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NCERT Solutions for Class 5th Maths Chapter 6 How the product changes, and number-pair grids — Multiplication Patterns

Book page 89–90 Updated on2026-09-19

Q1.
Notice how the multiplier, multiplicand, and products are changing. What is the relationship of the new product with the original product? a) 16 × 44 = 704; 1) 8 × 88 = 704; 2) 8 × 22 = 176; 3) 16 × 22 = ___; 4) 32 × 44 = ___
Answer

3) 16 × 22 = 352 and 4) 32 × 44 = 1,408.

Start from 16 × 44 = 704 and watch what happens to each number.

ProblemWhat changedEffect on the productAnswer
16 × 44 = 704the original—704
1) 8 × 8816 halved, 44 doubledno change704
2) 8 × 22both halvedone quarter of 704176
3) 16 × 2244 halved onlyhalf of 704352
4) 32 × 4416 doubled onlydouble of 7041,408
3) 16 × 22
Step 1. 22 is half of 44.
Step 2. So the product is half of 704.
Step 3. 704 ÷ 2 = 352

4) 32 × 44
Step 1. 32 is double 16.
Step 2. So the product is double 704.
Step 3. 704 × 2 = 1,408
The rule to remember: Change one number and the product changes the same way. Halve one number, the product halves. Double one, it doubles. Halve both and the product is quartered, because it is halved twice.
Check 3) directly: 16 × 22 = 16 × 20 + 16 × 2 = 320 + 32 = 352 ✓
Q2.
b) 12 × 32 = 384; 1) 6 × 16 = ___; 2) 24 × 16 = ___; 3) 24 × 64 = ___; 4) 12 × 16 = ___
Answer

1) 96 2) 384 3) 1,536 4) 192

ProblemWhat changedEffect on 384Answer
12 × 32 = 384the original—384
1) 6 × 16both halvedone quarter96
2) 24 × 1612 doubled, 32 halvedno change384
3) 24 × 64both doubledfour times1,536
4) 12 × 1632 halved onlyhalf192
1) 384 ÷ 2 = 192, then 192 ÷ 2 = 96
3) 384 × 2 = 768, then 768 × 2 = 1,536
4) 384 ÷ 2 = 192
Check 3) directly: 24 × 64 = 24 × 60 + 24 × 4 = 1,440 + 96 = 1,536 ✓
Why doubling both makes it four times bigger: The first doubling makes it twice as big. The second doubling doubles that again. Twice of twice is four times.
Q3.
Observe and complete the given pattern: 1 × 1 = 1; 11 × 11 = 121; 111 × 111 = 12,321; 1111 × 1111 = ___
Answer

1111 × 1111 = 1,234,321.

  1. Look at the answers so far. 1, then 121, then 12321. The digits climb up and then climb back down.
  2. Count the ones. With one 1 the top of the hill is 1. With two 1s the top is 2. With three 1s the top is 3.
  3. So with four 1s the top will be 4. Climb up 1, 2, 3, 4 and back down 3, 2, 1.
  4. The answer is 1234321, written with commas as 1,234,321.
1 × 1 = 1
11 × 11 = 121
111 × 111 = 12,321
1111 × 1111 = 1,234,321
(and 11111 × 11111 = 123,454,321)
Why the hill shape: When you multiply 1111 by 1111 you are adding 1111 shifted along four times. In the middle column four 1s line up and give 4. Towards the edges fewer line up, so you get 3, 2 and 1.
Q4.
Observe and complete: 11 × 12 = 132; 11 × 34 = 374; 11 × 56 = 616; 11 × 78 = 858; 11 × 54 = ___; 11 × 82 = ___
Answer

11 × 54 = 594 and 11 × 82 = 902.

The book shows the trick beside the pattern: for 11 × 12, the hundreds digit is 1 × 1, the tens digit is 1 + 2, and the ones digit is 1 × 2.

