Pre-Algebra Workbook 4: Fractions I – Simplifying & Equivalent Forms

Find the greatest common factor (GCF) of $12$ and $18$ and divide both the numerator and denominator by that number.

Find the greatest common factor (GCF) of $12$ and $18$

$$12 = 2^2 \cdot 3, \quad 18 = 2 \cdot 3^2$$

The common prime factors are $2$ and $3$ so the GCF is $2 \cdot 3 = 6$

Divide numerator and denominator by $6$

$$\dfrac{12}{18} = \dfrac{12 \div 6}{18 \div 6} = \dfrac{2}{3}$$

So the simplified form is $\dfrac{2}{3}$

The correct choice is B. $\dfrac{2}{3}$.

Look for a common factor of $45$ and $60$ You can use prime factorization or find the GCF by listing factors.

Use prime factorization:

$$45 = 3^2 \cdot 5, \quad 60 = 2^2 \cdot 3 \cdot 5$$

The common primes are $3$ and $5$ so the GCF is $3 \cdot 5 = 15$

Divide numerator and denominator by $15$

$$\dfrac{45}{60} = \dfrac{45 \div 15}{60 \div 15} = \dfrac{3}{4}$$

So the simplified form is $\dfrac{3}{4}$

The correct choice is C. $\dfrac{3}{4}$.

To find an equivalent fraction, multiply or divide both the numerator and denominator by the same nonzero number.

Start with $\dfrac{3}{5}$ and check each option to see if it can be obtained by multiplying both numerator and denominator by the same number.

• $\dfrac{6}{10}: 3 \to 6$ (multiply by $2$) and $5 \to 10$ (multiply by $2$), so this is equivalent.
• $\dfrac{9}{20}: 3 \to 9$ (multiply by $3$), but $5 \to 20$ (multiply by $4$), so not equivalent.
• $\dfrac{15}{16}: 3 \to 15$ (multiply by $5$), but $5 \to 16$ (not multiply by $5$), so not equivalent.
• $\dfrac{5}{3}$ is actually the reciprocal, not equivalent.

So the fraction equivalent to $\dfrac{3}{5}$ is $\dfrac{6}{10}$

The correct choice is A. $\dfrac{6}{10}$.

Think about how to get from $6$ to $12$ in the denominator, and then apply the same change to the numerator.

Start with $\dfrac{5}{6}$ and compare the denominators.

To go from $6$ to $12$ multiply by $2$

To keep the fractions equivalent, multiply the numerator by the same number:

$$5 \times 2 = 10$$

So the missing numerator is $10$ and the fraction is $\dfrac{10}{12}$

The correct choice is C. 10.

Convert the fraction to a decimal by dividing $3 \div 5$ then multiply by $100$ to get the percent.

First convert $\dfrac{3}{5}$ to a decimal:

$$3 \div 5 = 0.6$$

Now convert the decimal to a percent by multiplying by $100$

$$0.6 \times 100 = 60\%$$

So $\dfrac{3}{5}$ is equal to $60\%$

The correct choice is B. $60\%$.

Start by writing $54\%$ as $\dfrac{54}{100}$ then simplify the fraction by dividing the numerator and denominator by their GCF.

Write $54\%$ as a fraction:

$$54\% = \dfrac{54}{100}$$

Find the greatest common factor of $54$ and $100$

$$54 = 2 \cdot 3^3, \quad 100 = 2^2 \cdot 5^2$$

The common factor is $2$ so divide numerator and denominator by $2$

$$\dfrac{54}{100} = \dfrac{54 \div 2}{100 \div 2} = \dfrac{27}{50}$$

So $54\%$ as a fraction in simplest form is $\dfrac{27}{50}$

The correct choice is C. $\dfrac{27}{50}$.

Compare the fractions by rewriting them with a common denominator, or convert both to decimals and compare the decimal values.

Find a common denominator for $\dfrac{3}{4}$ and $\dfrac{4}{5}$

The least common multiple of $4$ and $5$ is $20$

Rewrite each fraction with denominator $20$

$$\dfrac{3}{4} = \dfrac{3 \times 5}{4 \times 5} = \dfrac{15}{20}, \quad \dfrac{4}{5} = \dfrac{4 \times 4}{5 \times 4} = \dfrac{16}{20}$$

Since $\dfrac{16}{20} > \dfrac{15}{20}$ we have $\dfrac{4}{5} > \dfrac{3}{4}$

The correct choice is B. $\dfrac{4}{5}$.

Either convert each fraction to a decimal or rewrite them with a common denominator so you can compare their sizes.

Convert each fraction to a decimal (or use a common denominator):

• $\dfrac{1}{2} = 0.5$
• $\dfrac{5}{8} = 0.625$
• $\dfrac{2}{3} \approx 0.666\dots$
• $\dfrac{3}{4} = 0.75$

From least to greatest:

$$\dfrac{1}{2}, \; \dfrac{5}{8}, \; \dfrac{2}{3}, \; \dfrac{3}{4}$$

The correct choice is the list that matches this order: A. $\dfrac{1}{2}, \dfrac{5}{8}, \dfrac{2}{3}, \dfrac{3}{4}$.

Compare $\dfrac{3}{4}$ and $\dfrac{5}{8}$ either by using a common denominator or by converting each to a decimal.

Compare $\dfrac{3}{4}$ and $\dfrac{5}{8}$

Use a common denominator of $8$

$$\dfrac{3}{4} = \dfrac{3 \times 2}{4 \times 2} = \dfrac{6}{8}, \quad \dfrac{5}{8} = \dfrac{5}{8}$$

Since $\dfrac{6}{8} > \dfrac{5}{8}$ we know that $\dfrac{3}{4} > \dfrac{5}{8}$

So Kurt ate more pizza.

The correct choice is A. Kurt.

Convert $\dfrac{18}{20}$ to a percent, or convert $90\%$ to a fraction with denominator $20$ then compare the two.

First convert $\dfrac{18}{20}$ to a percent.

Simplify the fraction:

$$\dfrac{18}{20} = \dfrac{9}{10}$$

Now write $\dfrac{9}{10}$ as a percent:

$$9 \div 10 = 0.9, \quad 0.9 \times 100 = 90\%$$

So Sam's score is $90\%$ and Alex's score is also $90\%$

They have the same score.

The correct choice is C. They had the same score.