Equivalent Fractions and Simplifying: Additional Practice (Set 2)

For $\frac{3}{4}$, the denominator $4$ becomes $12$ by multiplying by $3$, so multiply the top by $3$: $3 \times 3 = 9$. For $\frac{5}{6}$, the denominator $6$ becomes $12$ by multiplying by $2$, so multiply the top by $2$: $5 \times 2 = 10$.

$\frac{3}{4} = \frac{9}{12}, \qquad \frac{5}{6} = \frac{10}{12}$

The new numerators are $9$ and $10$.

The greatest common factor of $56$ and $70$ is $14$, since $56 = 14 \times 4$ and $70 = 14 \times 5$.

$\frac{56}{70} = \frac{56 \div 14}{70 \div 14} = \frac{4}{5}$

Since $4$ and $5$ share no factor above $1$, $\frac{4}{5}$ is in lowest terms.

The greatest common factor of $60$ and $84$ is $12$, since $60 = 12 \times 5$ and $84 = 12 \times 7$.

$\frac{60}{84} = \frac{60 \div 12}{84 \div 12} = \frac{5}{7}$

Since $5$ and $7$ share no factor above $1$, $\frac{5}{7}$ is in lowest terms.

Each equivalent of $\frac{3}{5}$ simplifies back to $\frac{3}{5}$. Check each.

$\frac{9}{15} = \frac{3}{5}, \quad \frac{12}{20} = \frac{3}{5}, \quad \frac{18}{30} = \frac{3}{5}$

But $\frac{20}{30}$ divides by $10$ to give $\frac{2}{3}$, not $\frac{3}{5}$. So $\frac{20}{30}$ is the one that is not equal.

Simplify $\frac{8}{12}$ by dividing out the greatest common factor of $8$ and $12$, which is $4$.

$\frac{8}{12} = \frac{8 \div 4}{12 \div 4} = \frac{2}{3}$

So the recipe uses $\frac{2}{3}$ of a cup.

First simplify $\frac{10}{12}$ by dividing by $2$: $\frac{10}{12} = \frac{5}{6}$. Now rewrite $\frac{5}{6}$ with denominator $18$ by multiplying top and bottom by $3$.

$\frac{5}{6} = \frac{5 \times 3}{6 \times 3} = \frac{15}{18}$

So $a = 15$.

The greatest common factor of $48$ and $72$ is $24$, since $48 = 24 \times 2$ and $72 = 24 \times 3$.

$\frac{48}{72} = \frac{48 \div 24}{72 \div 24} = \frac{2}{3}$

Since $2$ and $3$ share no factor above $1$, $\frac{2}{3}$ is in lowest terms.

The greatest common factor of $84$ and $120$ is $12$, since $84 = 12 \times 7$ and $120 = 12 \times 10$.

$\frac{84}{120} = \frac{84 \div 12}{120 \div 12} = \frac{7}{10}$

Since $7$ and $10$ share no factor above $1$, $\frac{7}{10}$ is in lowest terms.

The denominator went from $8$ to $56$, which is multiplying by $7$ since $8 \times 7 = 56$. Multiply the numerator by the same $7$.

$\frac{5}{8} = \frac{5 \times 7}{8 \times 7} = \frac{35}{56}$

So the missing numerator is $35$.

The greatest common factor of $50$ and $80$ is $10$, since $50 = 10 \times 5$ and $80 = 10 \times 8$.

$\frac{50}{80} = \frac{50 \div 10}{80 \div 10} = \frac{5}{8}$

Since $5$ and $8$ share no factor above $1$, $\frac{5}{8}$ is in lowest terms.

Cross-multiply: pair each numerator with the other fraction's denominator.

$12 \times 20 = 240 \qquad 15 \times 16 = 240$

The cross products are equal, so $\frac{12}{15} = \frac{16}{20}$. (Both also simplify to $\frac{4}{5}$.)

Each equivalent of $\frac{2}{3}$ is $\frac{2k}{3k}$, and its numerator and denominator add to $2k + 3k = 5k$. Set this equal to $25$.

$5k = 25 \;\Rightarrow\; k = 5$

So the fraction is $\frac{2 \times 5}{3 \times 5} = \frac{10}{15}$, whose parts $10$ and $15$ add to $25$.