Equivalent Fractions

A fraction is a division of a whole into equal parts: the bottom number (denominator) says how many parts the whole is cut into, and the top (numerator) says how many you take. So $\frac{3}{4}$ means cut into 4, take 3.

Here's the key idea: if you cut every part into the same number of smaller pieces, you have more pieces but the same amount. Cut each of the $4$ parts in two and you get $8$ parts, and the $3$ you took become $6$ — so $\frac{3}{4} = \frac{6}{8}$. Nothing about the amount changed; only how finely you sliced it. That's why equivalents matter everywhere in this course: to compare or add fractions you first re-slice them to a common size.

Multiply top and bottom by the same number. To rewrite $\frac{2}{3}$ with a denominator of $12$, ask what times $3$ gives $12$ — that's $4$ — then do the same to the top:

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

The two bars below hold the same shaded amount; the bottom one is just sliced into more pieces. The dashed line shows the shaded edges landing in exactly the same place — that's what "equal" looks like.

It works going the other way too. $\frac{1}{2}$ and $\frac{3}{6}$ are the same amount — six small slices, three of them taken:

The whole trick is "same factor, top and bottom." Find the factor first, then apply it. Predict before you check.