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State Board · Class 7 · Mathematics

Lesson 2: Multiplication and Division of Integers

5 concepts · 40 practice questions with answers · Free to read

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Recall: addition and subtraction of integers

From daily life

You have 50 rupees and you buy something that costs 80 rupees. You are 30 rupees short, so 50 − 80 = −30. Debt, a temperature below zero and a floor below the ground floor are all "falling short" things, and we write them with negative numbers.

Simple definition

Integers are the numbers ..., −3, −2, −1, 0, 1, 2, 3, ... They include positive numbers, negative numbers and zero.

Opposite numbers: 5 and −5 are opposite numbers. Their sum is 0: 5 + (−5) = 0. An opposite number is also called an additive inverse. Fractions have opposites too: 25 + −25 = 0.

Number lineNumber line from −5 to 7 in steps of 1. A jump from −2 to 4 labelled +6.−5−4−3−2−101234567+6
  • On the number line, adding a positive number is a jump to the right and adding a negative number is a jump to the left. In the figure, the jump of +6 from −2 lands on 4, so (−2) + 6 = 4.

Rule 1: when the signs are the same — add the numbers without their signs and keep the same sign.

  • Easy: (−8) + (−5) = −13 (8 + 5 = 13, sign negative).
  • Medium: (−17) + (−6) = −23.
  • Exam type: (−25) + (−38) = −63.

Rule 2: when the signs are different — subtract the smaller number from the bigger one (ignore the signs) and give the answer the sign of the bigger one.

  • Easy: 12 + (−7) = 5 (12 − 7 = 5, the bigger number 12 is positive).
  • Medium: (−15) + 9 = −6 (15 − 9 = 6, the bigger number 15 is negative).
  • Exam type: (−28) + 45 = 17 (45 − 28 = 17, the bigger number 45 is positive).

Rule 3: subtraction — change the sign of the number being subtracted and then add.

  • Easy: 7 − 11 = 7 + (−11) = −4.
  • Medium: (−6) − (−10) = (−6) + 10 = 4.
  • Exam type: 5 − (−12) − 9 = 5 + 12 − 9 = 17 − 9 = 8.
Live figure

Recall: addition and subtraction of integers

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Remember

Adding two negative numbers gives an even more negative answer (debt on top of debt). Subtracting means adding the opposite number.

Common mistake

Writing −3 or 13 as the answer of (−8) + (−5). When the signs are the same you add the numbers: 8 + 5 = 13, and the sign is negative, so −13.

Shortcut

Two minus signs side by side make a plus: 5 − (−12) = 5 + 12 = 17.

Spot the difference

SituationWhat to doExample
Same signsAdd the numbers, keep the sign(−8) + (−5) = −13
Different signsSubtract smaller from bigger, take the sign of the bigger12 + (−7) = 5
SubtractionAdd the opposite number5 − (−12) = 17

Practice questions

Easy
  1. 1(−7) + (−6) = ?
  2. 29 + (−4) = ?
  3. 36 + (−6) = ?
Medium
  1. 4(−12) + 5 = ?
  2. 54 − 10 = ?
  3. 6(−3) − (−8) = ?
Challenging
  1. 78 − (−6) + (−9) = ?
  2. 8In a village the morning temperature was −4 °C. By noon it rose by 11 degrees, and at night it fell by 9 degrees. What was the temperature at night?
Show answers (try them yourself first)
  1. 1−13 2) 5 3) 0, because 6 and −6 are opposite numbers.
  2. 4−7 5) −6 6) 5
  3. 78 − (−6) = 8 + 6 = 14; 14 + (−9) = 5.
  4. 8−4 + 11 = 7; 7 − 9 = −2, so the night temperature is −2 °C.

Multiplication as repeated addition

From daily life

You borrowed 4 rupees each from 5 friends. Each debt is −4. Total debt = (−4) + (−4) + (−4) + (−4) + (−4) = −20. We took −4 five times, so 5 × (−4) = −20.

Money borrowed from each friend−4−4−4−4−4
Simple definition

Multiplication means adding the same number many times. If the number being added is negative, the answer is negative.

On the number line: Jump −3 from 0 four times and you land on −12. So 4 × (−3) = −12. Swapping the number of jumps and the length of a jump does not change the answer: (−3) × 4 = −12.

Number lineNumber line from −14 to 2 in steps of 1. A jump from 0 to −3 labelled −3. A jump from −3 to −6 labelled −3. A jump from −6 to −9 labelled −3. A jump from −9 to −12 labelled −3.−14−12−10−8−6−4−202−3−3−3−3
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Multiplication as repeated addition

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Rule 1: positive × negative = negative — write the negative number as many times as needed and add.

