NCERT Solutions Ganita Prakash (Part 1) Chapter 1 –15 (How Long is the Product?)1.5 Patterns in Products — In-text Questions

Book page 14 Updated on2026-09-05

Q1.

In each of the following boxes, the multiplications produce interesting patterns. Evaluate them to find the pattern. Extend the multiplications based on the observed pattern. (11 × 11, 111 × 111, 1111 × 1111; 66 × 61, 666 × 661, 6666 × 6661; 3 × 5, 33 × 35, 333 × 335; 101 × 101, 102 × 102, 103 × 103)

Answer

Work each one out and the pattern jumps out.

Multiplication	Product	Multiplication	Product
11 × 11	121	66 × 61	4,026
111 × 111	12,321	666 × 661	4,40,226
1111 × 1111	12,34,321	6666 × 6661	4,44,02,226
Extend: 11111 × 11111	12,34,54,321	Extend: 66666 × 66661	4,44,40,22,226
3 × 5	15	101 × 101	10,201
33 × 35	1,155	102 × 102	10,404
333 × 335	1,11,555	103 × 103	10,609
Extend: 3333 × 3335	1,11,15,555	Extend: 104 × 104	10,816

Patterns:
1's: the product counts up and back down — 1, 121, 12321, 1234321, 123454321
6's: 4026 → 440226 → 44402226 — one extra 4, one extra 2 each time
3 & 5: 15 → 1155 → 111555 — as many 1's as 5's
101, 102, 103: 10201, 10404, 10609 — the middle jumps 2, 4, 6 and the end 1, 4, 9 (the squares)

Why it happens: In the 1's pattern, 1111 × 1111 is really (1000 + 100 + 10 + 1) added to itself in four shifted rows, so the columns pile up 1, 2, 3, 4, 3, 2, 1 — and since no column reaches 10, there is no carrying and the counts show up as digits. The pattern breaks at 1111111111 × 1111111111 because a column would then hold 10.

Check it yourself: 333 × 335 = 333 × 335 = 1,11,555 ✓ — and notice 3333 × 3335 = 1,11,15,555 keeps four 1's and four 5's.

Q2.

Observe the number of digits in the two numbers being multiplied and their product in each case. Is there any connection between the numbers being multiplied and the number of digits in their product?

Answer

Yes. If one number has m digits and the other has n digits, the product has either (m + n − 1) or (m + n) digits — never anything else.

Multiplication	m + n	Digits in product
11 × 11 = 121	2 + 2 = 4	3 = m + n − 1
1111 × 1111 = 12,34,321	4 + 4 = 8	7 = m + n − 1
66 × 61 = 4,026	2 + 2 = 4	4 = m + n
666 × 661 = 4,40,226	3 + 3 = 6	6 = m + n
3 × 5 = 15	1 + 1 = 2	2 = m + n

Why it happens: The smallest m-digit number is 10^m−1 and the largest is just under 10^m. So the product lies between 10^m−1 × 10ⁿ⁻¹ = 10^m+n−2 and 10^m × 10ⁿ = 10^m+n. That leaves room for exactly two possible digit counts: m + n − 1 or m + n.

Q3.

Roxie says that the product of two 2-digit numbers can only be a 3- or a 4-digit number. Is she correct?

Answer

Yes, Roxie is correct.

Smallest 2-digit × smallest 2-digit = 10 × 10 = 100 → 3 digits
Largest 2-digit × largest 2-digit = 99 × 99 = 9,801 → 4 digits
So every such product lies between 100 and 9,801
→ it must have 3 or 4 digits

Why it happens: This matches the rule above with m = n = 2: the product has m + n − 1 = 3 digits or m + n = 4 digits. Roxie's own argument uses 100 × 100 = 10,000 as a ceiling, which is a neat trick — since both numbers are below 100, the product must be below 10,000, i.e. at most 4 digits.

Q4.

Should we try all possible multiplications with 2-digit numbers to tell whether Roxie’s claim is true? Or is there a better way to find out?

Answer

There is a far better way — check only the smallest and the largest case. There is no need to test all 8,100 pairs.

All 2-digit numbers lie between 10 and 99.
So every product lies between 10 × 10 = 100
and 99 × 99 = 9,801.
Both ends have 3 or 4 digits → all products do

Why it happens: Multiplication is “monotonic” — making either factor bigger makes the product bigger. So squeezing the factors between their smallest and largest values automatically squeezes the product between the two extreme products. Checking two cases settles all of them.

Tip: This is a very useful habit in mathematics — instead of testing endless examples, find the extreme cases and reason about the range in between.

Q5.

Can multiplying a 3-digit number with another 3-digit number give a 4-digit number?

Answer

No. The product always has 5 or 6 digits.

Smallest case: 100 × 100 = 10,000 → 5 digits
Largest case: 999 × 999 = 9,98,001 → 6 digits
The smallest possible product, 10,000, is already a 5-digit number.

Why it happens: By the rule, m + n − 1 = 5 and m + n = 6 for two 3-digit numbers. A 4-digit answer would have to be under 10,000, but no pair of 3-digit numbers can multiply to less than 10,000.

Q6.

Can multiplying a 4-digit number with a 2-digit number give a 5-digit number?

Answer

Yes.

Smallest case: 1000 × 10 = 10,000 → 5 digits ✓
Largest case: 9999 × 99 = 9,89,901 → 6 digits
Example: 1234 × 12 = 14,808, a 5-digit number

Why it happens: Here m = 4 and n = 2, so the product has m + n − 1 = 5 digits or m + n = 6 digits. Both are possible, and the 5-digit case happens whenever the two numbers are near the small end of their ranges.

Q7.

Observe the multiplication statements below. Do you notice any patterns? See if this pattern extends for other numbers as well. (1-digit × 1-digit = 1-digit or 2-digit; 2-digit × 1-digit = 2-digit or 3-digit; 2-digit × 2-digit = 3-digit or 4-digit; 3-digit × 3-digit = 5-digit or 6-digit; 5-digit × 5-digit = ___ or ___; 8-digit × 3-digit = ___ or ___; 12-digit × 13-digit = ___ or ___)

Answer

The blanks follow the same rule: the answer has (m + n − 1) or (m + n) digits.

m-digit	×	n-digit	=	m + n − 1	or	m + n
1-digit	×	1-digit	=	1-digit	or	2-digit
2-digit	×	1-digit	=	2-digit	or	3-digit
2-digit	×	2-digit	=	3-digit	or	4-digit
3-digit	×	3-digit	=	5-digit	or	6-digit
5-digit	×	5-digit	=	9-digit	or	10-digit
8-digit	×	3-digit	=	10-digit	or	11-digit
12-digit	×	13-digit	=	24-digit	or	25-digit

Check 5-digit × 5-digit:
10,000 × 10,000 = 10,00,00,000 → 9 digits
99,999 × 99,999 = 9,99,98,00,001 → 10 digits ✓

Why it happens: Notice the odd-looking jump: 2-digit × 2-digit gives 3 or 4 digits, but 3-digit × 3-digit gives 5 or 6 — never 4. That is because m + n − 1 = 5 for three-digit numbers, so a 4-digit answer is simply out of reach.

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