Multiplying decimals as fractions

The idea

Multiplying decimals is the same as multiplying their corresponding fractions with denominators 10, 100, 1000, …: multiply the numerators, multiply the denominators, and write the result as a decimal.

Five pens

Arshad buys 5 pens at ₹9.5 each (9 rupees and 50 paisa). The cost is 9.5 × 5, which is 9.5 added 5 times.

How much should he pay the shopkeeper?

• ₹45.5
• ₹47.5
• ₹475

Take the fraction route

Every decimal is a fraction with 10, 100, 1000 … at the bottom. Turn both numbers into fractions, multiply the fractions, then write the answer as a decimal.

The same two moves every time

A decimal is a fraction whose bottom number is 10, 100, 1000 … To multiply fractions we multiply the tops and multiply the bottoms. Then we write the fraction as a decimal.

Multiplying decimals is the same as multiplying their fractions (with denominators 10, 100, 1000, …): multiply the numerators, multiply the denominators, and write the result as a decimal.

• Product | Through fractions
• 9.5 × 5 | 95/10 × 5/1 = 475/10 = 47.5
• 12.5 × 7.5 | 125/10 × 75/10 = 9375/100 = 93.75
• 0.827 × 2 | 827/1000 × 2/1 = 1654/1000 = 1.654

Notes

To multiply decimals, multiply their fractions: tops times tops, bottoms times bottoms, then write the result as a decimal.

Check yourself

Find 2.4 × 3. Use 24/10 × 3/1.

Answer: 7.2

24/10 × 3/1 = 72/10 = 7.2.

Write 0.6 × 0.7 as fractions and multiply. Which is right?

A notebook costs ₹23.6. What do 50 notebooks cost?

Answer: 1180 rupees

236/10 × 50/1 = 11800/10 = 1180. A decimal times a counting number can come out as a counting number.

To find 0.3 × 0.8, which is the correct first step?

Find 1.5 × 0.4 through fractions. Type the answer as a decimal.

Answer: 0.6

15/10 × 4/10 = 60/100 = 0.60 = 0.6.

• 6/10 × 7/10 = 42/10 = 4.2. The bottoms must be multiplied too: 10 × 10 = 100, not 10.
• 6/10 × 7/10 = 42/100 = 0.42 — correct. Yes. The tops multiply (6 × 7 = 42) and the bottoms multiply (10 × 10 = 100).
• 6/10 + 7/10 = 13/10 = 1.3. That adds the fractions. A product means multiply.
• Write 3/1 × 8/1. The point matters: 0.3 is 3 tenths, not 3 ones.
• Write 3/10 × 8/100. 0.8 has one digit after the point, so its bottom is 10, not 100.
• Write 3/10 × 8/10 — correct. 0.3 is 3 tenths and 0.8 is 8 tenths, so both have 10 at the bottom.