Mixed numbers and improper fractions

2 3/4 cups of flour is 11 quarter cups, which is why 2 3/4 and 11/4 are the same amount.

Two ways to write the same amount

A mixed number is a whole number and a proper fraction written side by side. 2 3/4 means 2 + 3/4: two whole cups and three quarters of another. Mixed numbers are how people usually talk about measurements, because the whole part tells you the size at a glance. The same amount written as a single fraction, 11/4, is an improper fraction. It is less readable but much easier to calculate with.

Mixed number to improper fraction

Each whole contains as many pieces as the denominator. In 2 3/4, each whole is 4 quarters, so two wholes are 2 × 4 = 8 quarters. Add the 3 quarters you already have and you get 11 quarters: 11/4. The rule is whole number times denominator, plus numerator, over the same denominator. 3 2/5 = (3 × 5 + 2)/5 = 17/5. 5 1/6 = (5 × 6 + 1)/6 = 31/6.

Improper fraction to mixed number

Going the other way is a division with remainder. For 17/3, ask how many whole groups of 3 fit into 17. 17 ÷ 3 = 5 remainder 2, so there are 5 wholes and 2 thirds left over: 17/3 = 5 2/3. For 47/8, 47 ÷ 8 = 5 remainder 7, so 47/8 = 5 7/8. If the remainder is 0, the fraction is a whole number: 24/6 = 4.

Simplify before or after converting, whichever is easier. 14/4 can be written 3 2/4 and then 3 1/2, or simplified first to 7/2 and then converted: 7 ÷ 2 = 3 remainder 1, so 3 1/2 again.

Adding mixed numbers

You can add the whole parts and the fraction parts separately. 1 1/2 + 2 3/4: wholes 1 + 2 = 3, fractions 1/2 + 3/4 = 2/4 + 3/4 = 5/4. The fraction part is more than 1, so carry: 5/4 = 1 1/4, and the total is 3 + 1 1/4 = 4 1/4. The other route is to convert everything to improper fractions: 3/2 + 11/4 = 6/4 + 11/4 = 17/4 = 4 1/4. Both give the same answer.

Subtracting mixed numbers and borrowing

Subtraction sometimes needs a borrow, just like subtracting whole numbers. In 5 1/4 − 2 3/4, you cannot take 3/4 from 1/4. Borrow one whole from the 5 and turn it into 4 quarters: 5 1/4 becomes 4 5/4. Now subtract: 4 5/4 − 2 3/4 = 2 2/4 = 2 1/2. Converting avoids borrowing altogether: 21/4 − 11/4 = 10/4 = 5/2 = 2 1/2.

Subtracting from a whole number works the same way. 3 − 1 2/3: borrow 1 from the 3 to get 2 3/3, then 2 3/3 − 1 2/3 = 1 1/3. As improper fractions, 9/3 − 5/3 = 4/3.

Comparing mixed numbers and reading time

To compare two mixed numbers, look at the whole parts first. 4 1/8 is bigger than 3 9/10, because 4 is bigger than 3 and a fraction part can never make up a whole. Only when the whole parts match do you compare the fraction parts, as in unit 5. For 3 2/5 and 3 3/8, the wholes are equal, and 2/5 = 16/40 while 3/8 = 15/40, so 3 2/5 is the larger.

Hours are a common place to meet mixed numbers. An hour has 60 minutes, so 3/4 of an hour is 60 ÷ 4 × 3 = 45 minutes, and 1 3/4 hours is 1 hour 45 minutes. In the other direction, 2 1/3 hours is 2 × 60 + 20 = 140 minutes.

Which form to use

Use improper fractions while you calculate, especially for multiplying and dividing in the next two units, where mixed numbers must always be converted first. Use mixed numbers when you report a measurement to a person: 2 1/2 hours reads more easily than 5/2 hours. Both forms are exact, and neither is more correct than the other.