Lesson 1.7: Decimals

New Concept

Decimals are fractions with denominators that are powers of 10, like $\frac{1}{10}$ or $\frac{1}{100}$. We'll master naming, rounding, all arithmetic operations, and converting between decimals, fractions, and percents.

What’s next

Next, you'll see these concepts in action. We'll start with interactive examples on decimal arithmetic, followed by challenge problems and practice cards.

Property

Decimals are another way of writing fractions whose denominators are powers of 10.

The “th” at the end of the name tells you that the number is smaller than one. To name a decimal, name the number to the left of the decimal, write “and” for the decimal point, name the number to the right, and state the place value of the last digit. To write a decimal from words, use “and” to place the decimal point, then write the number parts accordingly.

Examples

• To name the decimal 8.9, you say "eight and nine tenths".
• To name the decimal -12.345, you say "negative twelve and three hundred forty-five thousandths".
• To write "thirty-one and seven hundredths" as a decimal, you write 31.07.

Explanation

Decimals are just special fractions where the bottom number is 10, 100, or another power of ten. The word “and” separates the whole part from the fractional part, telling you exactly where the decimal point goes.

Property

Rounding decimals is very much like rounding whole numbers. To round a decimal:

1. Locate the given place value.
2. Look at the digit to the right of that place value.
3. If this digit is 5 or greater, add 1 to the digit in the rounding place value. If it is less than 5, do not change the digit.
4. Rewrite the number, removing all digits to the right of the rounding digit.

Examples

• To round 14.582 to the nearest tenth, look at the 8. Since it is 5 or greater, the 5 rounds up to 6. The answer is 14.6.
• To round 7.149 to the nearest hundredth, look at the 9. Since it is 5 or greater, the 4 rounds up to 5. The answer is 7.15.
• To round 56.38 to the nearest whole number, look at the 3. Since it is less than 5, the 6 stays the same. The answer is 56.

Explanation

Rounding makes numbers simpler and easier to use. Just look at the digit to the right of your target place. If it's 5 or more, round up; if it's 4 or less, let it rest. It's a quick way to estimate!

Property

To add or subtract decimals, we line up the decimal points. By lining up the decimal points this way, we can add or subtract the corresponding place values.

1. Write the numbers so the decimal points line up vertically.
2. Use zeros as place holders, as needed.
3. Add or subtract the numbers as if they were whole numbers. Then place the decimal point in the answer under the decimal points in the given numbers.

Examples

• To add $18.6 + 4.52$, write the numbers vertically, aligning the decimal points: $18.60 + 4.52 = 23.12$.
• To subtract $40 - 15.85$, place a decimal and zeros after 40: $40.00 - 15.85 = 24.15$.
• To calculate $-8.3 + 4.1$, subtract the smaller absolute value from the larger one ($8.3 - 4.1 = 4.2$) and keep the sign of the number with the larger absolute value. The answer is $-4.2$.

Explanation

Adding or subtracting decimals is all about alignment. By lining up the decimal points, you make sure you are combining tenths with tenths and hundredths with hundredths. It keeps your place values straight for an accurate answer.

Property

To multiply decimals, determine the sign, then multiply the numbers as if they are whole numbers. The number of decimal places in the product is the sum of the decimal places in the factors. To multiply by a power of 10, move the decimal point to the right the same number of places as there are zeros in the power of 10.

To divide decimals, first make the divisor a whole number by moving the decimal point to the right. Move the decimal point in the dividend the same number of places. Then, divide as usual and place the decimal point in the quotient directly above the decimal point in the dividend.

Examples

• To multiply $(0.5)(1.1)$, first multiply $5 \times 11 = 55$. Since there are two total decimal places in the factors, the product is $0.55$.
• To multiply $4.78$ by $1000$, move the decimal point 3 places to the right to get $4780$.
• To divide $8.4 \div 0.2$, move the decimal one place to the right in both numbers to get $84 \div 2$, which equals $42$.

Property

To convert a decimal to a fraction, write the digits to the right of the decimal point as the numerator. The denominator is the place value of the last digit.
To convert a fraction to a decimal, divide the numerator by the denominator. A decimal that repeats has a bar over the repeating digits.
A percent is a ratio whose denominator is 100. To convert a percent to a decimal, move the decimal point two places to the left. To convert a decimal to a percent, move the decimal point two places to the right and add a percent sign.

Examples

• To convert $0.65$ to a fraction, write it as $\frac{65}{100}$, which simplifies to $\frac{13}{20}$.
• To convert the fraction $\frac{3}{4}$ to a decimal, divide 3 by 4 to get $0.75$.
• To convert $47\%$ to a decimal, move the decimal point two places to the left to get $0.47$. To convert $0.9$ to a percent, move the decimal point two places to the right to get $90\%$.

Explanation

Decimals, fractions, and percents are different outfits for the same number. The fraction bar means divide, which turns a fraction into a decimal. Percent means 'per hundred', so moving the decimal two places is the quick trick to switch forms.