Equivalent Fractions and Lowest Terms | Grade 7 Math | LevelKite

One amount of a whole can have more than one fraction name. In this lesson you will see why, and you will learn to choose the simplest name.

A fraction such as $\frac{3}{4}$ describes part of a whole, which is the complete thing you are measuring or sharing. The bottom number, the denominator, tells you how many equal parts make up the whole. The top number, the numerator, tells you how many of those parts you count. The parts must all be equal. In this lesson every numerator and denominator is a counting number: $1, 2, 3$ and so on.

The bar below shows one whole with $4$ equal parts. Shading covers $3$ of them, so the shaded amount is $\frac{3}{4}$ of the bar.

One whole with

4

equal parts and

3

of them shaded:

\frac{3}{4}

of the bar.

Now picture the same bar with each of its $4$ parts cut into $2$ equal pieces. You add nothing and remove nothing, so the whole is the same and the shaded amount is the same. Only the way you count it changes.

The whole now has $4 \times 2 = 8$ equal parts.
The $3$ shaded parts have become $3 \times 2 = 6$ shaded parts.

The bar below shows the result: shading covers $6$ of the $8$ equal parts, so the shaded amount is $\frac{6}{8}$ of the bar. Both bars stand for the same whole. Each fourth (one of the $4$ equal parts) in the first bar matches $2$ eighths (two of the $8$ equal parts) in the second. That is why $\frac{3}{4}$ and $\frac{6}{8}$ name the same amount of this whole.

The same whole with

8

equal parts and

6

of them shaded:

\frac{6}{8}

of the bar.

Key concept

Equivalent fractions name the same amount

Equivalent fractions are fractions that name the same amount of the same whole, with the whole split into equal parts each time.

Let $a$ be a numerator, $b$ its denominator and $n$ any counting number. Multiplying both $a$ and $b$ by $n$ gives an equivalent fraction, because you cut each of the $b$ equal parts into $n$ smaller equal parts. The whole then has $b \times n$ parts and you count $a \times n$ of them, so the amount does not change. With the bars, $\frac{3}{4} = \frac{3 \times 2}{4 \times 2} = \frac{6}{8}$ .

You may also divide both numbers by the same counting number, as long as it divides both exactly, with no remainder: that is the same rule read from right to left. Zero is not allowed for $n$ : you cannot cut a whole into $0$ parts.

\frac{a}{b} = \frac{a \times n}{b \times n} \quad \text{for counting numbers } a, b, n

Worked example

Write an equivalent fraction

Write a fraction that is equivalent to $\frac{2}{5}$ and has a denominator of $20$ . Both fractions describe the same whole.

Find the counting number that turns the denominator 5 into 20. It is 4, because 5 times 4 is 20. So you cut each of the 5 equal parts of the whole into 4 smaller equal parts.
$5 \times 4 = 20$
Multiply the numerator by the same number, 4. Each of the 2 parts you count also splits into 4 smaller parts, so you now count 8 small parts and the amount stays the same.
$2 \times 4 = 8$
Write the new fraction. You multiplied the numerator and the denominator by the same counting number, so it names the same amount of the same whole.
$\frac{2}{5} = \frac{2 \times 4}{5 \times 4} = \frac{8}{20}$

Answer

The fraction $\frac{8}{20}$ is equivalent to $\frac{2}{5}$ and has a denominator of $20$ .

Worked example

Decide whether fractions are equivalent

Each fraction here describes part of the same whole. Is $\frac{10}{15}$ equivalent to $\frac{2}{3}$ ? Is $\frac{8}{9}$ equivalent to $\frac{2}{3}$ ?

Start with the first question. Rewrite two thirds with the denominator 15, so that both fractions use parts of the same size. The counting number that turns 3 into 15 is 5, because 3 times 5 is 15.
$3 \times 5 = 15$
Multiply the numerator and the denominator of two thirds by 5. This cuts each third into 5 equal parts, so the amount of the whole stays the same. The result matches the first fraction exactly.
$\frac{2}{3} = \frac{2 \times 5}{3 \times 5} = \frac{10}{15}$
Now the second question. Rewrite two thirds with the denominator 9. Multiply the denominator by 3, because 3 times 3 is 9, and multiply the numerator by 3 as well, so the amount stays the same.
$\frac{2}{3} = \frac{2 \times 3}{3 \times 3} = \frac{6}{9}$
Compare this result with the second fraction. Both fractions count ninths, the parts you get when you cut the same whole into 9 equal parts, so the parts are the same size. Two thirds is 6 of those parts, but the second fraction is 8 of them. Different counts of same-size parts are different amounts.
$\frac{6}{9} \neq \frac{8}{9}$

Answer

The fraction $\frac{10}{15}$ is equivalent to $\frac{2}{3}$ . The fraction $\frac{8}{9}$ is not, because $\frac{2}{3}$ of the whole is $6$ ninths, not $8$ ninths.

