CBSE Class 7 Mathematics Operations with Integers Assignment Set 01

Short Answer Questions

Question 1. Rakesh thinks of two numbers. Their sum is 20 and their difference (first number − second number) is 6. Find the two numbers by trying suitable pairs. Then write the pair whose sum is 20 but whose difference is −6.
Answer:
Try pairs that add up to 20 and check the difference: 15 and 5 give 10; 14 and 6 give 8; 13 and 7 give 6.
So the numbers are 13 and 7, since 13 + 7 = 20 and 13 − 7 = 6.
For a difference of −6, swap them: the first is 7 and the second is 13, since 7 + 13 = 20 and 7 − 13 = −6.
In simple words: 13 and 7 work; swapping them changes the difference from 6 to −6.

Teacher's Note:
1. Swapping the order changes only the sign of the difference.
2. Make a small table of trial pairs.

Question 2. Using the token model (green token = +1, red token = −1), explain how you would find (+6) − (+13).
(i) How many zero pairs must be put in before the positives can be removed?
(ii) What is the final answer?
Answer:
Put in 6 green tokens. We now have to remove 13 greens, but only 6 are there.
(i) 13 − 6 = 7 more greens are needed, so we put in 7 zero pairs (7 greens and 7 reds). The value of the bag does not change.
(ii) Now remove all 13 greens; 7 red tokens are left. So (+6) − (+13) = −7.
Check by adding the inverse: 6 + (−13) = −7.
In simple words: Add 7 zero pairs, take out 13 greens, and 7 reds remain: −7.

Teacher's Note:
1. Zero pairs are worth 0, so they never change the value.
2. Subtracting is the same as adding the inverse.

Question 3. A carrom coin is at 0 on a number line. Rightward movement is taken as positive and leftward as negative. The first strike moves it 7 units to the right and the second strike moves it 11 units to the left. Write the movements as integers, form the addition statement, and give the final position of the coin.
Answer:
First movement = +7, second movement = −11.
Final position = 7 + (−11) = −4.
The coin stops at −4, that is, 4 units to the left of 0.
In simple words: 7 right and 11 left leave the coin 4 units left of 0.

Teacher's Note:
1. Direction gives the sign, distance gives the size.
2. Add the two integers in order.

Question 4. State the rule for the sign of the product of two integers in all three possible cases, and give one example of each case.
Answer:
The size of the product is always the product of the sizes; only the sign needs a rule.
Positive × positive = positive. Example: 6 × 4 = 24.
Negative × negative = positive. Example: (−6) × (−4) = 24.
One positive and one negative = negative. Example: 6 × (−4) = −24 and (−6) × 4 = −24.
In simple words: Same signs give a positive product; different signs give a negative product.

Teacher's Note:
1. Learn the rule as 'same signs, positive'.
2. Give an example for every case.

Question 5. Find the following products and state the sign rule used in each case.
(i) 12 × (−7)
(ii) (−15) × (−4)
Answer:
(i) 12 × (−7) = −84. One factor is positive and the other negative, so the product is negative.
(ii) (−15) × (−4) = 60. Both factors are negative, so the product is positive.
In both, the size (84 and 60) is just the product of the sizes.
In simple words: Different signs give −84; two negatives give +60.

Teacher's Note:
1. Decide the sign before multiplying.
2. Size and sign are found separately.

Question 6. Find the following quotients and state the sign rule used in each case.
(i) (−91) ÷ 7
(ii) (−108) ÷ (−9)
Answer:
(i) (−91) ÷ 7 = −13. The dividend is negative and the divisor positive, so the quotient is negative. Check: 7 × (−13) = −91.
(ii) (−108) ÷ (−9) = 12. Both are negative, so the quotient is positive. Check: (−9) × 12 = −108.
In simple words: Different signs give −13; same signs give +12.

Teacher's Note:
1. Division follows the same sign rules as multiplication.
2. Check each answer by multiplying back.

Question 7. Write the additive inverse of (i) −26 and (ii) 40. Then find the integer whose product with −1 is −17, and state the general rule for −1 × a.
Answer:
(i) The additive inverse of −26 is 26, since (−26) + 26 = 0.
(ii) The additive inverse of 40 is −40, since 40 + (−40) = 0.
The required integer is 17, because (−1) × 17 = −17.
Rule: (−1) × a = −a for every integer a; multiplying by −1 gives the additive inverse.
In simple words: Inverses are 26 and −40; (−1) × 17 = −17.

