Class 7 Maths Ganita Prakash Part 2 Chapter 2 MCQs – Operations with Integers

For the pair −4 and −3:
Sum = −4 + (−3) = −7
Difference = −4 − (−3) = −4 + 3 = −1
Both conditions are satisfied. Reversing the pair would keep the sum but change the difference to 1.

A rightward movement is positive and a leftward movement is negative.
Final position = 5 + (−7)
= −2
The 7-unit leftward movement cancels the 5-unit rightward movement and takes the coin another 2 units left of 0.

Let the second movement be b.
−4 + b = 5
b = 5 − (−4) = 9
From −4, the coin needs 4 units to reach 0 and another 5 units to reach 5. Both movements are rightward.

The movements can be grouped in consecutive pairs.
(1 − 2) + (3 − 4) + (5 − 6) + (7 − 8) + (9 − 10)
= −1 − 1 − 1 − 1 − 1
= −5
So the coin finishes 5 units left of 0.

Subtracting an integer is the same as adding its additive inverse.
The additive inverse of −12 is 12.
4 − (−12) = 4 + 12 = 16
The two minus signs here mean that a negative number is being subtracted.

Each zero pair adds one positive and one negative token without changing the total value.
Extra positive tokens needed = 18 − 7 = 11
So 11 zero pairs are enough. After removing 18 positive tokens, 11 negative tokens remain.

Each turn adds −2 to the bag.
Total = (−2) + (−2) + (−2) + (−2)
= 4 × (−2)
= −8
There are 8 negative tokens and no positive tokens to cancel them.

Removing 2 negative tokens from 2 zero pairs leaves 2 positive tokens.
Doing this 4 times leaves 4 × 2 = 8 positive tokens.
Therefore, (−4) × (−2) = 8. The product of these two negative integers is positive.

The magnitudes of the factors are still 123 and 456, so the product has magnitude 56088.
Both factors are negative, making their product positive.
Thus, (−123) × (−456) = 56088 without repeating the long multiplication.

The first few products are −12, −9, −6 and −3.
−9 − (−12) = 3
−6 − (−9) = 3
So each product is 3 greater than the previous one. This continues through 0 and into positive products.

Multiplying by −1 changes an integer to its additive inverse.
For example, (−1) × 5 = −5 and (−1) × (−5) = 5.
For a = 0, the product is still 0. Hence, (−1) × a = −a for every integer.

Only the order of the two factors has changed.
(−30) × 12 = −360
12 × (−30) = −360
The products are equal. The rule a × b = b × a is called the commutative property of multiplication.

Marks for wrong answers = 20 × (−2) = −40
Total score = 150 + (−40) = 110
The negative marks reduce the total instead of being added as positive marks.

Downward movement is negative.
Movement in 45 minutes = 45 × (−3) = −135 m
Final position = 15 + (−135) = −120 m
So the elevator finishes 120 m below ground after travelling down from its initial positive position.

There is one negative factor, so the product is negative.
8 × 14 = 112
112 × 18 = 2016
2016 × (−15) = −30240
The sign is decided separately from the multiplication of the magnitudes.

Division can be checked using multiplication.
(−4) × 25 = −100
Therefore, (−100) ÷ (−4) = 25.
The dividend and divisor are both negative, so the quotient is positive. The magnitudes give 100 ÷ 4 = 25.

Total change = 10 × (−5) = −50°C
Final temperature = 32 + (−50)
= −18°C
The drop is greater than the starting temperature above zero, so the temperature passes below 0°C and ends at −18°C.

White-bag profit = 3000 × ₹8 = ₹24000
Grey-bag result = 5000 × (−₹5) = −₹25000
Net result = ₹24000 − ₹25000 = −₹1000
A negative net result represents a loss, so the overall loss is ₹1000.

Loss on grey bags = 6400 × ₹5 = ₹32000
To have no net profit or loss, the white-bag profit must also be ₹32000.
Required white bags = 32000 ÷ 8 = 4000
Then the positive and negative amounts cancel.

The dividend equals the quotient multiplied by the divisor.
x = 7 × (−8) = −56
Check: (−56) ÷ (−8) = 7
Both dividend and divisor are negative, so their quotient is positive as required.

Associativity changes the grouping of factors without changing their order.
Left side = (−15) × 4 = −60
Right side = 5 × (−12) = −60
The equal result shows that either pair of adjacent factors may be multiplied first.

Two factors of −1 multiply to 1.
(−1) × (−1) × (−1) × (−1) × (−1)
= 1 × 1 × (−1)
= −1
There is one unpaired negative factor because the number of factors is odd.

The outside factor −2 multiplies each term inside the brackets.
(−2) × (4 + (−3)) = (−2) × 4 + (−2) × (−3)
= −8 + 6 = −2
This also matches (−2) × 1 after adding inside the brackets first.

Every term in the new expression is the additive inverse of the corresponding original term.
So the new expression is −1 times the original expression.
New value = (−1) × (−4) = 4
There is no need to add all the terms again.

Output = (−10) + (−12) − (−9)
= −22 + 9
= −13
Subtracting −9 means adding 9. Adding −9 instead would give a different result because it would not follow the machine’s stated rule.

Marks from correct answers = 15 × 4 = 60
Marks lost = 60 − 40 = 20
Each incorrect answer loses 2 marks.
Incorrect answers = 20 ÷ 2 = 10
Thus, Anita answered 15 + 10 = 25 questions altogether.

The integers −3, −2 and −1 are consecutive.
(−3) × (−2) = 6
6 × (−1) = −6
The first two negative factors give a positive product, and the third makes it negative. Their product is therefore −6.

Value of 10 positive coins = 10 × 13 = 130 pibs
Value of 5 negative coins = 5 × (−9) = −45 pibs
Total = 130 + (−45) = 85 pibs
So these coins together make the required positive total.

Numerator = 280 × (−7) = −1960
Denominator = (−8) × (−35) = 280
Quotient = −1960 ÷ 280 = −7
The denominator is positive because it is a product of two negatives; the final quotient is negative.

Since −547 = −548 + 1, distributivity gives:
(−547) × 972 = ((−548) + 1) × 972
= (−548) × 972 + 972
= −532656 + 972 = −531684
Increasing one factor by 1 increases this product by 972.