Equivalent Fractions and Simplifying

When two fractions name the same amount

Two fractions are equivalent when they stand for the same number: the same point on the number line, the same amount of pizza. The clearest way to see it is with the bars you met in the last lesson. Draw one whole cut into $2$ equal parts with $1$ shaded, and below it the same whole cut into $4$ equal parts with $2$ shaded. The shaded region is identical in both: exactly half the bar is coloured.

To go from the top bar to the bottom one, we cut each of the $2$ parts into $2$ smaller parts. That doubles the number of parts in the whole, so the denominator goes from $2$ to $4$. But that same cutting also doubles the number of shaded parts, because each shaded part got cut in two as well. So the numerator goes from $1$ to $2$. The shaded amount never moved; we only drew more cut lines. That is why $\frac{1}{2}$ and $\frac{2}{4}$ are the same number:

You can keep going. Cut each part into three instead, and the same half of the bar is now $\frac{3}{6}$; into five, and it is $\frac{5}{10}$. Every one of these is just a finer description of the same shaded region:

You can do that cutting yourself. In the figure below, the top bar is fixed at one half. The bottom bar is the same whole, and you choose how many pieces each of its parts is cut into.

Watch the edge of the shaded region as you change the cut. It does not move. The numerator and the denominator both climb, so the fraction gets new names, but the amount of bar coloured in stays exactly where it started. That is what makes all these fractions equal.

Why multiplying top and bottom by the same number is allowed

The picture suggests a rule: to rewrite a fraction, multiply the numerator and the denominator by the same number.

Try that on $\frac{2}{5}$ with the multiplier $3$. Multiplying top and bottom by $3$ turns it into $\frac{6}{15}$, and the bars say the same thing. Cut each of the $5$ parts into $3$ pieces and the whole holds $15$ pieces. Each of the $2$ shaded parts becomes $3$ pieces too, so $6$ pieces are shaded. The extra cuts moved no shading, so $\frac{2}{5}$ and $\frac{6}{15}$ are the same number.

Now a different fraction and a different multiplier. Take $\frac{4}{7}$, a whole cut into $7$ equal parts with $4$ of them shaded, and cut every part in two. The whole was in $7$ parts and each became $2$ pieces, so it now holds $7 \times 2 = 14$ pieces. The $4$ shaded parts each became $2$ pieces too, so $4 \times 2 = 8$ pieces are shaded. You drew new lines but added and removed no area, so $\frac{4}{7} = \frac{8}{14}$.

Both times the multiplier did the same two jobs. It cut every part of the whole, and it cut every shaded part by the identical amount. That is why the top and the bottom both have to be multiplied, and any starting fraction with any nonzero multiplier behaves the same way.

Building rule. For any whole numbers $a$ and $b$ with $b$ not zero, and any nonzero whole number $c$, $\frac{a}{b} = \frac{a \times c}{b \times c}.$

This is also why the multiplier has to hit both numbers. Change only one of them and the picture no longer matches the fraction you started with. The requirement that $c$ is not zero matters too: you cannot cut each part into $0$ pieces, and a denominator of $0$ is undefined.

Building an equivalent fraction

The most common job is to rewrite a fraction so it has a particular denominator you need. Rewriting to a required denominator is exactly what you will do when adding fractions in the next lesson.

To rewrite $\frac{2}{3}$ with a denominator of $12$, ask ”$3$ times what gives $12$?” Since $3 \times 4 = 12$, the multiplier is $4$. Now multiply the numerator by that same $4$:

The rule guarantees these name the same number, because we multiplied top and bottom by the same $4$. That gives the method: find what the old denominator was multiplied by, then multiply the numerator by that same number. The fixed idea is that whatever you do to the bottom, you must do to the top. Cutting every part into the same number of equal pieces is what keeps the value the same.

Simplifying: the rule run in reverse

The building rule works in both directions. Building up cut each part into smaller pieces; simplifying runs the same picture backwards, grouping the small pieces back into larger ones.

Take $\frac{6}{8}$. Both $6$ and $8$ are divisible by $2$, so divide each by $2$:

Look at what that division does to the picture. The whole was cut into $8$ parts, and gathering those parts two at a time leaves $4$ larger parts. The $6$ shaded pieces gather into $3$ shaded larger parts. No area was added or removed, only cut lines erased, so the shaded amount is unchanged.

That same gathering works for any number that divides both the numerator and the denominator, which gives the rule.

Simplifying rule. If a whole number $c$ divides both $a$ and $b$, then $\frac{a}{b} = \frac{a \div c}{b \div c}.$

The number $c$ you divide by must be a common factor of the numerator and denominator, a factor of both. That requirement is exactly the idea from the factors chapter. Dividing by a common factor is the only way to make a fraction simpler while keeping its value.

Lowest terms and the one-step shortcut

A fraction is in lowest terms (also called simplest form) when the numerator and denominator have no common factor larger than $1$. At that point there is nothing left to divide out, so the fraction cannot be written with smaller whole numbers. For example $\frac{3}{4}$ is in lowest terms, because the only factor $3$ and $4$ share is $1$.

You could simplify $\frac{18}{24}$ by dividing out common factors one at a time: divide by $2$ to get $\frac{9}{12}$, then by $3$ to get $\frac{3}{4}$. That works, but it takes several steps and you have to keep checking whether you are done. There is a faster way that always finishes in a single division, and it uses the greatest common factor from the previous chapter.

For $\frac{18}{24}$, the greatest common factor of $18$ and $24$ is $6$, so

and you are guaranteed to be finished. If $3$ and $4$ still shared a factor, that factor times $6$ would divide both $18$ and $24$. But $6$ is already the biggest number that does. If you instead divide by a common factor that is not the greatest, you simplify the fraction but you are not done yet.

So the shortcut is: divide the numerator and the denominator by their GCF, once.

Checking whether two fractions are equal

Sometimes you are handed two fractions and asked whether they are equal, with no obvious cutting between them. One reliable test is to simplify both to lowest terms. Compare $\frac{6}{9}$ and $\frac{10}{15}$. The first simplifies by $3$ to $\frac{2}{3}$, and the second simplifies by $5$ to $\frac{2}{3}$. Same simplest form, so they are equal. That test works because equivalent fractions have the same lowest-terms form. If the simplified versions match the fractions are equal, and if they differ the fractions are not.

A second test avoids simplifying entirely. To compare $\frac{4}{6}$ and $\frac{6}{9}$, multiply the numerator of each by the denominator of the other:

Those two products are equal, and so are the fractions. The reason is that both of them, rewritten over the shared denominator $6 \times 9$, become $\frac{4 \times 9}{6 \times 9}$ and $\frac{6 \times 6}{9 \times 6}$. With equal denominators, the two fractions are equal exactly when the numerators are equal, and those numerators are the two products just multiplied out. In general, $\frac{a}{b}$ and $\frac{c}{d}$ are equal exactly when the cross products match, that is when $a \times d = b \times c$. This test is especially handy with larger numbers, where simplifying both might take longer than one multiplication on each side.

Common mistakes