Teacher's Note:
1. The inverse has the opposite sign.
2. Multiplying by −1 gives the inverse.

Question 8. Verify that the multiplication of integers is commutative by using the integers −9 and 7. Write both products and the general statement of the property.
Answer:
(−9) × 7 = −63 (one factor negative).
7 × (−9) = −63 (one factor negative).
Both products are equal, so (−9) × 7 = 7 × (−9).
General statement: for any integers a and b, a × b = b × a.
In simple words: Both orders give −63, so the order does not matter.

Teacher's Note:
1. Compute both sides separately.
2. Write the property in letters at the end.

Question 9. Copy and complete this pattern, and then describe in one sentence what happens to the product each time the multiplier decreases by 1.
3 × (−6) = −18
2 × (−6) = ___
1 × (−6) = ___
0 × (−6) = ___
(−1) × (−6) = ___
(−2) × (−6) = ___
Answer:
2 × (−6) = −12; 1 × (−6) = −6; 0 × (−6) = 0; (−1) × (−6) = 6; (−2) × (−6) = 12.
Each time the multiplier decreases by 1, the product increases by 6.
Continuing the same pattern below zero makes the product of two negative integers positive.
In simple words: The products go up by 6 each step: −12, −6, 0, 6, 12.

Teacher's Note:
1. The step is the size of the multiplicand.
2. The pattern must not break at zero.

Question 10. Explain how the division (−84) ÷ 12 can be answered by turning it into a multiplication question. Write the multiplication statement you use and give the quotient.
Answer:
(−84) ÷ 12 asks: which integer multiplied by 12 gives −84?
Multiplication statement: 12 × ___ = −84.
12 × 7 = 84, and the product must be negative, so the missing integer is −7: 12 × (−7) = −84.
So (−84) ÷ 12 = −7.
In simple words: 12 × (−7) = −84, so the quotient is −7.

Teacher's Note:
1. Division is the reverse of multiplication.
2. Get the size from the table, then fix the sign.

Question 11. Given that 316 × 24 = 7584, write the values of the following without doing any multiplication, and give a reason for each sign.
(i) (−316) × 24
(ii) 316 × (−24)
(iii) (−316) × (−24)
Answer:
Changing signs does not change the size 7584.
(i) (−316) × 24 = −7584, as one factor is negative.
(ii) 316 × (−24) = −7584, as one factor is negative.
(iii) (−316) × (−24) = 7584, as both factors are negative.
In simple words: Same size 7584 every time; only the sign changes.

Teacher's Note:
1. Use the given product.
2. Count the negative factors.

Question 12. Show that the distributive property holds for (−8) × (5 + (−9)) by evaluating the expression in two different ways.
Answer:
Bracket first: 5 + (−9) = −4, so (−8) × (−4) = 32.
Distributing: (−8) × 5 + (−8) × (−9) = −40 + 72 = 32.
Both ways give 32, so a × (b + c) = a × b + a × c holds here.
In simple words: Both methods give 32.

Teacher's Note:
1. Watch the signs in each product.
2. (−8) × (−9) is positive.

Question 13. Evaluate (−2) × 7 × (−5) by grouping the numbers in two different ways. Which property of integer multiplication does your work illustrate?
Answer:
Grouping the first two: ((−2) × 7) × (−5) = (−14) × (−5) = 70.
Grouping the last two: (−2) × (7 × (−5)) = (−2) × (−35) = 70.
Both give 70, which shows the associative property: (a × b) × c = a × (b × c).
In simple words: Either grouping gives 70, so grouping does not matter.

Teacher's Note:
1. Associative is about brackets, not order.
2. Two negative factors make the answer positive.

Question 14. Answer the following about the special integers 1 and 0.
(i) Write the values of 1 × (−43) and 0 × (−43).
(ii) Write the values of (−43) ÷ 1 and 0 ÷ (−43).
Answer:
(i) 1 × (−43) = −43, because 1 leaves every integer unchanged; 0 × (−43) = 0, because any integer times 0 is 0.
(ii) (−43) ÷ 1 = −43, since 1 × (−43) = −43; 0 ÷ (−43) = 0, since (−43) × 0 = 0.
In simple words: 1 keeps the number the same; 0 times anything is 0.

Teacher's Note:
1. 0 ÷ a number is allowed, but a number ÷ 0 is not.
2. Check each division by multiplying.

