NCERT Solutions for Class 7 Maths Chapter 10 Operations with Integers (PDF) – 2027

The two numbers are 18 and 7.

Work down the table the way the book does. Keep the sum fixed at 25 and watch how the difference changes.

The numbers are 7 and 18 — the same pair as before, but swapped.

Use the same two half-steps every time.

To find the two integers from the sum S and the difference D:

Sample turn: your partner says “sum 12, difference –30”.

The coin ends at 7.

So the coin is 4 + 3 = 7 units from 0, to the right.

Yes. If the first strike moves the coin a units to the right and the second strike moves it b units to the right, then

where P is the distance of the coin from the starting point 0.

The formula is the same one: P = a + b. Only the meaning of a and b widens.

• Take a rightward movement as positive.
• Take a leftward movement as negative.

Then a is the first movement (positive if the strike is to the right, negative if to the left), b is the second movement in the same sense, and

The coin ends at –2.

So the coin is 2 units to the left of 0.

The second movement is +9, that is, 9 units to the right.

Check: (–4) + 9 = 5 ✓

The coin ends at –5.

The ten movements are 1, –2, 3, –4, 5, –6, 7, –8, 9, –10. Pair each odd movement with the negative one just after it.

In each figure a is the first movement, b is the second, and P = a + b is where the coin stops.

So the movement with the greater magnitude decides on which side of 0 the coin stops.

You need 11 zero pairs, and what is left is –11.

(a) 7 – 18 = 7 + (–18)

(b) 4 – (–12) = 4 + 12

Both sides leave the same bag, so each statement is true.

There are 8 positives in the bag.

The picture shows 2 positive tokens being dropped in, and that is done 4 times.

Here 4 is the multiplier (how many times), 2 is the multiplicand (what is being put in each time), and 8 is the product.

Yes. Just change the colour of the tokens you drop in.

Start with an empty bag. 4 × (–2) means placing 2 negatives into the bag, 4 times. Negatives are red tokens.

4 × (–6) = –24 and 9 × (–7) = –63.

(–4) × 2: a negative multiplier means we remove tokens. So remove 2 positives (2 green tokens) from the bag, 4 times.

Because the multiplicand decides which colour is handled, and here the multiplicand is +2.

In (–4) × 2, read the two numbers separately.

The product is positive: (–4) × (–2) = 8.

The multiplier –4 says remove, 4 times; the multiplicand –2 says 2 red tokens. The bag is empty, so first put in zero pairs, as in subtraction.

Read the multiplier for the action and the multiplicand for the colour.

The magnitude never changes — only the sign does.

Here is a rule in two steps.

1. Magnitudes: multiply the two numbers as if both were positive.
2. Sign: if the two signs are the same, the product is positive. If they are different, the product is negative.

All three give the same answer, –8. The sets look different, but they are worth the same.

Every token set worth 4 gives the same product, 20.

Yes. Removing 2 positives is the same as adding 2 negatives.

Both routes empty into the same bag, so both descriptions of (–4) × 2 are correct.

The pattern is: every unit decrease in the multiplier drops the product by the multiplicand.

Yes. The steps of –3 simply carry on past zero.

Each line is still 3 less than the one above it.

The staircase turns the other way: every unit decrease in the multiplier now raises the product by the multiplicand’s magnitude.

Yes — and this is where the last sign rule appears.

Each line is 3 more than the one above it, so the products march up through the positive numbers.

Multiply the magnitudes, then fix the sign by counting negatives.

Yes. 1 × a = a for every integer a, positive or negative.

Use the token model. The multiplier is 1, so you place the multiplicand into the empty bag just once.

So in general,

–1 × a = –a — the additive inverse of a — for every integer a.

Yes. Swapping them never changes the product.

The two sides of each pair agree every time.

The filled table is:

What we notice: the product is the same when the multiplier and multiplicand are swapped.

Yes, always. The magnitude of a product depends only on the two magnitudes.

For example, both (–15) × (–8) and (–8) × (–15) have magnitude 15 × 8 = 120.

No. Whatever the signs are, they give the same result before and after swapping.

• If both are positive, the product is positive before and after the swap.
• If both are negative, the product is positive before and after the swap.
• If one is positive and the other negative, the product is negative before and after the swap — swapping only decides which of the two is written first.

Maximum = 250 marks. Minimum = –100 marks.

The best case is all 50 answers correct.

The worst case is all 50 answers wrong.

The elevator will be at –120 m, that is, 120 metres below the ground.

Playing the game as in the book’s example, the product of the circled numbers is –30,240.