Since one amount has many fraction names, it helps to agree on the simplest one. A fraction is in lowest terms when $1$ is the only counting number that divides both its numerator and its denominator exactly.

Three ideas make this easier to say:

A factor of a number is a counting number that divides it exactly, with no remainder. The factors of $8$ are $1$ , $2$ , $4$ and $8$ .
A common factor of two numbers is a factor of both.
The greatest common factor (GCF) is the largest common factor.

So a fraction is in lowest terms when the GCF of its numerator and denominator is $1$ . For example, $\frac{7}{12}$ is in lowest terms, because the only common factor of $7$ and $12$ is $1$ . But $\frac{6}{8}$ is not, because $2$ divides both $6$ and $8$ exactly.

To write a fraction in lowest terms, divide its numerator and its denominator by their GCF.

Why can you do this? It runs the multiplying rule backwards. Take $\frac{6}{8}$ . The common factors of $6$ and $8$ are $1$ and $2$ , so the GCF is $2$ . Because $6 = 3 \times 2$ and $8 = 4 \times 2$ , dividing both numbers by $2$ undoes a multiplication of both by $2$ :

$\frac{6}{8} = \frac{6 \div 2}{8 \div 2} = \frac{3}{4}$

In terms of parts, every $2$ of the $8$ equal parts of the whole join into $1$ larger part, so $6$ of $8$ small parts become $3$ of $4$ larger equal parts. The new fraction names the same amount of the same whole. The only common factor of $3$ and $4$ is $1$ , so $\frac{3}{4}$ is in lowest terms.

Why the greatest common factor? It finishes the job in one step. If the new numbers still shared a factor greater than $1$ , you could have divided by an even larger common factor, so the one you used would not have been the greatest.

The table shows the same steps for three more fractions.

Each row divides both numbers by their GCF, so the new fraction names the same amount of the same whole.

Fraction	Common factors	Divide both numbers by the GCF	Lowest terms
$\frac{20}{25}$	$1$ and $5$ , so the GCF is $5$	$\frac{20 \div 5}{25 \div 5}$	$\frac{4}{5}$
$\frac{16}{24}$	$1$ , $2$ , $4$ and $8$ , so the GCF is $8$	$\frac{16 \div 8}{24 \div 8}$	$\frac{2}{3}$
$\frac{4}{9}$	$1$ only, so the GCF is $1$	Dividing by $1$ changes nothing	$\frac{4}{9}$ , already in lowest terms

Worked example

Write a fraction in lowest terms

Write $\frac{36}{60}$ in lowest terms.

List the factors of the numerator and of the denominator, so you can find the ones they share. The factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18 and 36. The factors of 60 are 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30 and 60.
Pick out the numbers that appear in both lists. The common factors are 1, 2, 3, 4, 6 and 12, so the greatest common factor is 12. Writing each number as a multiplication by 12 confirms that 12 divides both.
$36 = 3 \times 12 \qquad 60 = 5 \times 12$
Divide the numerator and the denominator by 12. This undoes a multiplication of both 3 and 5 by 12: every 12 of the 60 equal parts of the whole join into 1 larger part. So the new fraction names the same amount of the same whole.
$\frac{36}{60} = \frac{36 \div 12}{60 \div 12} = \frac{3}{5}$
Check the result. The factors of 3 are 1 and 3, and the factors of 5 are 1 and 5. The only common factor is 1, so the fraction is in lowest terms.
If you do not spot 12, divide by smaller common factors one at a time. Dividing both numbers by 2 gives 18 and 30. Dividing both by 2 again gives 9 and 15. Dividing both by 3 gives 3 and 5. Each division keeps the amount the same. Keep going until 1 is the only common factor: stopping at 18 and 30, or at 9 and 15, would be too early.
$\frac{36}{60} = \frac{18}{30} = \frac{9}{15} = \frac{3}{5}$

Answer

In lowest terms, $\frac{36}{60}$ is $\frac{3}{5}$ .

Here is the idea to keep: a fraction's numbers can change while its amount stays the same. When you multiply the numerator and the denominator by the same counting number, or divide both by a common factor, you get an equivalent fraction, a new name for the same amount of the same whole. Dividing by the greatest common factor gives the name in lowest terms, where $1$ is the only common factor left.

Next, you will use equivalent fractions to give fractions the same denominator, so that you can compare and order them.