Convert Between Fractions and Terminating Decimals

A decimal uses place value to count parts of a whole. The first place to the right of the decimal point counts tenths. The second counts hundredths. The third counts thousandths.

Ten hundredths make 1 tenth. So the 75 hundredths on your strip are 7 tenths and 5 hundredths, and you write them as $0.75$. Now the shaded amount has three names: $\frac{3}{4}$, $\frac{75}{100}$ and $0.75$.

The place of the last digit tells you the size of the parts a decimal counts. $0.75$ ends in the hundredths place, so it counts hundredths. A decimal that ends in the tenths place counts tenths, and each tenth is 10 times as large as a hundredth of the same whole.

Every decimal in this lesson is between 0 and 1 and has digits that end. A decimal whose digits end is a terminating decimal.

Start by converting fractions to decimals. The first method is the one you used on the strip: rename the fraction in tenths, hundredths or thousandths.

Equivalent fractions name the same amount of the same whole. To make one, multiply the numerator and the denominator by the same whole number greater than 1. This cuts every part into that many smaller equal parts. You count more parts, but each part is smaller by the same factor, so the amount stays the same. On the strip, you multiplied by 25: $\frac{3}{4} = \frac{3 \times 25}{4 \times 25} = \frac{75}{100} = 0.75$.

This method works when the denominator is a factor of 10, 100 or 1000. Find the number that multiplies the denominator to make 10, 100 or 1000. Multiply the numerator by that same number. Then write the result with place value.

The second method is to divide. Use it when you cannot quickly turn the denominator into 10, 100 or 1000.

A fraction also means its numerator divided by its denominator. Here is why. Share 3 equal wholes among 8 groups. Cut each whole into eighths. That makes $3 \times 8 = 24$ eighths, so each group gets $24 \div 8 = 3$ eighths. This shows that $3 \div 8 = \frac{3}{8}$ of one whole.

To write the answer as a decimal, divide one place at a time. When a place leaves a remainder, an amount left over, rename it in the next smaller place: 1 one is 10 tenths, 1 tenth is 10 hundredths, and 1 hundredth is 10 thousandths. Renaming does not change the amount. It gives you parts of one size to share.

For every fraction in this lesson, the division reaches a remainder of 0, so the digits end. The result is a terminating decimal. Some fractions never reach a remainder of 0. You will work with those in the next unit.

Now convert the other way. A terminating decimal already names a fraction of the same whole, because each place names a part size.

Look at $0.25$. The 2 counts tenths and the 5 counts hundredths. Those parts are different sizes, so you cannot just add 2 and 5. Rename first: 2 tenths is 20 hundredths. Then $20 + 5 = 25$ hundredths, which is $\frac{25}{100}$.

You can write that fraction with larger parts. The number 25 is a common factor of 25 and 100, which means it divides both. When you divide both numbers by 25, you gather the hundredths into larger equal parts, and the amount stays the same: $\frac{25}{100} = \frac{25 \div 25}{100 \div 25} = \frac{1}{4}$.

The only common factor of 1 and 4 is 1, so $\frac{1}{4}$ is in simplest form.

Those moves give you three steps. Use them for a terminating decimal between 0 and 1.

1. Find the place of the last digit. It gives the denominator: tenths give 10, hundredths give 100, and thousandths give 1000.
2. Count the decimal in that part size. Read the digits after the decimal point as one whole number. That is the numerator.
3. Simplify. Divide the numerator and the denominator by a common factor until their only common factor is 1.

A 0 after the decimal point holds its place but adds no parts. $0.05$ has 0 tenths and 5 hundredths, so it is 5 hundredths, or $\frac{5}{100}$. The table that follows shows the three steps for five decimals.

Keep this idea: a fraction and a terminating decimal can name the same amount of the same whole. Converting changes how you write the amount, not how much there is.

• Fraction to decimal: rename the fraction in tenths, hundredths or thousandths when the denominator is a factor of 10, 100 or 1000, or divide the numerator by the denominator.
• Decimal to fraction: the place of the last digit gives the denominator. Count the parts for the numerator, then simplify.

When you count, compare or check amounts of the same whole, line up the place values so every part is the same size.

Next, you will convert between decimals and mixed numbers. You will also work with fractions whose decimals repeat instead of ending.