Mixed Numbers

Wholes and a part

Here is one full bar, and next to it 3 quarters of another bar. The first bar is one whole, 1. The second bar is not full: it is cut into 4 equal pieces and 3 of them are colored, so it is $\frac{3}{4}$3/4.

Altogether that is 1 whole and $\frac{3}{4}$3/4 more: $1+\frac{3}{4}$1 + 3/4.

One full bar, and 3 of the 4 quarters of a second bar.

The mixed number

A whole number and a fraction written together is called a mixed number. This one is written $1\frac{3}{4}$1 3/4 and read "one and three quarters". It means $1+\frac{3}{4}$1 + 3/4.

The whole number counts the full bars. The fraction counts what there is of the bar that is not full. So $2\frac{3}{4}$2 3/4 is two full bars and 3 quarters of a third bar, and $1\frac{1}{2}$1 1/2 is one full bar and half of another.

The full bar is the 1 and the part is the $\frac{3}{4}$3/4: together, $1\frac{3}{4}$1 3/4.

From an improper fraction to a mixed number

The amount $\frac{7}{4}$7/4 is seven quarters. To write it as a mixed number, pack the quarters into wholes. Four quarters make one whole, so the first 4 quarters fill one whole, and 7 − 4 = 3 quarters are left over. $\frac{7}{4}=1\frac{3}{4}$7/4 = 1 3/4: the same amount, written a second way.

The mixed number shows the size at a glance. $1\frac{3}{4}$1 3/4 is plainly between 1 and 2, closer to 2.

Each square is a quarter, and a tray holds 4 of them. Pour in quarters until there are 7: one tray is full and 3 quarters are loose, so $\frac{7}{4}=1\frac{3}{4}$7/4 = 1 3/4.

More than one whole

Pack 11 quarters the same way. The first 4 fill one whole and the next 4 fill a second whole, which uses 8 quarters. 11 − 8 = 3 quarters are left, so $\frac{11}{4}=2\frac{3}{4}$11/4 = 2 3/4.

Two mistakes are common. One is to swap the two numbers, writing $3\frac{1}{4}$3 1/4 for $1\frac{3}{4}$1 3/4. The whole number counts full wholes, and 7 quarters fill only 1 of them. The other is to count too many wholes, writing $2\frac{3}{4}$2 3/4 for $\frac{7}{4}$7/4. Two wholes would take 8 quarters, and there are only 7.

Quarters 1 to 4 fill the first bar and 5 to 8 fill the second. Quarters 9, 10 and 11 are the $\frac{3}{4}$3/4 left over: $\frac{11}{4}=2\frac{3}{4}$11/4 = 2 3/4.