CBSE Class 7 Mathematics Operations with Integers Worksheet Set 01

Very Short Answer Questions

Question 1. In the token model, what do we get when one green token and one red token are put together? What is such a pair called?
Answer:
A green token (+1) and a red token (−1) cancel each other, so together they stand for 0. Such a pair is called a zero pair.

Teacher's Note:
1. +1 + (−1) = 0.
2. Zero pairs can be added to a bag without changing its value.

Question 2. Write the additive inverse of −23.
Answer:
The additive inverse of −23 is 23, because (−23) + 23 = 0. We write −(−23) = 23.

Teacher's Note:
1. A number and its additive inverse add up to 0.
2. The inverse has the same size and the opposite sign.

Question 3. Find the value of 7 × (−12).
Answer:
7 × 12 = 84, and one factor is negative, so the product is negative.
7 × (−12) = −84.

Teacher's Note:
1. Different signs give a negative product.
2. Find the size first, then the sign.

Question 4. Find the value of (−7) × (−8).
Answer:
7 × 8 = 56, and both factors are negative, so the product is positive.
(−7) × (−8) = 56.

Teacher's Note:
1. Same signs give a positive product.
2. Two negatives make a positive.

Question 5. A carrom coin at 0 is struck 5 units to the right and then 9 units to the left. Write its final position.
Answer:
The movements are +5 and −9, so the final position is 5 + (−9) = −4, that is, 4 units to the left of 0.

Teacher's Note:
1. Right is positive, left is negative.
2. Add the movements to get the position.

Question 6. Find the value of (−45) ÷ 9.
Answer:
Different signs give a negative quotient: (−45) ÷ 9 = −5, because 9 × (−5) = −45.

Teacher's Note:
1. Check by multiplying back.
2. The sign rules for division match those for multiplication.

Question 7. Write the value of (−1) × a for any integer a.
Answer:
(−1) × a = −a, the additive inverse of a.

Teacher's Note:
1. Multiplying by −1 only changes the sign.
2. If a is negative, −a is positive.

Question 8. What is the sign of the product of two negative integers? Give one example.
Answer:
The product of two negative integers is positive. Example: (−6) × (−4) = 24.

Teacher's Note:
1. Same signs give a positive product.
2. Any correct example earns the mark.

Question 9. Find the value of (−236) × 0.
Answer:
Any integer multiplied by 0 gives 0, so (−236) × 0 = 0.

Teacher's Note:
1. Zero has no sign.
2. The size of the other factor does not matter.

Question 10. Name the Indian mathematician who first wrote down the rules for multiplying and dividing positive and negative numbers, and state the word he used for positive quantities.
Answer:
Brahmagupta, in his Brahmasphutasiddhanta (628 CE). He called positive quantities dhana, meaning fortune.

Teacher's Note:
1. Negative quantities were called rina, meaning debt.
2. Remember the year 628 CE.

Question 11. The sum of two numbers is 20 and their difference is −6. Find the two numbers.
Answer:
7 + 13 = 20 and 7 − 13 = −6, so the numbers are 7 and 13 (7 first, 13 second).

Teacher's Note:
1. A negative difference means the first number is smaller.
2. Check both the sum and the difference.

Question 12. Find the value of (−54) ÷ (−6).
Answer:
Both are negative, so the quotient is positive: (−54) ÷ (−6) = 9, because (−6) × 9 = −54.

Teacher's Note:
1. Same signs give a positive quotient.
2. Check by multiplication.

Question 13. In the token model, what does (−3) × 4 ask us to do? Write the value of the product.
Answer:
It asks us to remove 4 green tokens from the empty bag, 3 times. We first put in zero pairs; after removing 12 greens, 12 red tokens are left.
So (−3) × 4 = −12.

Teacher's Note:
1. A negative multiplier means take out.
2. Zero pairs make removal possible.

Question 14. Continue the pattern 4 × (−3) = −12, 3 × (−3) = −9, 2 × (−3) = −6, 1 × (−3) = −3, 0 × (−3) = 0 and write the values of (−1) × (−3) and (−2) × (−3).
Answer:
Each time the multiplier goes down by 1, the product goes up by 3.
So (−1) × (−3) = 3 and (−2) × (−3) = 6.

Teacher's Note:
1. Follow the step of the pattern beyond zero.
2. This pattern explains why negative × negative is positive.

Question 15. Given that 134 × 27 = 3618, write the values of (−134) × 27 and (−134) × (−27).
Answer:
(−134) × 27 = −3618 (signs different) and (−134) × (−27) = 3618 (signs the same).

Teacher's Note:
1. The size 3618 does not change.
2. Only the sign has to be decided.

Question 16. Name the property of integer multiplication used in writing (−9) × 5 = 5 × (−9).
Answer:
The commutative property of multiplication: a × b = b × a.

