Lesson 8 of 10 · Grades 3 to 5
Mixed numbers and improper fractions
Some fractions are worth more than one whole. You can write them as an improper fraction, where the numerator is at least as big as the denominator, or as a mixed number, which pairs a whole number with a proper fraction. 5/4 and 1 1/4 name the same amount.
When a fraction is worth more than one
Most of the fractions you have met are worth less than one whole. A proper fraction has a numerator smaller than its denominator, like 3/4 or 5/8. But fractions can also be worth one whole or more. When the numerator is equal to or bigger than the denominator, the fraction is called an improper fraction. 4/4, 5/3 and 7/2 are improper fractions.
There is a second way to write an amount bigger than one. A mixed number has a whole number and a proper fraction together: 1 3/4, or 2 1/2. Both names describe the same amount. The mixed number 1 3/4 means one whole plus three quarters, and the improper fraction 7/4 means seven quarters. If you put seven quarters together, four of them make a whole and three are left over, so the two names agree.
Improper fraction to mixed number
Turn 7/4 into a mixed number. Ask how many whole groups of 4 quarters fit into 7 quarters. Divide 7 by 4: the answer is 1 with remainder 3. The quotient, 1, becomes the whole number. The remainder, 3, becomes the numerator of the fraction part, and the denominator stays 4. So 7/4 = 1 3/4. The remainder must always be smaller than the denominator, so the fraction part is a proper fraction.
Mixed number to improper fraction
Turn 2 1/2 into an improper fraction. First work out how many halves are in the 2 wholes. Each whole is 2 halves, so 2 wholes are 2 x 2 = 4 halves. Then add the extra half from the fraction part: 4 + 1 = 5 halves. So 2 1/2 = 5/2. In short: multiply the whole number by the denominator, add the numerator, and put the result over the same denominator.
Worked example
Convert 3 2/5 to an improper fraction. Multiply the whole number by the denominator: 3 x 5 = 15. Add the numerator: 15 + 2 = 17. Put 17 over the denominator 5: the answer is 17/5. To check, convert back: 17 divided by 5 is 3 remainder 2, so 17/5 = 3 2/5. The round trip returns to where you started.
Convert 13/6 to a mixed number. Divide 13 by 6: 2 remainder 1. The whole number is 2, and the fraction part is 1/6, so 13/6 = 2 1/6.
Watch out for this
2 1/2 does not mean 2 x 1/2. When a whole number sits next to a fraction, the hidden operation is addition: 2 1/2 means 2 + 1/2. Multiplying would give 1, which is much smaller than 2 1/2. Read mixed numbers with plus in mind. Also keep the fraction part proper; 1 5/4 is an unusual way to write 2 1/4, and it is better to carry the extra whole across.
Remember
- A proper fraction is less than 1; an improper fraction is 1 or more.
- A mixed number pairs a whole number with a proper fraction.
- Improper to mixed: divide, then use the remainder over the denominator.
- Mixed to improper: whole x denominator + numerator, over the denominator.
Try it
Answer the five questions below. Work them out first, then open each answer to check.
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Write 3/2 as a mixed number.
Show answer
1 1/2. Two halves make one whole, and one half is left over.
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Write 2 1/4 as an improper fraction.
Show answer
9/4. 2 wholes are 8 quarters, and 8 + 1 = 9 quarters.
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Write 7/3 as a mixed number.
Show answer
2 1/3. 7 divided by 3 is 2 remainder 1.
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Write 3 1/2 as an improper fraction.
Show answer
7/2. 3 wholes are 6 halves, and 6 + 1 = 7 halves.
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How many quarters are in 2 3/4?
Show answer
11 quarters. 2 wholes are 8 quarters, and 8 + 3 = 11.
Puzzle · Pizza boxes
Puzzle time
Each pizza is cut into 4 equal slices, and each pizza comes in its own box. A class eats 11 slices. They finish each pizza before they open the next box. How many boxes did they open?
Show a hint
Write 11/4 as a mixed number. How many whole pizzas is that? Is there part of one more pizza?
Show the answer
Answer: 3. They opened 3 boxes. 11/4 = 2 3/4. That is 2 whole pizzas and 3/4 of a third pizza. The class had to open the third box, even though they finished only 2 pizzas.