Multiplying decimal by decimal

The general rule for multiplying two decimals:

1. Multiply ignoring the decimal points.
2. The answer has as many decimal places as the sum of decimal places in the two inputs.

Idea

So for $0.7 \times 0.3$: ignore points, $7 \times 3 = 21$. Each input has $1$ decimal place; total $1 + 1 = 2$. So the answer has $2$ decimal places: $0.21$.

Check: $0.7 \times 0.3 = \tfrac{7}{10} \times \tfrac{3}{10} = \tfrac{21}{100} = 0.21$. ✓

Why does this work? Each decimal place in the input is a factor of $\tfrac{1}{10}$. So $1$ decimal place in each input contributes a factor of $\tfrac{1}{100}$ in the result , meaning the result has $2$ decimal places.

Estimation first. Before computing, round each input to the nearest whole number and multiply mentally. Use the estimate to check your decimal point placement.

Example: $4.5 \times 2.1$. Estimate: $5 \times 2 = 10$. Now multiply: $45 \times 21 = 945$. Decimal places: $1 + 1 = 2$. So answer is $9.45$. Estimate was $10$ , close to $9.45$. ✓

Multiplying multiple decimals. Apply the rule pairwise. Total decimal places in answer = sum across all inputs.

Worked examples

Example 1. Compute $0.6 \times 0.4$.

$6 \times 4 = 24$. Total decimal places: $1 + 1 = 2$. Answer: $0.24$.

Example 2. Compute $1.5 \times 2.4$.

$15 \times 24 = 360$. Total decimal places: $1 + 1 = 2$. Answer: $3.60 = 3.6$. Estimate check: $1.5 \times 2.5 \approx 3.75$. ✓

Example 3. Compute $0.25 \times 0.04$.

$25 \times 4 = 100$. Total decimal places: $2 + 2 = 4$. Answer: $0.0100 = 0.01$. Verify: $\tfrac{1}{4} \times \tfrac{1}{25} = \tfrac{1}{100} = 0.01$. ✓

Example 4. Compute $1.23 \times 4$.

$123 \times 4 = 492$. Decimal places: $2 + 0 = 2$. Answer: $4.92$.

Try it yourself

1. Compute $0.2 \times 0.5$.
2. Compute $1.4 \times 1.5$.
3. Compute $2.7 \times 0.3$.
4. Compute $0.08 \times 0.05$.
5. Compute $4.2 \times 1.25$.
6. Compute $5.6 \times 0.1$.
7. Compute $0.001 \times 1000$.
8. Compute $3.14 \times 2.0$.

Activity

Use a calculator to verify the "sum of decimal places" rule for $10$ random pairs of decimals. You will see it holds every time.