In the four rounds shown in the book the circled numbers are –6, 14, 20 and 18.

The product is –30,240 again — every single time.

Try three completely different sets of circles.

The answer is –30,240 once more.

The grid printed for this round has exactly the same sixteen numbers as the first grid, so whichever numbers you circle, the product comes out the same.

The magic is in the arrangement. Each number is (its row number) × (its column number), like a multiplication table. One circle in every row and every column means every game multiplies the same eight hidden numbers, so the product never changes.

Yes, we can make more. Choose 4 row numbers and 4 column numbers and fill each cell with their product — for example, rows 1, –2, 4, 3 and columns 5, –1, 2, –4:

Division follows exactly the same sign rule as multiplication.

1. Magnitudes: divide as if both numbers were positive.
2. Sign: same signs give a positive quotient, different signs a negative quotient.

In symbols, for positive integers a and b with b ≠ 0,

The room will be at –18 °C.

A fall of 5 °C each hour is a change of –5 °C per hour.

Write profit as +8 per white bag and loss as –5 per grey bag.

(a) The company makes a loss of ₹1,000.

(b) The company must sell 4,000 bags of white cement.

Turn each one round into a division (or, for the last two, into a multiplication).

The value is –60, and no, the grouping does not matter.

The grouping never changes the answer. Integer multiplication is associative.

So for any three integers a, b and c,

Yes — every order gives –60.

The three numbers can be arranged in 6 orders, and each arrangement can be bracketed in 2 ways.

Every order gives –1800.

Yes. An even number of –1s gives +1; an odd number of –1s gives –1.

Count the negative factors.

• An even number of negative factors → the product is positive.
• An odd number of negative factors → the product is negative.
• If any factor is 0, the product is 0.

Yes. Both sides come to 10.

This is the distributive property, and it works for integers just as it did for whole numbers.

It holds. Both sides give –2.

Two more checks of your own choosing:

Yes, this will always happen. For any integers a, b and c,

Yes. Use the hint: multiplying by –4 means adding the inverse of the multiplicand, 4 times.

The multiplicand is the set 2 + (–3): 2 greens and 3 reds. Its inverse is 2 reds and 3 greens. Place this inverse set 4 times:

Machine 1 works out (first number) + (second number) – (third number), that is, a + b – c.

The answer is –13.

Machine 2 works out –(a × b) – c, which can also be written as (–a) × b – c. The blank is –111.

a machine you can copy. Show your friends only the three inputs and the result, and keep the rule hidden.

The hidden rule is (a × b) – c.

Multiplying by –1 gives the additive inverse, so the answer is always the additive inverse of the number given.

The value is 4.

Every single sign in the second expression is the opposite of the sign in the first. So the second expression is the first one multiplied by –1.

Most starts end in the loop 1 → –2 → –1 → 4 → 2 → 1.

• An odd number is always followed by an even number, and the sign flips.
• Not every start reaches 1: 0 stays at 0, and starts such as 13, –15 or –17 fall into a longer loop of 31 numbers.

(a) Anita had 10 incorrect answers, and the test has 25 questions.

(b) Anil had 15 incorrect answers, and he left 5 questions unanswered.

The machine works out a – (b × c) — the first number minus the product of the other two. The missing star is –70.

The expression is 8 + 4 × (–5), and its value is –12 °C.

So the temperature will be 12 °C below zero.

(a) –3, –2, –1.   (b) 4, 5, 6.

Yes — +85 and every total below can be made. With m coins of +13 and n coins of –9, the total is 13 × m – 9 × n.

(h) Yes: 122 × 13 – 2 × 9 = 1586 – 18 = 1568, so use 122 coins of +13 and 2 coins of –9.

Parts (b) and (c) do not divide exactly, so their values are fractions, not integers.

Work out each value first.

Increasing order:

Use the distributive property to adjust the known product by one step at a time.

The value is –5382 — exactly the same as the given product.

(a) Maximum = 60. One expression that reaches it:

(b) Minimum = –60. One expression that reaches it:

Remember that in (a) the multiplication is done before the addition.

(a) ___ + ___ × ___ = –36

(b) ( ___ – ___ ) × ___ = 12

(c) ( ___ – ( ___ – ___ )) = –1

Terhüchü is a board game to play, not a sum. In short:

• Move: on your turn, move one coin along a line to a neighbouring empty point.
• Capture: jump over an opponent’s coin on a neighbouring point to the empty point just beyond it; the jumped coin is removed. Several jumps in one move are allowed.
• Win: capture all your opponent’s coins, or leave them no legal move. If a draw seems unavoidable, the player with more coins wins.