Teacher's Note:
1. The factors have only been swapped.
2. Commute means move around.

Question 17. Name the property shown by (−4) × (6 + (−2)) = (−4) × 6 + (−4) × (−2).
Answer:
The distributive property of multiplication over addition: a × (b + c) = a × b + a × c.

Teacher's Note:
1. −4 is shared with both terms in the bracket.
2. Both sides equal −16.

Question 18. The temperature of a storage room is 5 °C and it falls by 3 °C every hour. Write the temperature after 6 hours.
Answer:
Change in 6 hours = 6 × (−3) = −18.
Temperature = 5 + (−18) = −13 °C.

Teacher's Note:
1. A fall is a negative change.
2. Add the total change to the start value.

Question 19. A lift goes down into a mine at 4 m per minute, starting from the ground level. Write its position after half an hour.
Answer:
Half an hour = 30 minutes. Position = 30 × (−4) = −120 m, that is, 120 m below the ground.

Teacher's Note:
1. Change the time to minutes first.
2. Going down is negative.

Question 20. Find the integer whose product with (−1) is 46.
Answer:
The integer is −46, because (−1) × (−46) = 46.

Teacher's Note:
1. Multiplying by −1 changes the sign.
2. So the answer is the additive inverse of 46.

Question 21. Find the value of (−6) × (−5) × (−2).
Answer:
(−6) × (−5) = 30, and 30 × (−2) = −60.
So the value is −60.

Teacher's Note:
1. Three negative factors give a negative product.
2. Multiply two at a time.

Question 22. In the magic grid of the chapter, every number is a row number multiplied by a column number. Why is the product of the circled numbers the same every time?
Answer:
Every play circles exactly one number from each row and each column. So the product always contains each row number once and each column number once.
Hence it always equals (product of all the row numbers) × (product of all the column numbers), whichever numbers are circled.

Teacher's Note:
1. Commutative and associative properties allow the factors to be rearranged.
2. The magic is in how the grid is built.

Question 23. Fill in the blank with an integer: ___ ÷ (−9) = 6.
Answer:
The blank is 6 × (−9) = −54. Check: (−54) ÷ (−9) = 6.

Teacher's Note:
1. Dividend = quotient × divisor.
2. Always check the sign.

Question 24. A shop makes a profit of ₹6 on every good tile sold and a loss of ₹4 on every cracked tile sold. Find its profit or loss on selling 500 good tiles and 200 cracked tiles.
Answer:
500 × 6 + 200 × (−4) = 3000 + (−800) = 2200.
The shop makes a profit of ₹2,200.

Teacher's Note:
1. Write a loss as a negative integer.
2. A positive total means profit.

Question 25. Without multiplying, write the sign of the product (−2) × (−5) × (−7) × (−1) × (−3) and give the reason.
Answer:
The product is negative, because there are 5 negative factors, and an odd number of negative factors gives a negative product.

Teacher's Note:
1. Count the negative factors.
2. Even count: positive; odd count: negative.

Question 26. Given that 38 − 45 + 12 − 7 = −2, write the value of −38 + 45 − 12 + 7 without working out the whole expression.
Answer:
Every term has been replaced by its additive inverse, so the value is the inverse of −2, that is, (−1) × (−2) = 2.

Teacher's Note:
1. Changing every sign changes the sign of the answer.
2. This is multiplying the whole expression by −1.

Question 27. Can (−15) ÷ 0 be found? Give a reason using multiplication.
Answer:
No. (−15) ÷ 0 asks which integer multiplied by 0 gives −15. Every integer times 0 gives 0, so no such integer exists. We can never divide by 0.

Teacher's Note:
1. Turn the division into a multiplication question.
2. Division by zero has no meaning.

Question 28. Is (−9) ÷ 2 an integer? Give a reason using multiplication.
Answer:
No. (−9) ÷ 2 asks which integer multiplied by 2 gives −9. 2 × (−4) = −8 and 2 × (−5) = −10, so no integer works.

Teacher's Note:
1. Any integer times 2 is even.
2. −9 is odd, so the division is not exact.

Question 29. In a country with coins of +13 pibs and −9 pibs only, show how an item costing exactly +4 pibs can be paid for.
Answer:
Hand over one +13 coin and one −9 coin: 13 + (−9) = 4 pibs.

Teacher's Note:
1. The total value is the sum of the coins.
2. A negative coin reduces the total.

Question 30. Find three consecutive integers whose product is −24.
Answer:
(−4) × (−3) = 12 and 12 × (−2) = −24.
The integers are −4, −3 and −2.

Teacher's Note:
1. 24 = 2 × 3 × 4.
2. Three negative factors make the product negative.