Decimal Operations Review

Decimals are fractions written in base-ten notation. You already use them every time you handle money, read a measuring tape, or check a thermometer. This page is a fluency review of the four basic operations with decimals, plus rounding and estimating. For the full foundational lesson, see Decimals in the Arithmetic section.

Adding and Subtracting Decimals

The key rule: line up the decimal points, then add or subtract as with whole numbers. Fill in trailing zeros so both numbers have the same number of decimal places.

Example 1: Add $14.7 + 3.256$

Line up the decimal points and pad with zeros:

$\begin{array}{r} 14.700 \\ +\;\; 3.256 \\ \hline 17.956 \end{array}$

Answer: $17.956$

Example 2: Subtract $20.5 - 8.73$

$\begin{array}{r} 20.50 \\ -\;\; 8.73 \\ \hline 11.77 \end{array}$

Answer: $11.77$

Estimation Check

Before computing, round each number to get a quick estimate. For Example 1: $15 + 3 = 18$, and the exact answer $17.956$ is close. This habit catches decimal-point placement errors, which are the most common mistake in decimal arithmetic.

Multiplying Decimals

Ignore the decimal points while multiplying, then place the decimal in the product.

Steps:

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

Example 3: Multiply $3.4 \times 2.15$

Step 1: Multiply $34 \times 215$:

$34 \times 215 = 34 \times 200 + 34 \times 15 = 6{,}800 + 510 = 7{,}310$

Step 2: Count decimal places: $3.4$ has 1, $2.15$ has 2. Total = 3 decimal places.

Step 3: Place the decimal: $7{,}310 \rightarrow 7.310 = 7.31$

Answer: $7.31$

Estimation check: $3 \times 2 = 6$, and $4 \times 2 = 8$. The answer $7.31$ falls right in that range.

Example 4: Multiply $0.06 \times 0.4$

$6 \times 4 = 24$. Total decimal places: $2 + 1 = 3$. Place the decimal: $0.024$.

Answer: $0.024$

This example shows why counting decimal places matters: the total number of decimal places in the product equals the sum of decimal places in the factors ($2 + 1 = 3$).

Why This Matters for Algebra

Multiplying decimals is the same as multiplying fractions with power-of-10 denominators. The expression $0.06 \times 0.4$ is equivalent to $\frac{6}{100} \times \frac{4}{10} = \frac{24}{1000}$. Seeing this connection helps when you work with scientific notation and polynomial coefficients later.

Dividing Decimals

The strategy: make the divisor a whole number by moving the decimal point, then divide normally.

Steps:

1. Move the decimal point in the divisor to the right until it becomes a whole number
2. Move the decimal point in the dividend the same number of places to the right
3. Divide as with whole numbers, placing the decimal point in the quotient directly above its new position in the dividend

Example 5: Divide $18.6 \div 0.3$

Step 1: Move the decimal in $0.3$ one place right to get $3$.

Step 2: Move the decimal in $18.6$ one place right to get $186$.

Step 3: Divide: $186 \div 3 = 62$.

Answer: $62$

Estimation check: $18.6 \div 0.3$ means “how many 0.3s fit in 18.6?” Since $0.3 \times 60 = 18$, the answer should be near 60. The answer 62 checks out.

Example 6: Divide $4.56 \div 0.12$

Move both decimals 2 places right: $456 \div 12 = 38$.

Answer: $38$

Example 7: Divide $7.5 \div 4$ (result with a decimal)

The divisor 4 is already a whole number. Divide:

$7.5 \div 4 = 1.875$

Check: $4 \times 1.875 = 7.5$. Correct.

Rounding Decimals

Rounding is essential for giving practical answers, especially in measurement and money contexts.

Steps:

1. Identify the place value you are rounding to
2. Look at the digit one place to the right
3. If that digit is 5 or greater, round up; otherwise, round down

Example 8: Round $3.2847$ to the nearest hundredth

The hundredths digit is 8. The digit to its right is 4 (less than 5), so round down.

Answer: $3.28$

Example 9: Round $12.695$ to the nearest tenth

The tenths digit is 6. The digit to its right is 9 (5 or greater), so round up.

Answer: $12.7$

Rounding and Money

Money is always rounded to the nearest hundredth (cent). If a calculation gives $14.8375, the practical answer is $14.84.

Estimating with Decimals

Estimation lets you catch major errors before they matter. The technique: round each number to one or two significant figures, then compute mentally.

If your exact answer is far from the estimate, double-check your decimal placement.

Real-World Application: Retail — Making Change

A customer buys items priced at $3.49, $12.75, and $0.89. They pay with a $20 bill. How much change do they receive?

Step 1 — Add the prices:

$\begin{array}{r} 3.49 \\ 12.75 \\ +\;\; 0.89 \\ \hline 17.13 \end{array}$

Step 2 — Subtract from $20.00:

$20.00 - 17.13 = 2.87$

The customer receives $2.87 in change.

Estimation check: roughly $3 + $13 + $1 = $17, so change is about $3. The answer $2.87 is reasonable.

Real-World Application: Nursing — IV Drip Rates

A nurse must administer 1.5 liters of saline over 6 hours. The flow rate in liters per hour is:

$1.5 \div 6 = 0.25 \text{ liters per hour}$

That is 250 mL per hour. If the administration set delivers 15 drops per mL, the drip rate is:

$0.25 \times 1000 \times 15 \div 60 = 62.5 \text{ drops per minute}$

A nurse would round to 63 drops per minute since partial drops cannot be delivered. Accurate decimal arithmetic is critical in clinical settings — a misplaced decimal point could mean a tenfold dosing error.

Common Mistakes to Avoid

1. Not lining up decimal points when adding or subtracting. This is the single most common decimal error. Always write the numbers vertically with the decimal points aligned.

2. Miscounting decimal places when multiplying. After multiplying the “whole number” product, count decimal places in both original factors — not just one of them.

3. Forgetting to move both decimal points when dividing. If you move the divisor’s decimal 2 places right, the dividend’s decimal must also move 2 places right.

4. Dropping trailing zeros too early. When subtracting $5.00 - 2.3$, write $5.00 - 2.30$ so both numbers have the same number of decimal places.

5. Not estimating first. A quick mental estimate catches errors like writing $731$ instead of $7.31$.

Practice Problems

Test your fluency. Click to reveal each answer.

Key Takeaways

• Line up decimal points for addition and subtraction — pad with trailing zeros as needed
• Count total decimal places in both factors when multiplying — the product gets that many decimal places
• Make the divisor a whole number when dividing — move both decimals the same number of places
• Always estimate first to catch decimal-placement errors
• Round to the appropriate place value for the context (hundredths for money, tenths for most measurements)
• For a deeper dive into any of these topics, see the full Decimals lesson in Arithmetic

Return to Pre-Algebra for more topics in this section.