Converting between fractions and decimals

When you reach for it

Eunji is sewing a quilt out of 25 fabric squares, 5 rows of 5. Nine of the squares are blue, so \displaystyle \frac{9}{25} of the quilt is blue. Her friend Benicio is sewing a quilt with 100 smaller squares, and 36 of his are blue. Which quilt is bluer? It is hard to tell from \displaystyle \frac{9}{25} and \displaystyle \frac{36}{100} as they stand. It is easy once both are written the same way.

Ngozi has a different puzzle on a hiking trail. Her trail map says the first lookout is \displaystyle \frac{5}{8} of a mile from the start. Her watch counts distance in decimals: 0.1, 0.2, 0.3 miles. What number will the watch show when she reaches the lookout?

Fractions and decimals are two ways of writing the same numbers. A fraction is a point on the number line (see Fractions on the number line), and so is a decimal, whose digits after the point count tenths, hundredths and thousandths (see Decimal place value and Tenths and hundredths). You reach for this lesson whenever a number arrives in one form and you need the other: to compare two amounts, to put a number on a decimal scale or watch, to add a fraction to a decimal, or to write a measurement in its simplest form.

This lesson takes three days. Part 1 renames a fraction so its bottom number is 10, 100 or 1000, and then reads off the decimal. Part 2 divides the top number by the bottom number, which works for every fraction. Part 3 goes the other way, from a decimal back to a fraction in lowest terms, and lists the pairs worth knowing by heart.

The procedure

Part 1: denominators of ten, one hundred and one thousand

A fraction whose denominator (bottom number) is 10, 100 or 1000 is already a decimal in disguise. You just read it in place value:

The denominator names the last place: tenths, hundredths or thousandths. The numerator (top number) says how many of them. So \displaystyle \frac{59}{1000} is 59 thousandths, and 59 thousandths ends in the thousandths place, the third place after the point. Its 5 lands in the hundredths place and a 0 holds the tenths place open. The lesson on Thousandths: reading, writing and expanded form shows why each place is worth one tenth of the place to its left.

Most fractions do not come with such a handy denominator. Eunji’s \displaystyle \frac{9}{25} has 25 on the bottom. But 25 \times 4 = 100, so she can rename the fraction. Multiplying the top and the bottom by the same number gives an equal fraction (see Equivalent fractions):

The quilt picture shows why the value does not change. Cut each of Eunji’s 25 squares into 4 small squares. Now the quilt has 100 small squares, and the blue part, which has not moved, covers 9 \times 4 = 36 of them.

So the two quilts in the puzzle are exactly as blue as each other: \displaystyle \frac{9}{25} and \displaystyle \frac{36}{100} are both 0.36.

To rename, ask: what whole number times the denominator makes 10, 100 or 1000? Some denominators that work, and the number to multiply by:

A mixed number such as \displaystyle 3\frac{1}{2} is a whole number and a fraction added together (see Mixed numbers and improper fractions). The whole number goes in front of the point and the fraction becomes the digits after it: \displaystyle 3\frac{1}{2} = 3 + 0.5 = 3.5.

Part 2: dividing the top by the bottom

Renaming needs a denominator that fits into 10, 100 or 1000. Division works for every fraction.

The fraction bar means divide: \displaystyle \frac{5}{8} = 5 \div 8. Here is a short reason. The number 5 \div 8 is the number that gives 5 when you multiply it by 8. And 8 copies of \displaystyle \frac{5}{8} make \displaystyle \frac{40}{8}, which is 5. So \displaystyle \frac{5}{8} is that number. The lesson on A fraction as division tells the full sharing story behind this; here we only use it.

The trouble is that 8 does not go into 5 even once. This is where the decimal point helps. Writing zeros after the point does not change a number: 5 = 5.0 = 5.00 = 5.000, since the zeros add no tenths, no hundredths and no thousandths. But 5.000 can be read as 50 tenths, or 500 hundredths, or 5000 thousandths, and those are numbers 8 can divide. Then the long division you know (divide place by place, starting at the left, and bring down what is left over) runs exactly as it does for whole numbers.

