Converting Fractions, Decimals, and Percents

Start with decimals. A terminating decimal is a decimal whose digits end, such as $0.45$ or $2.375$. Each digit to the right of the decimal point has a place value: it counts tenths, hundredths, thousandths, and so on. These are fractions whose denominator (bottom number) is a power of ten: $10$, $100$, $1000$, and so on.

So a terminating decimal already tells you a fraction. In $0.45$, the 4 counts tenths and the 5 counts hundredths. Rewrite the 4 tenths as 40 hundredths, then add:

$0.45 = \frac{4}{10} + \frac{5}{100} = \frac{40}{100} + \frac{5}{100} = \frac{45}{100}$

This gives a method for any terminating decimal less than 1: write its digits after the decimal point over the power of ten for the last digit's place.

A fraction from place value is often not in simplest form, so simplify it. Divide the numerator (top number) and the denominator by their greatest common factor, the largest whole number that divides both. For $\frac{45}{100}$ that factor is 5:

$\frac{45}{100} = \frac{45 \div 5}{100 \div 5} = \frac{9}{20}$

Dividing both by the same nonzero number gives an equivalent fraction, so the value stays the same.

If the decimal is greater than 1, the digits before the decimal point count whole ones. Keep them as the whole part and use the method on the digits after the point. The result is a mixed number: a whole number plus a fraction less than 1, such as $2\frac{3}{8}$.

Now go the other way. A fraction bar means division: $\frac{a}{b} = a \div b$ for numbers $a$ and $b$ with $b \neq 0$. The denominator is never 0, because division by zero is undefined.

So to write a fraction as a decimal, divide the numerator by the denominator. For a mixed number, keep the whole part and divide only for the fraction part.

Sometimes you can skip the long division. If an equivalent fraction has a power of ten as its denominator, place value finishes the job:

$\frac{9}{20} = \frac{9 \times 5}{20 \times 5} = \frac{45}{100} = 0.45$

When $a$ and $b$ are integers (whole numbers and their opposites) and $b \neq 0$, the number $\frac{a}{b}$ is a rational number. When you divide to find its decimal form, one of two things happens:

• A remainder (the amount left over) of 0 appears, so the decimal terminates.
• A remainder you have seen before comes back, so the same digits repeat forever. This is a repeating decimal. Mark the repeating digits with a bar: $0.8333\ldots = 0.8\overline{3}$.

One of these must happen. Dividing by a positive whole number $b$ can leave only $b$ different remainders, the whole numbers from 0 to $b - 1$. So if 0 never appears, a remainder has to come back.

A repeating decimal has no last digit, so the place-value method does not work on it. If you round it, write $\approx$, which means "is approximately equal to", because the rounded value is close but not exact.

A decimal that neither terminates nor repeats is irrational: no fraction of integers equals it, so this lesson does not convert such decimals.

Now add the third form. You saw that a percent counts parts out of 100. So the sign $\%$ means "divided by 100": $p\% = \frac{p}{100}$ for any number $p$. Since 100 is not zero, this division always works.

To go from a percent to a decimal, carry out that division: $45\% = \frac{45}{100} = 45 \div 100 = 0.45$.

To go from a decimal to a percent, undo it by multiplying by 100. That counts how many hundredths the number holds: $0.08 = \frac{0.08 \times 100}{100} = \frac{8}{100} = 8\%$. Multiplying and then dividing by 100 leaves the value unchanged.

A percent can be more than $100\%$ when the amount is more than one whole: $1.5 = \frac{150}{100} = 150\%$.

A percent always describes part of a stated whole. If a tank is $\frac{3}{4}$ full, it is also $0.75$ full and $75\%$ full, because all three describe the same tank. But $75\%$ of a larger tank is more water, so check that the forms you compare refer to the same whole.

Takeaway. A fraction, a decimal and a percent are three forms of one number, and converting never changes its value.

• Place value turns a terminating decimal into a fraction or mixed number.
• Dividing by the nonzero denominator turns a fraction into a decimal. For a fraction of integers, that decimal terminates or repeats. Mark a repeat with a bar and a rounded value with $\approx$.
• A percent is a number of hundredths. Divide by 100 to leave percent form, and multiply by 100 to enter it.

You can now move among all three forms. Next, you will use these conversions to compare and order fractions, decimals and percents.