Question 15. Without multiplying out fully, decide the sign of each product and then give its value. Finally, write the rule you used.
(i) (−1) × (−1) × (−1) × (−1) × (−1)
(ii) (−2) × (−3) × (−4) × (−5)
Answer:
(i) Five negative factors, an odd number, so the product is negative: −1.
(ii) Four negative factors, an even number, so the product is positive: 2 × 3 × 4 × 5 = 120.
Rule: an even number of negative factors gives a positive product; an odd number gives a negative product.
In simple words: Five negatives give −1; four negatives give +120.

Teacher's Note:
1. Count the negative signs first.
2. Pairs of negatives make positives.

Question 16. Replace each blank with an integer so that the statement is true, and show the check for each.
(i) (−8) × ___ = 72
(ii) ___ ÷ (−7) = 11
(iii) ___ × (−6) = −54
Answer:
(i) The blank is −9. Check: (−8) × (−9) = 72.
(ii) The blank is 11 × (−7) = −77. Check: (−77) ÷ (−7) = 11.
(iii) The blank is 9. Check: 9 × (−6) = −54.
The size comes from the table and the sign from the sign rules.
In simple words: The blanks are −9, −77 and 9.

Teacher's Note:
1. Always put the answer back to check.
2. The sign rule decides the sign of the blank.

Question 17. The temperature inside a cold room is 7 °C and the cooling unit lowers it by 4 °C every hour.
(i) Write an expression using integers for the temperature after h hours, and find the temperature after 5 hours.
(ii) After how many hours will the temperature be −9 °C?
Answer:
(i) Each hour the change is −4, so after h hours the temperature is 7 + h × (−4). For h = 5: 7 + 5 × (−4) = 7 − 20 = −13 °C.
(ii) We need 7 + h × (−4) = −9, so h × (−4) = −16 and h = (−16) ÷ (−4) = 4. The temperature is −9 °C after 4 hours.
In simple words: It drops 4 °C an hour: −13 °C after 5 hours and −9 °C after 4 hours.

Teacher's Note:
1. A fall is a negative change.
2. Check: 7 − 16 = −9.

Question 18. In a test, +5 marks are given for every correct answer and −2 marks for every wrong answer. Rohit attempted every question and scored 46 marks, out of which 14 answers were correct. Find the number of wrong answers and the total number of questions in the test.
Answer:
Marks from 14 correct answers = 14 × 5 = 70.
Let the number of wrong answers be w; they give w × (−2) marks.
70 + w × (−2) = 46, so w × (−2) = −24 and w = 12. There were 12 wrong answers.
Total questions = 14 + 12 = 26. Check: 70 + 12 × (−2) = 70 − 24 = 46.
In simple words: He lost 24 marks, which is 12 wrong answers; 14 + 12 = 26 questions.

Teacher's Note:
1. Find the marks lost first.
2. Every question was attempted, so add correct and wrong.

Question 19. Answer both parts, using the idea that p ÷ q asks “which number multiplied by q gives p?”
(i) Is (−23) ÷ 5 an integer? Justify your answer.
(ii) Why is (−23) ÷ 0 not allowed?
Answer:
(i) No. We need an integer that gives −23 when multiplied by 5. But 5 × (−4) = −20 and 5 × (−5) = −25, and −23 lies between them, so no integer works.
(ii) It asks which number multiplied by 0 gives −23. Every number times 0 is 0, never −23, so there is no answer. Division by zero is not allowed.
In simple words: No integer times 5 gives −23, and nothing times 0 gives −23.

Teacher's Note:
1. Look for the two multiples on either side.
2. Division by 0 never has an answer.

Question 20. A student claims that (−5) × (−3) should be −15 “because both numbers are negative”. Use the pattern obtained by decreasing the multiplier one step at a time in the table of −3 to explain why (−5) × (−3) must be 15.
Answer:
Table of −3: 2 × (−3) = −6, 1 × (−3) = −3, 0 × (−3) = 0.
Each time the multiplier goes down by 1, the product goes up by 3.
Continuing: (−1) × (−3) = 3, (−2) × (−3) = 6, (−3) × (−3) = 9, (−4) × (−3) = 12, (−5) × (−3) = 15.
The pattern must not break, so the answer is 15. The student's −15 would make the products suddenly fall instead of rising.
In simple words: The table keeps rising by 3, so (−5) × (−3) = 15.

Teacher's Note:
1. Patterns explain the sign rules.
2. Two negatives give a positive product.