Ngozi divides 5.000 by 8, one place at a time:

In each row, the digit of the answer, the amount used and the amount left over are all counted in the unit that row names: the second row reads “50 tenths divided by 8 is 6 tenths, using 48 tenths, with 2 tenths left over”. Each “have” is the amount left over, cut into ten times as many pieces of the next smaller size, which is what bringing down a 0 does. When the amount left over is 0, the division is finished:

The decimal point in the answer goes directly above the point in 5.000. The reason is in the table: tenths divided by 8 give tenths, hundredths give hundredths, so each digit of the answer sits in the same place as the digits it came from. Her watch will read 0.625 miles at the lookout.

You can check the answer by going back the other way: \displaystyle 0.625 = \frac{625}{1000}, and dividing the top and the bottom by 125 gives \displaystyle \frac{5}{8}. Another check is to multiply: 0.625 \times 8 should be 5.

If the denominator has two digits, the same steps work, with the estimating of each digit that Dividing by a two-digit number practices. If the fraction is more than 1, such as \displaystyle \frac{13}{4}, the division gives a whole number part first: 13 \div 4 is 3 with 1 left over, and the 1 continues as 10 tenths.

Part 3: from a decimal back to a fraction

Going from a decimal to a fraction is reading aloud. Say the decimal in place value words, and write down what you said. The last digit’s place is the denominator:

The answer is right, but often not finished. A fraction is in lowest terms when its top and bottom share no whole-number factor except 1. To get there, divide the top and the bottom by a number that goes into both, and repeat until no such number is left. Dividing both by the same number gives an equal fraction, the reverse of what Part 1 did. Since 10, 100 and 1000 are made by multiplying 2s and 5s, the numbers worth trying first are 2 and 5, and their products such as 4, 8, 25 and 125.

For \displaystyle 0.875 = \frac{875}{1000}, both numbers end in 5 or 0, so divide by 5, then by 5 again, then by 5 once more:

Now 7 and 8 share no factor but 1, so \displaystyle 0.875 = \frac{7}{8}. Dividing by 125 in one step gives the same answer faster, if you spot it.

A few pairs come up so often that they are worth knowing by heart:

They are easier to remember as a family than one by one. A quarter is half of a half, so \displaystyle \frac{1}{4} = 0.5 \div 2 = 0.25. An eighth is half of a quarter, so \displaystyle \frac{1}{8} = 0.25 \div 2 = 0.125, and every other eighth is a count of those: \displaystyle \frac{3}{8} is 3 times 0.125. A fifth is two tenths, so \displaystyle \frac{1}{5} = 0.2 and \displaystyle \frac{3}{5} = 0.6.

One look ahead. Every fraction so far in this lesson gives a decimal that stops. Not all of them do. Try 1 \div 3: 10 tenths divided by 3 is 3 tenths with 1 tenth left over, then 10 hundredths divided by 3 is 3 hundredths with 1 left over, and the same 1 comes back at every step. The digits 0.333\ldots never stop. Such decimals have their own notation, in Terminating and repeating decimals, and Which fractions give terminating decimals explains which fractions stop and which do not.

Worked examples

Examples 1 and 2 go with Part 1, Examples 3 and 4 with Part 2, and Examples 5 to 7 with Part 3.

Boundaries

• The denominator cannot be 0. A fraction such as \displaystyle \frac{5}{0} is not a number, because no number times 0 gives 5. Every fraction here has a denominator that is a counting number.
• Renaming to 10, 100 or 1000 does not always work. It works only when some whole number times the denominator makes 10, 100, 1000, or a larger number like them. That happens for 2, 4, 5, 8, 20, 25 and many more, but never for 3, 6, 7, 9 or 12 when the fraction is in lowest terms. Simplify first: \displaystyle \frac{3}{6} cannot be renamed to tenths as it stands, but \displaystyle \frac{3}{6} = \frac{1}{2} = 0.5.
• Division always works, but it does not always stop. For \displaystyle \frac{1}{3} the same remainder comes back forever. This lesson writes only decimals that stop; the others are in Terminating and repeating decimals, and the rule for telling them apart is in Which fractions give terminating decimals.
• Decimal to fraction, as done here, is for decimals that end. Reading off “how many tenths, hundredths or thousandths” needs a last digit. A decimal that goes on forever needs another method.
• Lowest terms changes how a fraction looks, not its value. Dividing the top and the bottom by the same number keeps the fraction equal, but dividing only one of them does not.

