Equivalent Fractions

The highest common factor of 15 and 20 is 5.

Dividing both parts by 5: $15÷5=3$ and $20÷5=4$, so $\frac{15}{20}=\frac{3}{4}$.

3 and 4 share no factor other than 1, so $\frac{3}{4}$ is fully simplified.

The numerator and denominator must be divided by the same number. Dividing them by different numbers, or dividing only one of them, changes the value of the fraction. And a fully simplified fraction does not have to have 1 on top: $\frac{3}{4}$, not $\frac{1}{4}$, is the answer.

The highest common factor of 18 and 27 is 9.

Dividing both parts by 9: $18÷9=2$ and $27÷9=3$, so $\frac{18}{27}=\frac{2}{3}$.

2 and 3 share no factor other than 1, so $\frac{2}{3}$ is fully simplified.

Simplifying means dividing both parts by the same factor. Subtracting 9 from each instead gives $\frac{9}{18}=\frac{1}{2}$, which is a different number, and the simplest form does not have to have 1 on top.

A fraction is in its simplest form when the numerator and denominator share no factor other than 1.

$\frac{5}{7}$ is simplest: the only factor 5 and 7 share is 1.

Each of the others can still be cancelled:

$\frac{4}{8}=\frac{1}{2}$ (divide both by 4: small numbers can still have a common factor)

$\frac{9}{12}=\frac{3}{4}$ (divide both by 3: a fraction can simplify even when one of the numbers is odd)

$\frac{6}{10}=\frac{3}{5}$ (divide both by 2: look for a shared factor, not for one number dividing into the other)

Multiplying the numerator and the denominator by the same number gives an equivalent fraction.

$7\times 6=42$ and $9\times 6=54$, so $\frac{7}{9}=\frac{42}{54}$.

Both parts must be multiplied by 6. Adding 6 to each instead gives $\frac{13}{15}$; multiplying only the denominator gives $\frac{7}{54}$; and scaling the top but keeping the original gap of 2 between top and bottom gives $\frac{42}{44}$. None of these is equal to $\frac{7}{9}$.

Write each number as a product of primes: $105=3\times 5\times 7$ and $150=2\times 3\times 5\times 5$.

The factors they share are 3 and 5, so the highest common factor is $3\times 5=15$.

Dividing both parts by 15: $105÷15=7$ and $150÷15=10$, giving $\frac{7}{10}$.

7 and 10 share no factor other than 1, so $\frac{7}{10}$ is fully simplified. Both parts must be divided by the same number, and the simplest form does not have to have 1 on top.

Undo the expansion by dividing both parts by 8: $96÷8=12$ and $144÷8=18$, giving $\frac{12}{18}$.

This is not yet fully simplified. The highest common factor of 12 and 18 is 6, and $12÷6=2$, $18÷6=3$, so the answer is $\frac{2}{3}$.

You can also go straight there: the highest common factor of 96 and 144 is 48, and $96÷48=2$, $144÷48=3$.

An expansion is undone by dividing both parts by 8, not by subtracting 8 from each, and both parts must be divided, not just the numerator.