Multiplying Decimals

Multiplying decimals uses a simple strategy: ignore the decimal points, multiply as whole numbers, then place the decimal point by counting the total number of decimal places in both factors.

The Method

1. Multiply the numbers as if they were whole numbers (ignore decimal points)
2. Count the total number of decimal places in both original factors
3. Place the decimal point in the product that many places from the right

Example 1: Multiply $4.5 \times 0.12$

Step 1: Multiply without decimals: $45 \times 12 = 540$

Step 2: Count decimal places: $4.5$ has 1, $0.12$ has 2. Total = 3 decimal places.

Step 3: Place the decimal 3 places from the right: $540 \rightarrow 0.540 = 0.54$

$4.5 \times 0.12 = 0.54$

Example 2: Multiply $3.6 \times 7$

Step 1: $36 \times 7 = 252$

Step 2: $3.6$ has 1 decimal place, $7$ has 0. Total = 1.

Step 3: $252 \rightarrow 25.2$

$3.6 \times 7 = 25.2$

Example 3: Multiply $0.25 \times 0.4$

Step 1: $25 \times 4 = 100$

Step 2: $0.25$ has 2 places, $0.4$ has 1. Total = 3.

Step 3: $100 \rightarrow 0.100 = 0.1$

$0.25 \times 0.4 = 0.1$

Example 4: Multiply $12.5 \times 3.2$

Step 1: $125 \times 32 = 4{,}000$

Step 2: $12.5$ has 1 place, $3.2$ has 1. Total = 2.

Step 3: $4{,}000 \rightarrow 40.00 = 40$

$12.5 \times 3.2 = 40$

Why This Works

The decimal places track the powers of 10. $4.5$ is really $\frac{45}{10}$ and $0.12$ is $\frac{12}{100}$. Multiplying:

$\frac{45}{10} \times \frac{12}{100} = \frac{540}{1{,}000} = 0.540$

The denominator $1{,}000$ has 3 zeros — corresponding to the 3 total decimal places.

Multiplying by Powers of 10

Multiplying by 10, 100, or 1,000 simply moves the decimal point to the right:

$3.45 \times 10 = 34.5$ $3.45 \times 100 = 345$ $3.45 \times 1{,}000 = 3{,}450$

Dividing by powers of 10 (covered in Dividing Decimals) moves it left.

Estimation Check

Always estimate to verify your decimal placement is correct:

• $4.5 \times 0.12$: roughly $5 \times 0.1 = 0.5$. Our answer of $0.54$ is close.
• $12.5 \times 3.2$: roughly $13 \times 3 = 39$. Our answer of $40$ is close.

If your estimate and your answer are wildly different, recheck your decimal placement.

Practice Problems

Test your understanding with these problems. Click to reveal each answer.

Key Takeaways

• Multiply as whole numbers, then place the decimal by counting total decimal places in both factors
• The total decimal places in the factors = total decimal places in the product
• Multiplying by 10, 100, 1000 moves the decimal point to the right
• Estimate to check that your decimal point is in the right place
• When the product of the whole numbers ends in zeros, be careful not to lose them before placing the decimal

Return to Arithmetic for more foundational math topics.