Mixed Numbers and Repeating Decimals

Start with a decimal and write it as a mixed number. The plan is to keep the whole part and change only the decimal part.

• The whole part comes from the digits to the left of the decimal point. It stays, because it already counts wholes.
• The decimal part comes from the digits to the right. It becomes a fraction. The place of its last digit names the size of the parts. In $0.4$ the last digit is in the tenths place, so $0.4 = \frac{4}{10}$. In $0.25$ the last digit is in the hundredths place, so $0.25 = \frac{25}{100}$.
• Then write the fraction in simplest form. That means 1 is the only whole number that divides both the numerator and the denominator.

This place-value method works for a decimal that ends, because it has a last digit.

Notice that $0.4$ and $0.25$ count parts of different sizes: tenths and hundredths. Writing two numbers as decimals does not make their parts the same size. So read the place value before you count.

Now go the other way. Start with a mixed number and write it as a decimal. The whole part stays again, and only the fraction part changes.

The fraction bar means divide: the numerator divided by the denominator. So divide to turn the fraction part into a decimal. Then add the whole part back. The two parts count the same whole, so you can add them. Line up the place values when you add: ones with ones, tenths with tenths, and so on.

Some divisions never reach a remainder of 0. Try $\frac{1}{3}$, which means $1 \div 3$.

• 3 is greater than 1, so the ones digit is 0. Trade 1 one for 10 tenths: $10 \div 3 = 3$ R $1$. The tenths digit is 3.
• Trade the 1 tenth that is left for 10 hundredths: $10 \div 3 = 3$ R $1$. The hundredths digit is 3.

The remainder is 1 again. When you divide one whole number by another, every trade past the ones place works the same way. So the same remainder always leads to the same next step, and the digit 3 comes up forever: $\frac{1}{3} = 0.333\ldots$

A repeating decimal is a decimal that never ends because, from some place on, one digit or one group of digits repeats forever. The three dots mean that the pattern keeps going. You can also draw a bar over the digits that repeat: $0.\overline{3}$ means $0.333\ldots$ When a group of digits repeats, the bar covers the group: $0.\overline{09}$ means $0.0909\ldots$ Put the bar over only the digits that repeat.

Now go from a repeating decimal back to a fraction. The place-value method does not work here. A repeating decimal has no last digit, so there is no last place to name the size of the parts.

In this lesson, use one method only. Recognize a familiar repeating decimal whose fraction you have checked with long division, like the ones in the table. For example, $0.\overline{6} = \frac{2}{3}$, because dividing 2 by 3 gives $0.666\ldots$ If you are not sure about a match, do the long division to check it.

The same idea works for a repeating decimal greater than 1, such as $2.\overline{3}$. Keep the whole part, 2. Recognize the repeating part, $0.\overline{3}$, as $\frac{1}{3}$. Both parts count the same whole, so you can join them:

Takeaway: A mixed number, a fraction and a decimal can all name the same amount of the same whole. Keep the whole part, convert only the part smaller than one whole, and line up place values when you add decimals. When long division keeps bringing back the same remainder, the decimal repeats. You can turn a familiar repeating decimal back into a fraction, such as $0.\overline{3} = \frac{1}{3}$, because long division has checked the match.

Next, you will put a mix of fractions and decimals in order.