Equivalent Fractions and Simplifying: Practice

Divide by the greatest common factor of $36$ and $48$. Both factor as $36 = 12 \times 3$ and $48 = 12 \times 4$, so the GCF is $12$.

$\frac{36}{48} = \frac{36 \div 12}{48 \div 12} = \frac{3}{4}$

Since $3$ and $4$ share no factor above $1$, $\frac{3}{4}$ is in lowest terms.

The greatest common factor of $42$ and $56$ is $14$, since $42 = 14 \times 3$ and $56 = 14 \times 4$.

$\frac{42}{56} = \frac{42 \div 14}{56 \div 14} = \frac{3}{4}$

Since $3$ and $4$ share no factor above $1$, $\frac{3}{4}$ is in lowest terms.

Rewrite both over the common denominator $28$.

$\frac{3}{4} = \frac{21}{28}, \qquad \frac{5}{7} = \frac{20}{28}$

With the same denominator, the larger numerator wins: $21 > 20$, so $\frac{3}{4} > \frac{5}{7}$.

Each equivalent of $\frac{4}{5}$ simplifies back to $\frac{4}{5}$. Check each.

$\frac{16}{20} = \frac{4}{5}, \quad \frac{12}{15} = \frac{4}{5}, \quad \frac{20}{25} = \frac{4}{5}$

But $\frac{15}{20}$ divides by $5$ to give $\frac{3}{4}$, not $\frac{4}{5}$. So $\frac{15}{20}$ is the one that is not equal.

The denominator went from $12$ to $60$, which is multiplying by $5$ since $12 \times 5 = 60$. Multiply the numerator by the same $5$.

$\frac{7}{12} = \frac{7 \times 5}{12 \times 5} = \frac{35}{60}$

So the missing numerator is $35$.

First simplify $\frac{15}{20}$ by dividing by $5$: $\frac{15}{20} = \frac{3}{4}$. Now rewrite $\frac{3}{4}$ with denominator $12$ by multiplying top and bottom by $3$.

$\frac{3}{4} = \frac{3 \times 3}{4 \times 3} = \frac{9}{12}$

So $a = 9$.

Rewrite both over the common denominator $18$.

$\frac{5}{6} = \frac{15}{18}, \qquad \frac{7}{9} = \frac{14}{18}$

With the same denominator, the larger numerator wins: $15 > 14$, so $\frac{5}{6} > \frac{7}{9}$.

The greatest common factor of $63$ and $81$ is $9$, since $63 = 9 \times 7$ and $81 = 9 \times 9$.

$\frac{63}{81} = \frac{63 \div 9}{81 \div 9} = \frac{7}{9}$

Since $7$ and $9$ share no factor above $1$, $\frac{7}{9}$ is in lowest terms.

Simplify $\frac{18}{24}$ by dividing out the greatest common factor of $18$ and $24$, which is $6$.

$\frac{18}{24} = \frac{18 \div 6}{24 \div 6} = \frac{3}{4}$

So $\frac{3}{4}$ of the students walk.

A fraction is in lowest terms when its numerator and denominator share no factor above $1$. Check each.

$\frac{14}{21} = \frac{2}{3}, \quad \frac{15}{25} = \frac{3}{5}, \quad \frac{12}{18} = \frac{2}{3}$

But $9$ and $16$ share only the factor $1$, so $\frac{9}{16}$ is already in lowest terms.

The greatest common factor of $54$ and $90$ is $18$, since $54 = 18 \times 3$ and $90 = 18 \times 5$.

$\frac{54}{90} = \frac{54 \div 18}{90 \div 18} = \frac{3}{5}$

Since $3$ and $5$ share no factor above $1$, $\frac{3}{5}$ is in lowest terms.

The greatest common factor of $72$ and $96$ is $24$, since $72 = 24 \times 3$ and $96 = 24 \times 4$.

$\frac{72}{96} = \frac{72 \div 24}{96 \div 24} = \frac{3}{4}$

Since $3$ and $4$ share no factor above $1$, $\frac{3}{4}$ is in lowest terms.