  1. 11 × 54. Split 54 into 5 and 4. Write 5, then 5 + 4 = 9, then 4. That gives 594.
  2. 11 × 82. Split 82 into 8 and 2. Write 8, then 8 + 2 = 10 — this is more than 9, so keep the 0 and carry the 1 to the 8, making 9. Then write 2. That gives 902.
Check both the long way
11 × 54 = 10 × 54 + 54 = 540 + 54 = 594 ✓
11 × 82 = 10 × 82 + 82 = 820 + 82 = 902 ✓
Why the middle digit is the sum: 11 × 54 is 54 + 540. Adding 54 to 540 puts the 5 and the 4 in the same tens column, so they add together. That is where 5 + 4 = 9 comes from.
Careful with 82: The short trick only works straight through when the two digits add to 9 or less. When they add to 10 or more, remember to carry, exactly as in 11 × 82 = 902.
Q5.
Observe and complete: 5 × 5 = 25; 15 × 15 = 225; 25 × 25 = 625; 35 × 35 = 1,225; 45 × 45 = ___; 55 × 55 = ___
Answer

45 × 45 = 2,025 and 55 × 55 = 3,025.

The book shows the trick: every answer ends in 25, and the front part comes from the tens digit.

  1. Take the tens digit. For 45 it is 4. For 55 it is 5.
  2. Multiply it by the next number up. 4 × 5 = 20. 5 × 6 = 30.
  3. Write 25 after it. 20 then 25 gives 2,025. 30 then 25 gives 3,025.
ProblemFront partEndAnswer
25 × 252 × 3 = 625625
35 × 353 × 4 = 12251,225
45 × 454 × 5 = 20252,025
55 × 555 × 6 = 30253,025
Check 45 × 45 the long way
Step 1. 45 × 40 = 1,800
Step 2. 45 × 5 = 225
Step 3. 1,800 + 225 = 2,025 ✓
Try this: Use the trick on 65 × 65. Front part 6 × 7 = 42, then 25. So 65 × 65 = 4,225. Check it the long way and see.
Q6.
Observe and complete: 1 × 9 + 1 = 10; 12 × 9 + 2 = 110; 123 × 9 + 3 = 1,110; 1,234 × 9 + 4 = ___
Answer

1,234 × 9 + 4 = 11,110.

  1. Multiply first. 1,234 × 9. Use the nearest multiple: 1,234 × 10 = 12,340, then take away one 1,234.
  2. 12,340 − 1,234 = 11,106.
  3. Now add 4. 11,106 + 4 = 11,110.
1 × 9 + 1 = 10
12 × 9 + 2 = 110
123 × 9 + 3 = 1,110
1,234 × 9 + 4 = 11,110
(and 12,345 × 9 + 5 = 111,110)
Look at the pattern of answers: 10, then 110, then 1110, then 11110. Each answer gains one more 1 at the front and always ends in 0. So you could have written the answer straight away — but doing the multiplication proves the pattern is real, not a lucky guess.
Q7.
Here are some numbers (8, 35, 75 / 32, 62, 13 / 66, 11, 14). Any two adjacent numbers in a row or a column are number pairs. Can you identify the pair whose product is the smallest and another pair whose product is the largest? Do you need to find every product or can you find this by looking at the numbers?
Answer

Smallest product: 11 × 14 = 154. Largest product: 35 × 75 = 2,625. No, you do not need to work out every product — you can see it by looking.

First, list which numbers touch each other in a row or a column.

PairProductPairProduct
8 and 352808 and 32256
35 and 752,62532 and 662,112
32 and 621,98435 and 622,170
62 and 1380662 and 11682
66 and 1172675 and 13975
11 and 1415413 and 14182

How to find it by looking:

  1. For the smallest product, hunt for two small numbers sitting next to each other. The smallest numbers in the grid are 8, 11, 13 and 14. Of these, 11 and 14 touch. Their product 154 must be the smallest.
  2. For the largest product, hunt for two big numbers sitting next to each other. The biggest numbers are 75, 66, 62 and 35. Of these, 35 and 75 touch, and 32 and 66 touch. 35 × 75 uses the largest number in the grid, so it wins.
  3. Only then check the few close rivals. 35 × 62 = 2,170 and 32 × 66 = 2,112 are both smaller than 2,625 ✓
Why 8 does not give the smallest product: 8 is the smallest number in the grid, but its neighbours are 35 and 32, which are both big. 8 × 32 = 256, still bigger than 154. It is the pair that matters, not one small number.
Check it yourself: 11 × 14 = 11 × 14 = 154. And 35 × 75 = 35 × 75 → double and halve: 70 × 37.5 is messy, so do it the plain way: 35 × 70 = 2,450 and 35 × 5 = 175. 2,450 + 175 = 2,625 ✓
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