Terminating and Repeating Decimals, and Back to Fractions | Grade 7 Math | LevelKite

In the last lesson you used long division to change fractions into decimals. Some divisions stopped, like $\frac{3}{4} = 0.75$ . Others never finished, like $\frac{1}{3}$ , which gave $0.333\ldots$ with no end. In this lesson you will name these two kinds of decimals, see why a fraction always gives one kind or the other, and change a decimal that stops back into a fraction.

Every number in this lesson is a rational number: a number you can write as a fraction $\frac{p}{q}$ , where $p$ and $q$ are integers (the whole numbers and their opposites: $\ldots, -2, -1, 0, 1, 2, \ldots$ ) and $q$ is not $0$ . The denominator $q$ cannot be $0$ because you cannot divide by $0$ .

You can write the opposite of $\frac{p}{q}$ in three ways, and all three name the same number: $-\frac{p}{q} = \frac{-p}{q} = \frac{p}{-q}$ . So a negative fraction may carry its sign in front, in the numerator or in the denominator. For example, $-\frac{3}{4} = \frac{-3}{4} = \frac{3}{-4} = -0.75$ .

A terminating decimal is a decimal that ends. It has a last digit, like the $5$ in $0.75$ .

A repeating decimal never ends. One digit, or a block of digits that are not all zeros, repeats forever. You mark the repeating part with a bar over it:

$0.\overline{3}$ means $0.3333\ldots$ , with the $3$ repeating forever.
$0.8\overline{3}$ means $0.8333\ldots$ , where only the $3$ repeats. The bar covers the $3$ alone because the $8$ appears just once.
$1.\overline{27}$ means $1.272727\ldots$ , with the block $27$ repeating forever.

The bar is part of the number. Without it, $0.3$ is only three tenths. Extra zeros at the end do not make a repeat: $0.750$ is the same number as $0.75$ , and it terminates.

A negative sign does not change the type: $-0.75$ terminates just as $0.75$ does, and $-0.\overline{3}$ repeats just as $0.\overline{3}$ does.

Read the type from the way the decimal is written

Decimal	What its digits do	Type
$0.6$	Ends at the tenths place	Terminating
$0.\overline{6}$	The $6$ repeats forever	Repeating
$0.666$	Ends at the thousandths place; it has no bar	Terminating
$-2.75$	Ends at the hundredths place	Terminating
$0.1\overline{6}$	After the $1$ , the $6$ repeats forever	Repeating
$-3.\overline{45}$	The block $45$ repeats forever	Repeating

Why does a fraction always give one of these two types? The remainders, the amounts left over at each step of the long division, tell you.

Take any rational number $\frac{p}{q}$ , with $p$ and $q$ integers and $q$ not $0$ . Divide without any negative signs, then attach a negative sign if the number is negative. The sign does not change the digits, because $-\frac{p}{q} = \frac{-p}{q} = \frac{p}{-q}$ .

A remainder is always a whole number less than the divisor, the number you divide by. When you divide by $6$ , the only possible remainders are $0$ , $1$ , $2$ , $3$ , $4$ and $5$ . Watch the remainders from the decimal point on, where every step brings down a $0$ . Exactly one of two things happens:

A remainder is $0$ . Nothing is left to divide, so the division stops. The decimal terminates.
No remainder is ever $0$ . The division never stops, but it has only a limited supply of nonzero remainders (five when the divisor is $6$ ). So a remainder must come up a second time. From there the same remainder gives the same next digit and the same next remainder, so the digits loop. The decimal repeats.

Worked example

Terminating or repeating?

Write $\frac{3}{8}$ and $-\frac{5}{6}$ as decimals. Tell whether each decimal terminates or repeats.

Divide 3 by 8. Each step brings down a 0, gives one digit and leaves a remainder. The divisor 8 goes into 30 three times with 6 left over, into 60 seven times with 4 left over, and into 40 five times with 0 left over.
$30 = 8 \times 3 + 6 \qquad 60 = 8 \times 7 + 4 \qquad 40 = 8 \times 5 + 0$
The last remainder is 0, so nothing is left to divide and the division stops. The digits 3, 7 and 5 are the whole decimal. A decimal that ends is a terminating decimal.
$\frac{3}{8} = 0.375$
Now look at the sign of the second number. Wherever the negative sign sits, the number is the same, so its digits are the digits of 5 divided by 6. Divide without the sign and attach it at the end.
$-\frac{5}{6} = \frac{-5}{6} = \frac{5}{-6}$
Divide 5 by 6. The divisor 6 goes into 50 eight times with 2 left over. Then it goes into 20 three times with 2 left over.
$50 = 6 \times 8 + 2 \qquad 20 = 6 \times 3 + 2$
The remainder 2 has come up a second time. The next step is 20 divided by 6 again, so it gives the digit 3 and the remainder 2 again, and this never ends. The 8 appears once and the 3 repeats, so the bar goes over the 3 alone. Attach the negative sign. A decimal with a digit that repeats forever is a repeating decimal.
$-\frac{5}{6} = -0.8333\ldots = -0.8\overline{3}$

Answer

The decimal $\frac{3}{8} = 0.375$ terminates, and the decimal $-\frac{5}{6} = -0.8\overline{3}$ repeats.

Key concept

A rational number's decimal terminates or repeats

A rational number is a number you can write as $\frac{p}{q}$ , where $p$ and $q$ are integers and $q$ is not $0$ . A negative sign can sit in front, in the numerator or in the denominator without changing the value: $-\frac{p}{q} = \frac{-p}{q} = \frac{p}{-q}$ .

When you divide $p$ by $q$ , the decimal does one of two things. It terminates if the division ends with a remainder of $0$ . It repeats if the division never ends, because a nonzero remainder comes back and a block of digits loops forever.

