) × (−6) = 10 × (−6) = −60

16. Fill in the blanks in 5 ways.

(a) 0 + 6 × (−6); 4 + (−5) × 8; −6 + (−5) × 6; 4 + 10 × (−4); −36 + 1 × 0

(b) (5 − 2) × 4; (2 − 5) × (−4); (1 − 7) × (−2); (10 − 4) × 2; (−3 − (−9)) × 2

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