Where this goes next

Once a fraction can be written as a decimal, two numbers in different forms can be compared on one scale, which Comparing and ordering decimals does carefully. The next lessons in this unit multiply and divide with decimals, starting with Multiplying decimals: the eighths and fifths from Part 3 make good checks there, since 8 \times 0.125 must come out to 1.

Hundredths lead straight to percents. A percent is a number of hundredths, so \displaystyle \frac{9}{25} = \frac{36}{100} is also 36 percent; What percent means: a rate per hundred and Converting among fractions, decimals and percents build on exactly the renaming of Part 1. Later, decimals that never stop get their own notation in Terminating and repeating decimals, and Which fractions give terminating decimals shows that whether a \div b stops depends only on the denominator, once the fraction is in lowest terms.

Exercises

Exercises 1 to 8 go with Part 1, 9 to 16 with Part 2, and 17 to 22 with Part 3. The reasoning group draws on all three.

Part 1: denominators of ten, one hundred and one thousand

1. Write each as a decimal: (a) \displaystyle \frac{7}{10}, (b) \displaystyle \frac{63}{100}, (c) \displaystyle \frac{209}{1000}.
2. Write each as a decimal: (a) \displaystyle \frac{8}{100}, (b) \displaystyle \frac{8}{1000}, (c) \displaystyle \frac{80}{1000}.
3. Rename each fraction with denominator 100, then write it as a decimal: (a) \displaystyle \frac{19}{20}, (b) \displaystyle \frac{17}{50}, (c) \displaystyle \frac{21}{25}.
4. Rename each fraction with denominator 1000, then write it as a decimal: (a) \displaystyle \frac{9}{125}, (b) \displaystyle \frac{37}{200}, (c) \displaystyle \frac{123}{500}.
5. Ngozi’s quilt has 50 squares, and 23 of them are green. What decimal part of the quilt is green?
6. Write each mixed number as a decimal: (a) \displaystyle 6\frac{1}{20}, (b) \displaystyle 2\frac{3}{25}.
7. Benicio cuts \displaystyle 1\frac{29}{40} yards of fabric. What length does the cutting machine show?
8. Write each as a decimal: (a) \displaystyle \frac{47}{10}, (b) \displaystyle \frac{1309}{100}.

Part 2: dividing the top by the bottom

9. Write \displaystyle \frac{3}{8} as a decimal by long division. Show what is left over at each step.
10. Write \displaystyle \frac{9}{20} as a decimal by dividing 9 by 20.
11. Write \displaystyle \frac{27}{4} as a decimal.
12. Write \displaystyle \frac{11}{8} as a decimal.
13. Eunji splits a 15 mile trail into 8 legs of equal length. How long is each leg, as a decimal?
14. Write \displaystyle \frac{33}{40} as a decimal by dividing.
15. Write \displaystyle \frac{7}{25} as a decimal in two ways, once by renaming and once by dividing, and show that the answers agree.
16. Write \displaystyle \frac{63}{12} as a decimal. (Look for a way to make the work shorter.)

Part 3: from a decimal back to a fraction

17. Write each as a fraction in lowest terms: (a) 0.6, (b) 0.85, (c) 0.004.
18. Write each as a fraction in lowest terms: (a) 0.72, (b) 0.016, (c) 0.15.
19. Write each as a mixed number in lowest terms: (a) 4.6, (b) 12.025.
20. A trail sign gives the distance to a waterfall as 2.35 miles. Write the distance as a mixed number in lowest terms.
21. From memory, without working it out: (a) write 0.4 as a fraction, (b) write \displaystyle \frac{3}{4} as a decimal, (c) write 0.125 as a fraction, (d) write \displaystyle \frac{1}{4} as a decimal.
22. Three quilters have each finished part of a quilt: \displaystyle \frac{3}{5}, 0.57 and \displaystyle \frac{5}{8}. Put the three amounts in order from least to greatest.

Reasoning

23. Explain why 0.5, 0.50, \displaystyle \frac{5}{10} and \displaystyle \frac{1}{2} all name the same point on the number line.
24. Benicio says that \displaystyle \frac{3}{12} cannot be written as a decimal by renaming, because no whole number times 12 makes 100 or 1000. Is he right that it cannot be renamed as it stands? Find its decimal anyway, and explain how.
25. Ngozi divides 1 by 6. Her first step leaves 4 tenths, her second step leaves 4 hundredths, and she notices a pattern. Explain why the division will never finish.