Decimals and fractions (age 11+)
Work interchangeably with terminating decimals and their corresponding fractions (such as 3.5 and 7/2 or 0.375 and 3/8); convert fluently between the two forms
What a learner can do afterwards
- Convert any terminating decimal to a fraction in simplest form
- Convert any fraction with a denominator whose prime factors are only 2 and 5 to a terminating decimal
- Explain why some fractions produce terminating decimals and others do not
1 · Read
You already know decimals and fractions can name the same number. Now you'll convert between them for any terminating decimal, simplify the fraction all the way, and understand why some fractions stop while others repeat forever.
To convert 0.375 to a fraction, write it over the matching power of ten: 0.375 = 375/1000. Both numbers share a factor of 125, so divide top and bottom by 125 to get 3/8.
To go the other way, look at the denominator. 8 = 2 x 2 x 2, so it only has 2s as prime factors. That means you can multiply top and bottom to reach a power of ten: 3/8 = 375/1000 = 0.375.
A fraction only becomes a terminating decimal when its simplest-form denominator is built from 2s and 5s alone. A denominator like 6 or 7 always leaves something over, so the decimal repeats no matter how far you divide.
If a fraction's simplest denominator has only 2s and 5s as prime factors, it terminates; convert by matching a power of ten, then simplify all the way.
More to read
1.5: Convert Between Decimals and Fractions
Decimals and fractions are two ways to write the same number. To turn a decimal into a fraction, look at the last digit after the point. If it sits in the tenths place, put the number over 10. If it is in the hundredths place, put it over 100. Then simplify. For example, 0.6 is six tenths, so it is 6/10, which simplifies to 3/5. Going the other way, turn the fraction into one with a denominator of 10, 100, or 1000. For 3/4, multiply the top and bottom by 25 to get 75/100, which is 0.75. Some fractions convert to decimals that end, like 7/8 = 0.875. Others go on forever, like 1/3. You will learn why soon, but first practice both directions until switching feels easy.
5 Ways to Convert Percents, Fractions, and Decimals
Changing between fractions and decimals is a skill you will use all the time. To change a fraction to a decimal, divide the top number by the bottom number. So 3/5 becomes 3 ÷ 5 = 0.6. A faster trick works when the denominator is 10, 100, or 1000: just read the digits after the point. 7/10 is 0.7 and 7/100 is 0.07. To change a decimal to a fraction, name the place of the last digit. 0.04 ends in the hundredths place, so it is 4/100, which simplifies to 1/25. After you convert, always check that the fraction is in simplest form, and try dividing it back to see if you get the decimal you started with.
Equivalent forms of fractions, decimals, and percents
The same amount can look different depending on how you write it. Half of a pizza can be 1/2, or 0.5, or 50 percent. All three mean exactly the same thing. Being able to swap between these forms is called finding equivalent representations. Practice by picking a fraction like 3/4 and writing it as a decimal: multiply by 25 to get 75/100, which is 0.75. Then pick a decimal like 0.2 and write it as a fraction: two tenths is 2/10, which simplifies to 1/5. The more you swap back and forth, the faster you get.
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2 · Watch
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Where it sits
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Where this leads
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14 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.