Equivalent Fractions and Lowest Terms | Grade 6 Math | LevelKite

Can two different-looking fractions name exactly the same amount of one whole? They can, and you are about to see why.

A fraction describes part of a whole. The whole is the complete object or amount that is being divided. The denominator (the bottom number) tells how many equal parts the whole is cut into. The numerator (the top number) tells how many of those parts you count.

Picture a strip of paper. Fold it in half and shade one part. You have shaded $\frac{1}{2}$ of the strip. Now fold the same strip in half again. Each half becomes $2$ smaller equal parts, so the strip has $4$ equal parts and $2$ of them are shaded. That is $\frac{2}{4}$ of the strip.

You did not shade anything new. The shaded amount is the same, but it has a new name. Fractions that name the same amount of the same whole are called equivalent fractions.

One strip, one shaded amount, three names

Times folded in half	Equal parts in the strip	Shaded parts	Fraction of the strip shaded
$1$	$2$	$1$	$\frac{1}{2}$
$2$	$4$	$2$	$\frac{2}{4}$
$3$	$8$	$4$	$\frac{4}{8}$

Read down the table. Each new fold splits every part into $2$ equal pieces. So the number of parts in the strip and the number of shaded parts are both multiplied by $2$ :

$\frac{1}{2} = \frac{1 \times 2}{2 \times 2} = \frac{2}{4}$ and $\frac{2}{4} = \frac{2 \times 2}{4 \times 2} = \frac{4}{8}$

Now read up the table. Undoing a fold joins every $2$ pieces into $1$ larger part. So both numbers are divided by $2$ :

$\frac{4}{8} = \frac{4 \div 2}{8 \div 2} = \frac{2}{4}$ and $\frac{2}{4} = \frac{2 \div 2}{4 \div 2} = \frac{1}{2}$

The size of the pieces changes, but the shaded amount does not. All three fractions name the same amount of the same strip.

Key concept

Same amount, same whole, different names

Equivalent fractions name the same amount of the same whole. To make an equivalent fraction, multiply the numerator and the denominator by the same whole number other than zero. Here is why: you cut every part of the whole into that many equal pieces. The number of pieces in the whole and the number you count are both multiplied by that number, so the amount does not change.

Dividing undoes multiplying. Divide both numbers by a whole number greater than $1$ that divides each of them exactly, and you join pieces back into larger equal parts. The amount still does not change.

In the formula, $a$ is the numerator, $b$ is the denominator (not zero), and $n$ is the whole number (not zero) you multiply by. Both fractions describe the same whole.

\frac{a}{b} = \frac{a \times n}{b \times n}

One amount of a whole has many fraction names. The name with the smallest whole numbers is the fraction in lowest terms, also called simplest form. It uses the fewest and largest equal parts that can name the amount.

A common factor of two whole numbers is a whole number that divides both of them exactly, with no remainder. A fraction is in lowest terms when the only common factor of its numerator and its denominator is $1$ . In the strip table, $\frac{1}{2}$ is the lowest-terms name.

To write a fraction in lowest terms:

Find a common factor of the numerator and the denominator that is greater than $1$ . Your divisibility rules help you spot one.
Divide the numerator and the denominator by that common factor. You join equal pieces into larger equal parts, so the new fraction names the same amount of the same whole.
Check again. Repeat until the only common factor is $1$ .

Worked example

Write a fraction in lowest terms

Write $\frac{18}{24}$ in lowest terms.

Look for a common factor greater than 1. Both 18 and 24 are even, so 2 divides both. Divide the numerator and the denominator by 2. This joins every 2 pieces into 1 larger part, so the fraction still names the same amount of the same whole.
$\frac{18}{24} = \frac{18 \div 2}{24 \div 2} = \frac{9}{12}$
Check again. The numerator 9 is odd, so 2 is no longer a common factor. Try 3. Both 9 and 12 can be divided by 3 exactly, because 9 is 3 times 3 and 12 is 3 times 4. Divide the numerator and the denominator by 3.
$\frac{9}{12} = \frac{9 \div 3}{12 \div 3} = \frac{3}{4}$
Check once more. A common factor of 3 and 4 must divide 3 exactly, and the only whole numbers that do are 1 and 3. But 3 does not divide 4 exactly, because 4 divided by 3 leaves a remainder of 1. The only common factor is 1, so the fraction is in lowest terms.
$4 = 3 \times 1 + 1$
See a shortcut. Dividing by 2 and then by 3 is the same as dividing by 6, because 2 times 3 is 6. So you can divide 18 and 24 by 6 in one step and reach the same fraction. A larger common factor can save steps, but always check that the only common factor left is 1.
$\frac{18}{24} = \frac{18 \div 6}{24 \div 6} = \frac{3}{4}$

Answer

In lowest terms, $\frac{18}{24}$ is $\frac{3}{4}$ . Both fractions name the same amount of the same whole.

Now go the other way. Instead of joining pieces, cut each part into smaller equal pieces. To find an equivalent fraction, multiply the numerator and the denominator by the same whole number other than zero. Multiplying by $2$ , $3$ , $4$ and so on gives a different name each time for the same amount of the same whole.

The table lists five more names for $\frac{3}{4}$ of the same whole. Look at the last row. It brings back $\frac{18}{24}$ from the first example, because multiplying by $6$ undoes dividing by $6$ .

