Converting Between Fractions and Decimals

A decimal is already a fraction

You saw in the place-value lesson that the digits after the decimal point are tenths, hundredths, and thousandths. That single fact is the whole method for turning a decimal into a fraction.

Take $0.49$. The $4$ is four tenths and the $9$ is nine hundredths, so written out it is $\tfrac{4}{10} + \tfrac{9}{100}$. Put both pieces over a hundred and add:

The two-digit string "$49$" lands directly on top of $100$, which is the shortcut. A decimal that ends always works this way, because its last place names a denominator every digit can be written over. The rule is short: the number of digits after the point is the number of zeros in the denominator.

A fraction read straight off the digits may not yet be in lowest terms, so the conversion has a second half. A fraction is in lowest terms when its top and bottom share no factor larger than $1$. Reduce by dividing the top and bottom by their greatest common factor, the same simplifying you did in the fractions chapter.

A number with a whole part converts the same way: leave the whole part alone and convert only the decimal piece. The result is a mixed number you can read off directly. For instance $2.6$ is $2\tfrac{6}{10} = 2\tfrac{3}{5}$.

A fraction is a division waiting to happen

Going the other direction, from a fraction to a decimal, rests on a fact you have known since the fractions chapter. The fraction bar means divide.

Take $\tfrac{3}{8}$. The bar says divide $3$ by $8$. Since $8$ does not go into $3$, write $0$ in the ones place, bring the point up, and continue dividing into $3.000$:

Eight goes into $30$ tenths three times ($3 \times 8 = 24$) with a remainder of $6$ tenths, which is $60$ hundredths. Eight goes into $60$ seven times ($56$) with a remainder of $4$ hundredths, which is $40$ thousandths. Eight goes into $40$ exactly five times with nothing left over, so the division ends and the decimal is $0.375$.

Every fraction goes the same way. The fraction $\tfrac{a}{b}$ is exactly $a \div b$, the size of each share when $a$ wholes are divided equally among $b$ groups. So to write a fraction as a decimal, carry out that division, and the decimal long division from the previous lesson is the tool.

Divide the numerator by the denominator, appending zeros after the point in the numerator whenever the remainder has not cleared. Keep the decimal point in the quotient lined up above the point in the dividend. The remainder does not always clear, and the last section of this lesson says which fractions it clears for.

A shortcut when the denominator already fits

Long division always works, but you can often skip it. The fraction $\tfrac{3}{4}$ is the classic case. A hundred is $4 \times 25$, so multiply the top and bottom by $25$:

That step uses equivalent fractions: multiplying the top and bottom by the same number does not change the value. So $\tfrac{75}{100}$ is still three quarters, now written with a bottom you can read as hundredths. Once the bottom is a hundred, the top is simply the hundredths, and the decimal is read off directly. This is the same conversion as the long division $3 \div 4 = 0.75$, just reached by scaling instead of dividing.

The general rule follows. If the denominator divides evenly into ten, a hundred, or a thousand, then scaling the fraction to that denominator turns it into a decimal. You choose the multiplier that makes the denominator into ten, a hundred, or a thousand. You can then read that decimal straight off, with no division of the numerator at all.

Choosing the multiplier is the whole skill here. Take $\tfrac{3}{5}$: multiplying top and bottom by $2$ gives $\tfrac{6}{10}$, which reads straight off as $0.6$. Multiplying by $3$ or by $4$ instead gives $\tfrac{9}{15}$ and $\tfrac{12}{20}$, equally true names for the same amount. But $15$ and $20$ are not ten, a hundred, or a thousand, so neither can be read off as a decimal without more scaling.

The picture below shows why scaling does not change the amount: cutting each of the five fifths into two makes ten tenths. The shaded portion is unchanged, so $\tfrac{2}{5}$ and $\tfrac{4}{10} = 0.4$ are the same number.

Terminating or repeating: it depends on the denominator

Some fractions, like $\tfrac{3}{8} = 0.375$, give a decimal that ends. Others do not. Try $\tfrac{1}{3}$ with long division: $1 \div 3$ gives $0.333\ldots$, where the remainder is $1$ at every step, so the $3$s never stop. We write a repeating decimal with a bar over the block of digits that repeats forever:

You can predict which happens before doing any division by looking at the prime factors of the denominator. The dividing line is clean: a fraction in lowest terms terminates exactly when its denominator has no prime factor other than $2$ and $5$. If any other prime, such as $3$ or $7$, survives in the denominator, the decimal repeats.

To apply the test, reduce the fraction to lowest terms first, then prime factorize the denominator. Reducing first matters: $\tfrac{6}{12}$ looks like it has the bad factor $3$. But it reduces to $\tfrac{1}{2}$, whose denominator is just $2$, so it terminates ($0.5$).

Common mistakes