Converting Among Ratios, Fractions, and Decimals

Ratios, fractions, and decimals are three different ways of writing the exact same kind of information: a comparison between numbers, or a part-to-whole relationship. Once you know how to move between these three forms, you can choose whichever one is easiest to work with for a given problem, whether that's a recipe, a map scale, a test score, or a sale price.

A ratio like \(3:4\) compares two quantities. That same comparison can be written as the fraction \(\frac{3}{4}\), and that fraction can be written as the decimal \(0.75\). Nothing about the underlying relationship changes; only the notation does. This is why converting among the three forms is really just a set of rewriting rules, not new math.

To write a ratio as a fraction, put the first term over the second term (for a two-term ratio) and simplify if possible.

Example: Convert the ratio \(6:8\) to a fraction.

Write it as \(\frac{6}{8}\), then simplify by dividing top and bottom by \(2\): \(\frac{6}{8} = \frac{3}{4}\).

This is the same idea used throughout the earlier lesson on what a ratio is: a ratio compares two quantities, and a fraction is simply one accepted way to express that comparison.

A fraction \(\frac{a}{b}\) means \(a \div b\). To convert, divide the numerator by the denominator.

Example: Convert \(\frac{3}{4}\) to a decimal.

Divide: \(3 \div 4 = 0.75\).

Example: Convert \(\frac{5}{8}\) to a decimal.

Divide: \(5 \div 8 = 0.625\).

Some fractions produce a repeating decimal, such as \(\frac{1}{3} = 0.333...\). When that happens, you can either round to a sensible number of decimal places or leave the repeating pattern in place, depending on what the problem asks for.

To convert a decimal to a fraction, write the decimal digits over the matching power of ten, then simplify.

Example: Convert \(0.6\) to a fraction.

There is one digit after the decimal point, so write \(\frac{6}{10}\). Simplify by dividing by \(2\): \(\frac{6}{10} = \frac{3}{5}\).

Example: Convert \(0.75\) to a fraction.

Two digits after the decimal point means the denominator is \(100\): \(\frac{75}{100}\). Simplify by dividing by \(25\): \(\frac{75}{100} = \frac{3}{4}\).

Once a fraction is fully simplified, its numerator and denominator become the two terms of the ratio.

Example: Convert \(\frac{3}{5}\) to a ratio.

The numerator and denominator become the ratio terms: \(3:5\).

This last step is especially handy for word problems, such as those covered in applications of ratios, where a computed fraction needs to be reported back in ratio form, for example when describing a mixing or scaling instruction.

Start with the ratio \(5:8\).

• As a fraction: \(\frac{5}{8}\)
• As a decimal: \(5 \div 8 = 0.625\)
• Back to a fraction: \(0.625 = \frac{625}{1000} = \frac{5}{8}\)
• Back to a ratio: \(5:8\)

Since every step lands back on the original ratio, this confirms each conversion was done correctly. Using this kind of check is a reliable way to catch mistakes before moving on to a related idea, such as proportions, which compare two ratios or fractions to each other.

Watch out for these frequent errors:

• Flipping the order of a ratio's terms when writing the fraction, for example turning \(3:4\) into \(\frac{4}{3}\) instead of \(\frac{3}{4}\).
• Forgetting to simplify a fraction or ratio to lowest terms before comparing it with another value.
• Miscounting decimal places when converting a decimal to a fraction, which changes the denominator's power of ten.
• Rounding a repeating decimal too early, which can make later calculations slightly inaccurate.

Keep these three moves in mind whenever you need to switch forms:

• Ratio \(a:b\) becomes fraction \(\frac{a}{b}\).
• Fraction \(\frac{a}{b}\) becomes decimal by dividing \(a \div b\).
• Decimal becomes fraction by writing the digits over the matching power of ten, then simplifying.

Related lessons

• Basic Math Help