  • Easy: 3 × (−6) = (−6) + (−6) + (−6) = −18.
  • Medium: 6 × (−4) = −24 (6 × 4 = 24, sign negative).
  • Exam type: 7 × (−14) = −98 (7 × 14 = 98).

Rule 2: negative × positive = negative — the order does not matter, so (−9) × 3 = 3 × (−9) = −27.

  • Easy: (−2) × 5 = 5 × (−2) = −10.
  • Medium: (−11) × 4 = −44.
  • Exam type: (−15) × 6 = −90.

Rule 3: multiplying by 0 — any number multiplied by 0 gives 0.

  • Easy: (−7) × 0 = 0.
  • Medium: 0 × 25 = 0.
  • Exam type: (−6) × 11 × 0 = 0 (if one number is 0, the product is 0).
Remember

Positive × negative and negative × positive both give a negative answer. Changing the order does not change the product.

Common mistake

Writing 12 as the answer of 4 × (−3). If you take a negative number four times, the debt grows, so the answer is −12.

Shortcut

First multiply the numbers and ignore the signs (4 × 3 = 12). Then add the sign: if exactly one number is negative, the answer is negative.

Spot the difference

MultiplicationMeaningAnswer
3 × (−6)−6 taken 3 times−18 (negative)
(−6) × 3same as 3 × (−6)−18 (negative)
3 × 66 taken 3 times18 (positive)
0 × (−6)−6 taken 0 times0

Practice questions

Easy
  1. 14 × (−8) = ?
  2. 2(−2) × 9 = ?
  3. 3(−8) × 0 = ?
Medium
  1. 4Write (−2) + (−2) + (−2) + (−2) + (−2) + (−2) as a multiplication and find the answer.
  2. 59 × (−7) = ?
  3. 6A pen costs 12 rupees. You buy 4 such pens on credit. How much do you owe in total? Write the answer as a negative number.
Challenging
  1. 7Starting from 0, you make jumps of −7 to reach −42. How many jumps are needed?
  2. 8A tank holds 40 litres of water. Because of a leak, 3 litres leak out every hour. How much water is left after 5 hours?
Show answers (try them yourself first)
  1. 1−32 2) −18 3) 0
  2. 46 × (−2) = −12.
  3. 5−63.
  4. 64 × (−12) = −48, that is a debt of 48 rupees.
  5. 76 jumps, because 6 × (−7) = −42.
  6. 840 + 5 × (−3) = 40 + (−15) = 25, so 25 litres.

Patterns and rules of multiplication

From daily life

A street vendor loses 6 rupees every day, so the daily record is −6. Over the next 4 days the total is 4 × (−6) = −24. And 4 days ago he was ahead by 24 rupees: (−4) × (−6) = 24. "Days in the past" and "loss" are both opposite, so the answer is positive.

Look at the pattern: When the multiplier goes down by one, the product always changes by the same amount.

  • 4 × 3 = 12, 4 × 2 = 8, 4 × 1 = 4, 4 × 0 = 0. When the multiplier goes down by 1, the product goes down by 4.
  • The same pattern continues: 4 × (−1) = −4, 4 × (−2) = −8. So positive × negative = negative.
  • Now the pattern of (−8): (−8) × 3 = −24, (−8) × 2 = −16, (−8) × 1 = −8, (−8) × 0 = 0. When the multiplier goes down by 1, the product goes up by 8.
  • The same pattern continues: (−8) × (−1) = 8, (−8) × (−2) = 16, (−8) × (−3) = 24. So negative × negative = positive.
(−6) × 3 =−18+6(−6) × 2 =−12+6(−6) × 1 =−6+6(−6) × 0 =0+6(−6) × (−1) =6+6(−6) × (−2) =12+6(−6) × (−3) =18
Live figure

Patterns and rules of multiplication

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Sign rule: When the signs are the same, the product is positive; when the signs are different, the product is negative.

A(+6) × (+3)= 18B(+6) × (−3)= −18C(−6) × (+3)= −18D(−6) × (−3)= 18
  • In the figure, A and D have the same signs (+ + and − −), so the answer is 18. B and C have different signs, so the answer is −18.
Live figure

Patterns and rules of multiplication

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Rule 1: negative × negative = positive — if both numbers are negative, multiply the numbers and write a positive answer.

  • Easy: (−7) × (−3) = 21.
  • Medium: (−12) × (−5) = 60.
  • Exam type: (−25) × (−4) = 100.

Rule 2: different signs = negative product — positive × negative or negative × positive, both give a negative answer.

  • Easy: 8 × (−6) = −48.
  • Medium: (−9) × 7 = −63.
  • Exam type: (−14) × 5 = −70.

