Adding Fractions with the Same Denominator

The pieces already match

Add $\frac{3}{8}$3/8 and $\frac{2}{8}$2/8. Both fractions have the denominator 8, so both are counted in eighths, and every piece is the same size. Nothing has to be rewritten before they are added.

Both bars are cut into 8 equal pieces. One has 3 pieces colored and the other has 2.

Count the pieces

Put the colored pieces together and count them: 3 + 2 = 5. There are 5 pieces, and each one is still an eighth, so $\frac{3}{8}+\frac{2}{8}=\frac{5}{8}$3/8 + 2/8 = 5/8.

Add the numerators and keep the denominator. The numerators count the pieces, so they are added. The denominator says what size the pieces are, and putting pieces together does not change their size, so it stays 8. Lengths work the same way: 3 meters and 2 meters make 5 meters, and they are still meters.

The line from 0 to 1 is cut into eighths. Hop 3 eighths, then 2 more eighths, and the hops end at $\frac{5}{8}$5/8.

The denominator stays

The usual mistake is to add the denominators as well: $\frac{3}{8}+\frac{2}{8}=\frac{5}{16}$3/8 + 2/8 = 5/16. Sixteenths are half the size of eighths, so $\frac{5}{16}$5/16 is less than the $\frac{3}{8}$3/8 you started with, and adding $\frac{2}{8}$2/8 cannot make the amount smaller.

Two halves show it plainly. $\frac{1}{2}+\frac{1}{2}$1/2 + 1/2 fills the whole bar: $\frac{2}{2}$2/2, which is 1. Adding the denominators gives $\frac{2}{4}$2/4, which is only a half again.

The bars open on the answer $\frac{2}{4}$2/4, which is the same length as one half. Choose the other answer: two halves fill the bar, so $\frac{1}{2}+\frac{1}{2}=\frac{2}{2}=1$1/2 + 1/2 = 2/2 = 1.

More pieces, and a whole

The same count works for three fractions or more: $\frac{1}{10}+\frac{3}{10}+\frac{4}{10}=\frac{8}{10}$1/10 + 3/10 + 4/10 = 8/10, because 1 + 3 + 4 = 8.

When the numerators add up to the denominator, the pieces fill the whole: $\frac{3}{5}+\frac{2}{5}=\frac{5}{5}$3/5 + 2/5 = 5/5, which is 1. And a sum can often be simplified: $\frac{1}{8}+\frac{3}{8}=\frac{4}{8}$1/8 + 3/8 = 4/8, and $\frac{4}{8}$4/8 in simplest form is $\frac{1}{2}$1/2.

3 fifths and 2 fifths make all 5 fifths of the bar, which is one whole.