Adding Fractions (Same Denominator)
Cut a cake into eight equal slices: you take three and a friend takes two, so five slices
are gone. Adding fractions answers exactly this — and when the pieces are all the same size,
it is as easy as counting.
A fraction is really two facts in one. The bottom number (the
denominator) tells you how many equal pieces the whole was cut into — so it
tells you the size of each piece. The top number (the
numerator) tells you how many of those pieces you have. So
\frac{3}{8} means "three pieces, each one an eighth".
Now here is the lovely part. When two fractions have the same denominator,
the pieces are all the same size. Adding them is then no harder than counting:
three same-size pieces plus two more same-size pieces is five same-size pieces. You
add the numerators and keep the denominator.
\frac{3}{8} + \frac{2}{8} = \frac{5}{8}
Three eighths plus two eighths is five eighths. The bottom number is the size of
the slice, and that size never changes when you put slices together — so it stays exactly
where it is. Only the count on top grows.
Cut a pizza into eight equal slices. You eat
\frac{3}{8} — three slices — and your friend eats
\frac{2}{8} — two slices. Together you have eaten
\frac{3}{8} + \frac{2}{8} = \frac{5}{8} of the pizza: five
slices gone. Notice nobody ever says "five sixteenths" — the slices did not get
smaller just because you counted more of them. They are still eighths.
The rule
In general, for any two fractions that share a denominator
n:
\frac{a}{n} + \frac{b}{n} = \frac{a + b}{n}
Read it out loud: "a nths plus b nths is (a + b) nths." The word
"nths" — the size of the piece — is said three times and never changes. That is the whole
idea in one line.
Subtraction works exactly the same way. Same-size pieces, so take the numerators apart and
keep the denominator:
\frac{5}{8} - \frac{2}{8} = \frac{3}{8}
Only the top changes; the bottom is the size of the slice and rides along unchanged.
The most common mistake is to add the bottoms too. Do not:
- \frac{1}{5} + \frac{2}{5} = \frac{3}{5}, not
\frac{3}{10}. The pieces stay fifths — you only add
the tops.
- Adding the denominators would actually make the answer smaller than what you
started with, which makes no sense: putting more pieces together can never give you
less. Tenths are tinier than fifths!
Rule of thumb: add only the tops; the bottom is the slice size and never moves.
See it built
One bar cut into equal pieces. First we shade some pieces in one colour, then a few more in
another. The shaded pieces simply add up — the slices never change size, so the bottom number
never moves. Step through it, and press Refresh for a fresh pair to add.
Three worked examples
Cover the answer, work it in your head, then check. Add the tops; keep the bottom.
- \frac{1}{6} + \frac{4}{6} = \frac{1 + 4}{6} = \frac{5}{6}
Five sixths. The 6 never moved.
- \frac{2}{7} + \frac{3}{7} = \frac{2 + 3}{7} = \frac{5}{7}
Two sevenths and three sevenths make five sevenths.
- \frac{2}{5} + \frac{3}{5} = \frac{2 + 3}{5} = \frac{5}{5} = 1
When the top reaches the bottom, all the pieces are shaded — that is one whole.
A snack bar is scored into six equal squares. Sam snaps off
\frac{2}{6} and Mia snaps off \frac{3}{6}.
How much of the bar is gone? Count the squares:
\frac{2}{6} + \frac{3}{6} = \frac{5}{6} — five of the six
squares. One square is left. The squares were never re-sized, so they are all still sixths.
See it explained
Sal Khan adds fractions that already share a denominator, just by adding the tops.
What if the bottoms are different?
This easy trick only works when the denominators match — the pieces have to be the
same size before you can count them together. Picture trying to add
\frac{1}{2} of a watermelon to \frac{1}{3}
of a watermelon:
A half-slice and a third-slice are different sizes, so you cannot just say "one piece plus
one piece is two pieces" — two pieces of what? First you re-cut both into matching
pieces (sixths work: \frac{1}{2} = \frac{3}{6} and
\frac{1}{3} = \frac{2}{6}), and only then add:
\frac{3}{6} + \frac{2}{6} = \frac{5}{6}. Making the bottoms match
first uses
equivalent fractions
— that is the next step.