Rule 3: multiplying by 1 and −1 — any number × 1 = the same number. Any number × (−1) = its opposite number.

  • Easy: (−9) × 1 = −9.
  • Medium: 9 × (−1) = −9.
  • Exam type: (−15) × (−1) = 15.

Rule 4: product of three or more numbers — count the negative numbers. An even count gives a positive answer; an odd count gives a negative answer.

  • Easy: (−2) × (−3) × (−4): three negatives (odd), so −24.
  • Medium: (−5) × (−2) × (−1) × (−3): four negatives (even), so 30.
  • Exam type: (−2) × 5 × (−3) × 4 × (−1): three negatives (odd), 2 × 5 × 3 × 4 × 1 = 120, so −120.
Remember

Same signs → positive answer; different signs → negative answer. Any number × 0 = 0.

Common mistake

Writing −21 as the answer of (−7) × (−3). If both are negative the answer is positive, that is 21.

Shortcut

Decide the sign first, then multiply the numbers. An even count of negative signs gives positive; an odd count gives negative.

Spot the difference

OperationOn two negative numbersAnswer
Addition(−4) + (−3)−7 (negative)
Multiplication(−4) × (−3)12 (positive)
Subtraction(−4) − (−3)−1

Practice questions

Easy
  1. 1(−4) × (−5) = ?
  2. 26 × (−13) = ?
  3. 3(−16) × 5 = ?
Medium
  1. 4(−11) × (−3) = ?
  2. 5(−2) × (−3) × (−5) = ?
  3. 6(−4) × 5 × (−2) = ?
Challenging
  1. 7(−3) × (−4) − (−5) × 2 = ?
  2. 8If you multiply (−1) by itself 25 times, what is the answer? What if you multiply it 26 times?
Show answers (try them yourself first)
  1. 120 2) −78 3) −80
  2. 433 5) (−2) × (−3) = 6; 6 × (−5) = −30. 6) (−4) × 5 = −20; (−20) × (−2) = 40.
  3. 7(−3) × (−4) = 12; (−5) × 2 = −10; 12 − (−10) = 12 + 10 = 22.
  4. 825 times: 25 negatives (odd), so −1. 26 times: 26 negatives (even), so 1.

Division of integers

From daily life

5 shopkeepers shared a debt of 30 rupees equally. How much debt does each one have? (−30) ÷ 5 = −6, so each has a debt of 6 rupees.

−30?????Shared equally
Simple definition

Division is the opposite operation of multiplication. (−30) ÷ 5 = −6 because (−6) × 5 = −30. The number being divided is the dividend, the number we divide by is the divisor, and the answer is the quotient.

Number lineNumber line from −14 to 2 in steps of 1. A jump from 0 to −4 labelled −4. A jump from −4 to −8 labelled −4. A jump from −8 to −12 labelled −4.−14−12−10−8−6−4−202−4−4−4
  • On the number line, jump −4 from 0 three times and you reach −12. So (−12) ÷ (−4) = 3, because 3 × (−4) = −12.
Live figure

Division of integers

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Rule 1: sign rule — divide the numbers first. If the signs are the same the answer is positive; if they are different the answer is negative.

  • Easy: (−42) ÷ 6 = −7.
  • Medium: (−36) ÷ (−9) = 4.
  • Exam type: 45 ÷ (−5) = −9.

Rule 2: check by multiplying — see whether answer × divisor = dividend.

  • (−42) ÷ 6 = −7. Check: (−7) × 6 = −42.
  • 45 ÷ (−5) = −9. Check: (−9) × (−5) = 45.

Rule 3: division and zero — 0 ÷ (any number except 0) = 0. But no number can be divided by 0.

  • Easy: 0 ÷ (−8) = 0, because 0 × (−8) = 0.
  • Medium: 9 ÷ 0 is not possible, because no number multiplied by 0 gives 9.

Rule 4: when the division is not exact — the answer is a fraction. We always write the denominator as positive.

  • Easy: 5 ÷ (−2) = −52.
  • Medium: (−14) ÷ 4 = −144 = −72.
  • Exam type: (−9) ÷ (−12) = 912 = 34.
Remember

The sign rule of division is the same as in multiplication: same signs → positive, different signs → negative. Division by 0 is not possible.

Common mistake

Writing −4 as the answer of (−36) ÷ (−9). If both are negative the answer is positive, that is 4.

Shortcut

Decide the sign first, then divide the numbers. Check the answer by multiplying.

