Patterns in Products

Class 7 Maths · Chapter 1: Large Numbers Around Us · NCERT Ganita Prakash section 1.5–1.6

Learn

A shortcut: multiplying by 5 is the same as dividing by 2 and multiplying by 10, because 5 = 10 ÷ 2. So 116 × 5 = (116 ÷ 2) × 10 = 58 × 10 = 580. In the same way 25 = 100 ÷ 4 and 125 = 1000 ÷ 8: 824 × 25 = (824 ÷ 4) × 100 = 20,600.

824 × 25 = (824 ÷ 4) × 100 = 20,600

Regroup to make it easy: 2 × 1768 × 50 = (2 × 50) × 1768 = 100 × 1768 = 1,76,800. And 125 × 40 × 8 × 25 = (125 × 8) × (40 × 25) = 1000 × 1000 = 10,00,000. Look for pairs that make 10, 100 or 1000.

125 × 8 = 1000 · 40 × 25 = 1000

How long is the product? Two 2-digit numbers: the smallest product is 10 × 10 = 100 (3 digits), and every product is less than 100 × 100 = 10,000 (5 digits). So the product has 3 or 4 digits. In general, an a-digit × b-digit product has (a + b − 1) or (a + b) digits: 3-digit × 3-digit gives 5 or 6 digits.

a-digit × b-digit → (a + b − 1) or (a + b) digits

Did you ever wonder…? Could everyone in a city of 1 crore 24 lakh fit into 1 lakh buses? If a bus holds 50, then 1 lakh × 50 = 50 lakh people, not enough. Work big questions in stages: travel 100 km a day → 36,500 km in a year → 3,65,000 km in 10 years, a little short of the 3,84,400 km to the Moon!

100 × 365 × 10 = 3,65,000 km

Explore: how many digits in the product?

1. Match each multiplication to the possible length of its product.

Remember

• × 5 = ÷ 2 then × 10; × 25 = ÷ 4 then × 100; × 125 = ÷ 8 then × 1000.
• An a-digit × b-digit product has (a + b − 1) or (a + b) digits; solve big questions in stages.

Quick check

Find quickly: 72 × 125

1. 9,000
2. 7,200
3. 90,000
4. 8,750

Think about it

Can you count 10 lakh coins in one day if you count 1 coin every second without stopping?

Practice

Practise with fresh questions at three levels (easy, medium, hard), some answered hands-on, with hints and worked answers. Get 4 of your last 5 right to finish the topic.