Equivalent Fractions and Lowest Terms | Grade 5 Math | LevelKite

Two fractions can look different and still name the same amount. We call them equivalent fractions: fractions that cover the same amount of the same whole.

A model lets you see this. The bar below is one whole cut into 3 equal parts, and 2 of the parts are shaded. In words, two thirds of the bar is shaded. As a fraction, that is $\frac{2}{3}$ . The numerator (the top number, 2) counts the shaded parts. The denominator (the bottom number, 3) counts all the equal parts in the whole.

Two of the 3 equal parts are shaded, so

\frac{2}{3}

of the bar is shaded.

Now cut each of the 3 parts into 2 equal pieces. The bar below shows the same whole after the cut.

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

In words, four sixths of the bar is shaded. As a fraction, that is $\frac{4}{6}$ . The cut added no shading and took none away.

Four of the 6 equal parts are shaded, so

\frac{4}{6}

of the same bar is shaded.

Compare the two bars. They show the same whole, and the shading covers the same amount of it. Only the parts are different: the second bar has twice as many parts, and each part is half as big.

So $\frac{2}{3}$ and $\frac{4}{6}$ are equivalent fractions, and you can write $\frac{2}{3} = \frac{4}{6}$ .

The whole has to be the same for this to work. Two thirds of a small bar is not the same amount as four sixths of a bigger bar.

The cut also gives you a rule, and it shows why the rule works. When you cut every part into 2 equal pieces, you multiply both numbers by 2. The numerator went from 2 to 4, and the denominator went from 3 to 6. You could cut every part into 3, 4 or more equal pieces in the same way. Each time, you multiply the numerator and the denominator by the same number, and the shaded amount does not change.

Dividing runs the cut backwards. Start with $\frac{4}{6}$ and join the sixths in pairs. The 6 parts become $6 \div 2 = 3$ bigger parts, and the 4 shaded parts become $4 \div 2 = 2$ shaded parts. You are back at $\frac{2}{3}$ , and the shaded amount is still the same. To keep whole numbers on the top and the bottom, divide by a number that goes into both exactly, as 2 goes into both 4 and 6.

The number you multiply or divide by can never be 0. Multiplying both numbers by 0 gives $\frac{0}{0}$ , which does not name an amount, and you cannot divide by 0.

Key concept

Same amount, different parts

Two fractions are equivalent when they cover the same amount of the same whole.

Multiply or divide the numerator and the denominator by the same nonzero number (any number except 0) and you get an equivalent fraction. The number of parts and the size of each part can change, but the amount stays the same.

\frac{2}{3} = \frac{2 \times 2}{3 \times 2} = \frac{4}{6}

Worked example

Find a missing numerator

Find the missing numerator: $\frac{3}{4} = \frac{?}{12}$ .

Compare the denominators. Ask what 4 was multiplied by to make 12. The answer is 3, so each fourth has been cut into 3 equal pieces called twelfths.
$4 \times 3 = 12$
Multiply the numerator by the same number. Each of the 3 fourths you are counting becomes 3 twelfths, so the amount stays the same.
$3 \times 3 = 9$
Write both multiplications in one number sentence. It shows that three fourths and nine twelfths are equivalent.
$\frac{3}{4} = \frac{3 \times 3}{4 \times 3} = \frac{9}{12}$

Answer

The missing numerator is 9, so $\frac{3}{4} = \frac{9}{12}$ .

Worked example

Decide whether fractions are equivalent

Is each fraction equivalent to $\frac{2}{3}$ ? Test $\frac{10}{15}$ and $\frac{9}{12}$ .

Test ten fifteenths by dividing. The number 5 goes into both 10 and 15 exactly, so divide the numerator and the denominator by 5. This joins the fifteenths in groups of 5 and keeps the amount the same. The result is two thirds, so ten fifteenths is equivalent to two thirds.
$\frac{10}{15} = \frac{10 \div 5}{15 \div 5} = \frac{2}{3}$
Test nine twelfths by writing two thirds in twelfths, so that both fractions count parts of the same size. Since 3 times 4 is 12, multiply the numerator by 4 as well.
$\frac{2}{3} = \frac{2 \times 4}{3 \times 4} = \frac{8}{12}$
Compare eight twelfths with nine twelfths. The parts are the same size, but one fraction counts 8 of them and the other counts 9. A different count of same-size parts is a different amount, so nine twelfths is not equivalent to two thirds.
$\frac{8}{12} < \frac{9}{12}$

Answer

The fraction $\frac{10}{15}$ is equivalent to $\frac{2}{3}$ , but $\frac{9}{12}$ is not.

Many fractions name the same amount, so it helps to agree on the simplest name. To find it, you need two new words.

A factor of a number is a whole number that divides it exactly, with nothing left over. The factors of 6 are 1, 2, 3 and 6. A common factor of two numbers is a factor of both. The common factors of 4 and 6 are 1 and 2.

A fraction is in lowest terms when 1 is the only common factor of its numerator and denominator. The fraction $\frac{4}{6}$ is not in lowest terms, because 2 is also a common factor of 4 and 6. Divide both numbers by 2 and you get $\frac{2}{3}$ . The only common factor of 2 and 3 is 1, so $\frac{2}{3}$ is in lowest terms.

To write a fraction in lowest terms, look for a common factor greater than 1 and divide the numerator and the denominator by it. Then check the new fraction and divide again if you can. Stop when 1 is the only common factor left. Each division gives an equivalent fraction, so the amount never changes.

Worked example

Write a fraction in lowest terms

Write $\frac{18}{30}$ in lowest terms.

Look for a common factor greater than 1. Both 18 and 30 are even, so 2 is a common factor. Divide the numerator and the denominator by 2. This gives an equivalent fraction with smaller numbers.
$\frac{18}{30} = \frac{18 \div 2}{30 \div 2} = \frac{9}{15}$
Check nine fifteenths for another common factor. The number 2 does not work now, because 9 is odd. But 3 goes into both 9 and 15 exactly, so divide both numbers by 3.
$\frac{9}{15} = \frac{9 \div 3}{15 \div 3} = \frac{3}{5}$
Check three fifths. The factors of 3 are 1 and 3. The factors of 5 are 1 and 5. The only common factor is 1, so the fraction is in lowest terms and you can stop.

Answer

In lowest terms, $\frac{18}{30}$ is $\frac{3}{5}$ .

Here is the idea to keep. Equivalent fractions are different names for the same amount of the same whole. You can make an equivalent fraction by multiplying or dividing the numerator and the denominator by the same nonzero number. That can change the number and size of the parts, but it never changes the amount. A fraction is in lowest terms when 1 is the only common factor of its numerator and denominator.

Next, you will use equivalent fractions to rewrite two fractions so that they share the same denominator. Then both are counted in parts of the same size.