Add 2/3 + 3/4 + 5/6 and simplify

Question

Add $\dfrac{2}{3} + \dfrac{3}{4} + \dfrac{5}{6}$ and simplify.

Solution — Step by Step

To add fractions with different denominators, we need a common denominator. The most efficient choice is the Least Common Multiple (LCM) of 3, 4, and 6.

Factors:

$3 = 3$
$4 = 2^2$
$6 = 2 \times 3$

LCM = $2^2 \times 3 = 12$ .

The LCD is 12.

\frac{2}{3} = \frac{2 \times 4}{3 \times 4} = \frac{8}{12}

\frac{3}{4} = \frac{3 \times 3}{4 \times 3} = \frac{9}{12}

\frac{5}{6} = \frac{5 \times 2}{6 \times 2} = \frac{10}{12}

\frac{8}{12} + \frac{9}{12} + \frac{10}{12} = \frac{8 + 9 + 10}{12} = \frac{27}{12}

$\dfrac{27}{12}$ can be simplified. GCD of 27 and 12:

$27 = 3^3$ and $12 = 2^2 \times 3$ . GCD = 3.

\frac{27}{12} = \frac{27 \div 3}{12 \div 3} = \frac{9}{4}

As a mixed number: $\dfrac{9}{4} = 2\dfrac{1}{4}$ .

\boxed{\frac{2}{3} + \frac{3}{4} + \frac{5}{6} = \frac{9}{4} = 2\frac{1}{4}}

Why This Works

Adding fractions requires a common denominator because we can only add “same-sized pieces.” If you have 2 pieces of a pizza cut into 3 slices, and 3 pieces of a pizza cut into 4 slices, you can’t just add “2 + 3 = 5 pieces” — the pieces are different sizes.

By converting to a common denominator (LCM), we ensure all pieces are the same size. Then adding numerators makes sense.

Choosing the LCM (rather than just multiplying all denominators: $3 \times 4 \times 6 = 72$ ) keeps numbers smaller and simplification easier.

Alternative Method

Using the product of denominators as the common denominator:

$\frac{2}{3} = \frac{2 \times 4 \times 6}{3 \times 4 \times 6} = \frac{48}{72}$ , $\frac{3}{4} = \frac{3 \times 3 \times 6}{72} = \frac{54}{72}$ , $\frac{5}{6} = \frac{5 \times 3 \times 4}{72} = \frac{60}{72}$

Sum = $\frac{48 + 54 + 60}{72} = \frac{162}{72} = \frac{9}{4}$ (dividing by GCD 18).

Same answer, more arithmetic. The LCM method is cleaner.

Common Mistake

The classic mistake is adding numerators AND denominators separately: $\frac{2}{3} + \frac{3}{4} = \frac{5}{7}$ . This is completely wrong — denominators don’t add. The denominator tells you the size of each piece; adding piece-sizes makes no sense. Always find the LCM first, then add only the numerators (keeping the common denominator).

Whenever you add fractions with denominators that are multiples of each other (like 3, 6, 12), the LCM is just the largest one. Here, 3 and 6 are both factors of 12, so the LCM of 4 and 12 is 12. This shortcut saves time — check if the larger denominators are multiples of the smaller ones before computing the full LCM.

Question

Solution — Step by Step

Why This Works

Alternative Method

Common Mistake

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