Mark a repeat with a bar. Never cut it off: the shortened decimal is a different number.

A terminating decimal changes back into a fraction through place value. You will do that next.

Now go the other way: start with a terminating decimal and write it as a fraction. Place value, the value a digit has because of where it sits, makes this possible. Each place after the decimal point is named for a denominator: the first place is tenths, the second is hundredths and the third is thousandths.

So a terminating decimal is already a fraction in disguise. Take $0.25$ . The $2$ is in the tenths place and the $5$ is in the hundredths place. Rename $2$ tenths as $20$ hundredths by multiplying the numerator and the denominator by $10$ , which keeps the value the same. Then both digits fit over one denominator:

0.25 = \frac{2}{10} + \frac{5}{100} = \frac{20}{100} + \frac{5}{100} = \frac{25}{100}

The digits $25$ ended up over $100$ because the last digit is in the hundredths place. The same thing happens with every terminating decimal, so you can follow these steps:

Find the place of the last digit. Count every decimal place, zeros included. In $0.07$ the $7$ is in the second place, the hundredths place.
Write the digits after the decimal point over the matching denominator: $10$ for one decimal place, $100$ for two, $1000$ for three, and so on. So $0.07 = \frac{7}{100}$ .
Simplify. Divide the numerator and the denominator by a common factor, a whole number greater than $1$ that divides both. Dividing both by the same nonzero number keeps the value the same. Stop when no whole number greater than $1$ divides both. The fraction is then in simplest form.

The result is an integer over an integer that is not $0$ , which is the form $\frac{p}{q}$ of a rational number.

These steps need a last digit, so they work only for terminating decimals. A repeating decimal has no last place to read.

Worked example

A decimal between 0 and 1

Write $0.45$ as a fraction in simplest form.

Find the place of the last digit. There are two digits after the decimal point, so the last digit, 5, is in the hundredths place. Write the digits 45 over 100.
$0.45 = \frac{45}{100}$
Simplify. Both 45 and 100 end in 5 or 0, so 5 divides both. Divide the numerator and the denominator by 5. Dividing both by the same nonzero number keeps the value the same.
$\frac{45}{100} = \frac{45 \div 5}{100 \div 5} = \frac{9}{20}$
Make sure the fraction is in simplest form. The only whole numbers greater than 1 that divide 9 are 3 and 9, and neither one divides 20. So no whole number greater than 1 divides both 9 and 20.
Check by dividing 9 by 20 with long division. The divisor 20 goes into 90 four times with 10 left over, and into 100 five times with 0 left over. The digits are 4 and 5, so you get back the decimal you started with.
$90 = 20 \times 4 + 10 \qquad 100 = 20 \times 5 + 0$

Answer

Written as a fraction in simplest form, $0.45$ is $\frac{9}{20}$ .

Two more cases use the same steps.

A whole-number part. Keep the whole number and convert only the decimal part. The result is a mixed number: a whole number and a fraction written side by side, meaning their sum. For example, $1.5 = 1 + \frac{5}{10} = 1\frac{5}{10} = 1\frac{1}{2}$ .

A negative decimal. A negative decimal is the opposite of a positive decimal. Convert the positive decimal, then put the negative sign on the result. The sign belongs to the whole answer: $-1.5 = -\left(1 + \frac{1}{2}\right) = -1\frac{1}{2}$ .

Worked example

A negative decimal with a whole-number part

Write $-2.375$ as a mixed number in simplest form. Then write it as a single fraction.

Set the sign aside. Negative 2.375 is the opposite of 2.375, so convert 2.375 first and put the negative sign on the result at the end.
$-2.375 = -(2.375)$
Keep the whole number 2 and convert only the decimal part. There are three decimal places, so the last digit is in the thousandths place. Write the digits 375 over 1000.
$2.375 = 2 + 0.375 = 2 + \frac{375}{1000}$
Simplify the fraction part. Both numbers end in 5 or 0, so 5 divides both. Divide the numerator and the denominator by 5, three times in a row. Each division keeps the value the same. No whole number greater than 1 divides both 3 and 8, so the fraction is in simplest form.
$\frac{375}{1000} = \frac{75}{200} = \frac{15}{40} = \frac{3}{8}$
Join the whole number and the fraction into a mixed number. Then put the negative sign back. The sign belongs to the whole mixed number.
$-2.375 = -\left(2 + \frac{3}{8}\right) = -2\frac{3}{8}$
Write the same number as a single fraction. Two wholes are 16 eighths, and 16 eighths plus 3 eighths is 19 eighths. The negative sign can sit in front, in the numerator or in the denominator.
$-2\frac{3}{8} = -\frac{19}{8} = \frac{-19}{8} = \frac{19}{-8}$
Check by dividing 19 by 8. The divisor 8 goes into 19 two times with 3 left over. From there the division is 3 divided by 8, which gave the digits 3, 7 and 5 and ended with a remainder of 0 in the first example. So 19 eighths is 2.375, and its opposite is the decimal you started with.
$\frac{19}{8} = 2 + \frac{3}{8} = 2 + 0.375 = 2.375$

Answer

As a mixed number in simplest form, $-2.375 = -2\frac{3}{8}$ . As a single fraction, $-2.375 = -\frac{19}{8}$ .

Here is the idea to keep. Every rational number $\frac{p}{q}$ , with $p$ and $q$ integers and $q$ not $0$ , has a decimal that either terminates or repeats. A terminating decimal has a last digit. A repeating decimal loops forever, so you mark it with a bar and never cut it off.

A terminating decimal changes back into a fraction through place value: put its decimal digits over $10$ , $100$ , $1000$ and so on to match the place of the last digit, then simplify. Keep any whole-number part and any negative sign.

Next, you will compare and order rational numbers and find the distance between two rational numbers on a number line.