Spot the difference

OperationIf the order is changedExample
MultiplicationSame answer(−6) × 2 = 2 × (−6) = −12
DivisionDifferent answer(−6) ÷ 2 = −3, but 2 ÷ (−6) = −13

Practice questions

Easy
  1. 1(−48) ÷ 6 = ?
  2. 272 ÷ (−8) = ?
  3. 3(−40) ÷ (−5) = ?
Medium
  1. 40 ÷ (−15) = ?
  2. 5Write the answer of 6 ÷ (−8) in simplest form. (Keep the denominator positive.)
  3. 6A trader lost a total of 1400 rupees in 7 days. If the loss was the same every day, what is the daily loss as a negative number?
Challenging
  1. 7(−90) ÷ (−5) ÷ (−3) = ?
  2. 8Which number multiplied by (−4) gives (−36)? Find it using division and check by multiplying.
Show answers (try them yourself first)
  1. 1−8 2) −9 3) 8
  2. 40 5) 6 ÷ (−8) = −68 = −34. 6) (−1400) ÷ 7 = −200.
  3. 7(−90) ÷ (−5) = 18; 18 ÷ (−3) = −6.
  4. 8(−36) ÷ (−4) = 9. Check: 9 × (−4) = −36.

Practice with division and multiplication

From daily life

In a building a lift is on floor 1. Each time it goes down 3 floors, and it does this 3 times. The ground floor is 0 and the floors below it are written −1, −2, ... So 1 + 3 × (−3) = 1 + (−9) = −8, that is, the lift stops 8 floors below the ground floor, at −8.

210−1−2−3−4−5−6−7−8−3−3−3

Order of operations: When there is more than one operation, do the brackets first, then multiplication and division (left to right), then addition and subtraction (left to right).

  • Easy: (−8) × 3 + 5 = −24 + 5 = −19.
  • Medium: 10 + 4 × (−3) = 10 + (−12) = −2.
  • Exam type: (−6) × (−4) + (−18) ÷ 3 = 24 + (−6) = 18.
  • With brackets: ((−6) − 6) ÷ 4 = (−12) ÷ 4 = −3.

Steps for word problems

  • Step 1: Write the numbers: debt, loss, down and decrease are negative; profit, up and increase are positive.
  • Step 2: Decide the operation from the table below.
  • Step 3: Solve using the sign rules.
  • Step 4: Write the answer in words and check that it makes sense.
WordsOperation
total, togetheraddition
difference, how much lesssubtraction
each, that many timesmultiplication
share equally, to eachdivision

Solved examples

  • Easy: A submarine goes down 12 metres every minute. In 5 minutes it goes 5 × (−12) = −60, that is 60 metres down.
  • Medium: The temperature is −6 °C. It rises by 2 degrees every hour. After 4 hours the temperature = −6 + 4 × 2 = −6 + 8 = 2 °C.
  • Exam type: A shopkeeper loses 35 rupees every day for the first 4 days and makes a profit of 20 rupees every day for the next 2 days. Overall result = 4 × (−35) + 2 × 20 = −140 + 40 = −100, that is a loss of 100 rupees.
Remember

Multiplication and division first, then addition and subtraction. Check the sign at every step.

Common mistake

Adding first in (−6) × (−4) + (−18) ÷ 3. Do the multiplication and division first: 24 and −6, then 24 + (−6) = 18.

Shortcut

To decide the sign in multiplication and division, count the negative signs: even → positive, odd → negative.

Spot the difference

OperationSame signsDifferent signs
MultiplicationPositive: (−6) × (−4) = 24Negative: (−6) × 4 = −24
DivisionPositive: (−6) ÷ (−3) = 2Negative: (−6) ÷ 3 = −2

Practice questions

Easy
  1. 136 ÷ (−4) = ?
  2. 2(−11) × 5 = ?
  3. 3(−45) ÷ 5 = ?
Medium
  1. 420 + 3 × (−6) = ?
  2. 5(−35) ÷ 7 − 2 = ?
  3. 6((−9) + 3) × (−4) = ?
Challenging
  1. 7A diver goes down 4 metres every 10 seconds. At what depth, in metres below the water surface, is the diver after 1 minute? Write the answer as a negative number.
  2. 8The product of two integers is −48, and the first number ÷ the second number = −3. Which numbers are they?
Show answers (try them yourself first)
  1. 1−9 2) −55 3) −9
  2. 43 × (−6) = −18; 20 + (−18) = 2. 5) (−35) ÷ 7 = −5; −5 − 2 = −7. 6) (−9) + 3 = −6; (−6) × (−4) = 24.
  3. 760 ÷ 10 = 6 times; 6 × (−4) = −24, so −24 metres.
  4. 812 and −4: 12 × (−4) = −48 and 12 ÷ (−4) = −3. Or −12 and 4: (−12) × 4 = −48 and (−12) ÷ 4 = −3.