Five more names for three fourths of the same whole

Multiply both numbers by	New numerator	New denominator	Equivalent fraction
$2$	$3 \times 2 = 6$	$4 \times 2 = 8$	$\frac{6}{8}$
$3$	$3 \times 3 = 9$	$4 \times 3 = 12$	$\frac{9}{12}$
$4$	$3 \times 4 = 12$	$4 \times 4 = 16$	$\frac{12}{16}$
$5$	$3 \times 5 = 15$	$4 \times 5 = 20$	$\frac{15}{20}$
$6$	$3 \times 6 = 18$	$4 \times 6 = 24$	$\frac{18}{24}$

Sometimes you need a fraction with one particular denominator. Then ask: what was the old denominator multiplied by to make the new one?

A multiple of a number is that number times a whole number. For example, $20$ is a multiple of $4$ because $4 \times 5 = 20$ .

If the new denominator is a multiple of the old one, divide the new denominator by the old one. The result is your multiplier, the number you multiply by. Multiply the numerator by it too, so the new fraction names the same amount of the same whole.
If it is not a multiple, write the fraction in lowest terms first. Then, if the new denominator is a multiple of the lowest-terms denominator, find the multiplier from there. Every step keeps the same amount of the same whole.

In the table, the name for $\frac{3}{4}$ with denominator $20$ is in the row where the multiplier is $20 \div 4 = 5$ . That name is $\frac{15}{20}$ .

Worked example

Find a fraction with a given denominator

Write $\frac{5}{8}$ as an equivalent fraction with a denominator of $24$ .

Find the multiplier. The denominator must change from 8 to 24, and 24 is a multiple of 8. Divide the new denominator by the old one. The result tells you how many smaller pieces each eighth is cut into.
$24 \div 8 = 3$
Multiply the numerator and the denominator by 3. Every eighth of the whole is cut into 3 equal pieces, so the 5 counted eighths become 15 counted twenty-fourths. The new fraction names the same amount of the same whole.
$\frac{5}{8} = \frac{5 \times 3}{8 \times 3} = \frac{15}{24}$
Check by undoing the multiplication. Divide the numerator and the denominator of the new fraction by 3. You get back the fraction you started with, so both fractions name the same amount of the same whole.
$\frac{15}{24} = \frac{15 \div 3}{24 \div 3} = \frac{5}{8}$

Answer

The fraction $\frac{5}{8}$ is equivalent to $\frac{15}{24}$ . Both name the same amount of the same whole.

Adding is not multiplying. A common mistake is to add the same number to the numerator and the denominator. Start with $\frac{1}{2}$ and add $2$ to each: $\frac{1 + 2}{2 + 2} = \frac{3}{4}$ . This is not an equivalent fraction.

To compare the two amounts, count parts of the same size in the same whole. Each half holds $2$ fourths, so $\frac{1}{2} = \frac{1 \times 2}{2 \times 2} = \frac{2}{4}$ . Then $2$ fourths is less than $3$ fourths.

The mistake fails because adding $2$ doubled the parts in the whole (from $2$ to $4$ ) but tripled the counted parts (from $1$ to $3$ ). With parts half as large, the same amount needs exactly twice as many counted parts, not three times as many. Multiply both numbers by the same whole number other than zero instead.

Worked example

Lowest terms first, then a new denominator

Let $x$ stand for a missing numerator. Find the whole number $x$ that makes $\frac{10}{15} = \frac{x}{12}$ true. Both fractions describe the same whole.

Look for a multiplier. The denominator must change from 15 to 12, but 12 is not a multiple of 15, because 15 times 1 is already more than 12. You cannot get twelfths by cutting each fifteenth into equal pieces. So write the fraction in lowest terms first.
Both 10 and 15 end in 0 or 5, so 5 divides both. Divide the numerator and the denominator by 5. The only whole numbers that divide 2 exactly are 1 and 2, and 2 does not divide 3 exactly because 3 is odd. The only common factor left is 1.
$\frac{10}{15} = \frac{10 \div 5}{15 \div 5} = \frac{2}{3}$
Look again. The new denominator, 12, is a multiple of the lowest-terms denominator, 3. Divide to find the multiplier.
$12 \div 3 = 4$
Multiply the numerator and the denominator by 4. Each third of the whole is cut into 4 equal pieces, so the amount does not change. The new numerator is the value of x.
$\frac{2}{3} = \frac{2 \times 4}{3 \times 4} = \frac{8}{12}$
Check. Write the new fraction in lowest terms by dividing both numbers by 4. It has the same lowest-terms form as the first fraction, so the two fractions name the same amount of the same whole.
$\frac{8}{12} = \frac{8 \div 4}{12 \div 4} = \frac{2}{3}$

Answer

The missing numerator is $x = 8$ , so $\frac{10}{15} = \frac{8}{12}$ . Both fractions name the same amount as $\frac{2}{3}$ of the same whole.

Takeaway. Equivalent fractions are different names for the same amount of the same whole. Multiply the numerator and the denominator by the same whole number greater than $1$ to cut the parts smaller. Divide both by a common factor greater than $1$ to join the parts into larger ones. Either way, the amount does not change. A fraction is in lowest terms when the only common factor of its numerator and denominator is $1$ . Two fractions of the same whole are equivalent when they have the same lowest-terms form.

Next, you will use these ideas to convert between improper fractions and mixed numbers and to find the least common denominator.