5.2 Decimal Operations

Decimal Number Operations

Decimal operations are how you handle adding, subtracting, multiplying, and dividing numbers that have decimal points. These come up constantly in money, measurement, and science problems. The biggest key to getting them right: understanding place value and knowing where to put that decimal point in your answer.

Addition and Subtraction of Decimals

The most common mistake with decimal addition and subtraction is misaligning the numbers. Here's the process:

1. Line up the decimal points vertically. Stack the numbers so every decimal point sits in the same column.
2. Add placeholder zeros so each number has the same number of decimal places. For example, if you're adding 0.5 + 1.23, rewrite 0.5 as 0.50.
3. Add or subtract as if they were whole numbers. Just work through the columns like normal.
4. Drop the decimal point straight down into your answer, directly below where it sits in the problem.

Multiplication with Decimal Numbers

Multiplication works differently from addition and subtraction. You don't need to align decimal points. Instead:

1. Ignore the decimals and multiply as whole numbers. Treat 2.5 × 1.2 as 25 × 12 = 300.
2. Count the total decimal places in both factors. 2.5 has one decimal place, and 1.2 has one decimal place, so that's two total.
3. Place the decimal point in your product so it has that many digits to its right. 300 becomes 3.00 (two decimal places).

So $2.5 \times 1.2 = 3.00$, which simplifies to 3.

Division Involving Decimals

Dividing by a decimal can feel tricky, but there's a clean method to handle it:

1. Make the divisor a whole number. Multiply both the divisor and the dividend by the same power of 10. Choose the power of 10 that eliminates all decimal places in the divisor. For example, $1.2 \div 0.25$ becomes $120 \div 25$ (both multiplied by 100).
2. Divide as usual. Place the decimal point in the quotient directly above where it sits in the dividend.
3. Add zeros to the dividend if needed. If you run out of digits before the division is complete, tack on zeros after the last decimal digit and keep going.

Decimals in Money Calculations

Money always uses two decimal places (dollars and cents), so keep these rules in mind:

• Write all money amounts with two decimal places. Five dollars is $5.00$, not just 5.
• Use the same decimal operation rules from above for any money calculation.
• Round your final answer to the nearest cent (the hundredths place) when multiplication or division gives you more than two decimal places.

To round to the nearest cent, look at the third decimal place:

• If it's 5 or greater, round up. $1.255$ becomes $1.26$.
• If it's less than 5, round down. $1.254$ becomes $1.25$.

Checking Decimal Operation Results

Estimating Results for Reasonableness

Before you trust your answer, do a quick estimate to see if it's in the right ballpark. This catches errors like a misplaced decimal point.

1. Round each number to make the math easy.

   • For addition and subtraction, round to the same place value. For instance, $0.25 + 1.8$ becomes roughly $0.3 + 1.8 = 2.1$.
   • For multiplication and division, rounding to the nearest whole number works well. $2.5 \times 1.2$ becomes roughly $3 \times 1 = 3$.

2. Do the operation with the rounded numbers to get your estimate.

3. Compare your estimate to your actual answer. If they're far apart, recheck your work. A misplaced decimal point will usually make your answer 10 or 100 times too big or too small, which the estimate will catch.

For expressions with multiple operations, follow PEMDAS (Parentheses, Exponents, Multiplication/Division, Addition/Subtraction) even when estimating.

Decimal Representation and Notation

Understanding Decimal Numbers

Decimal numbers are part of the base-10 system, where each place value is 10 times larger than the one to its right. To the left of the decimal point, you have ones, tens, hundreds. To the right, you have tenths, hundredths, thousandths.

Every decimal can also be written as a fraction. For example, 0.75 is the same as $\frac{75}{100}$, which simplifies to $\frac{3}{4}$.

Notation Methods

• Standard notation is the everyday way you write decimals: 0.00345.
• Scientific notation is useful for very large or very small numbers. It expresses a number as a value between 1 and 10 multiplied by a power of 10.

For example, 0.00345 in scientific notation is $3.45 \times 10^{-3}$. You moved the decimal point 3 places to the right to get 3.45, so the exponent is -3. For large numbers, the exponent is positive: 45,000 becomes $4.5 \times 10^{4}$.