Equivalent Fractions - SSAT Middle Level Quantitative

1. What is an equivalent fraction to $\frac{3}{5}$ with denominator $20$?

   $\frac{12}{20}$. To reach denominator 20, multiply numerator and denominator of $\frac{3}{5}$ by 4, yielding an equivalent fraction.

2. What value of $x$ makes $\frac{7}{x}=\frac{21}{30}$?

   $x=10$. Cross-multiplying the equation results in $7\times30=21x$, yielding $x=10$ for equivalent fractions.

3. What value of $x$ makes $\frac{2}{7}=\frac{x}{21}$?

   $x=6$. Solving the proportion by cross-multiplying gives $2\times21=7x$, so $x=6$ makes the fractions equivalent.

4. What is the rule for creating an equivalent fraction from $\frac{a}{b}$ using a nonzero integer $k$?

   Multiply: $\frac{a}{b}=\frac{a\cdot k}{b\cdot k}$ for $k\neq 0$. Multiplying both the numerator and denominator by the same nonzero integer $k$ preserves the value of the fraction, creating an equivalent one.

5. What is an equivalent fraction to $\frac{11}{15}$ with denominator $45$?

   $\frac{33}{45}$. Multiplying numerator and denominator of $\frac{11}{15}$ by 3 achieves denominator 45 while preserving equivalence.

6. Which fraction is equivalent to $\frac{4}{11}$: $\frac{12}{33}$ or $\frac{12}{34}$?

   $\frac{12}{33}$. Cross products confirm $\frac{12}{33}$ equals $\frac{4}{11}$ since $4\times33=11\times12=132$, unlike $\frac{12}{34}$.

7. Find and correct the error: Is $\frac{8}{12}$ simplified to $\frac{2}{3}$ correct?

   Correct; $\frac{8}{12}=\frac{2}{3}$. Dividing both numerator and denominator by 4 correctly simplifies $\frac{8}{12}$ to its lowest terms.

8. What is the simplest form of $\frac{18}{24}$?

   $\frac{3}{4}$. Dividing numerator and denominator by their greatest common divisor of 6 simplifies $\frac{18}{24}$ to its lowest terms.

9. Find and correct the error: Is $\frac{6}{10}$ simplified to $\frac{3}{10}$ correct?

   Incorrect; $\frac{6}{10}=\frac{3}{5}$. The simplification is wrong because dividing only the numerator by 2 ignores the denominator, but both must be divided by 2 for equivalence.

10. What value of $x$ makes $\frac{5}{12}=\frac{x}{36}$?

    $x=15$. Cross-multiplying the proportion yields $5\times36=12x$, solving to $x=15$ for equivalence.

11. What is the rule for simplifying $\frac{a}{b}$ by dividing by a common factor $k$?

    Divide: $\frac{a}{b}=\frac{a\div k}{b\div k}$ if $k$ divides both. Dividing both the numerator and denominator by a common factor $k$ simplifies the fraction while maintaining its equivalent value.

12. What is an equivalent fraction to $\frac{7}{8}$ with numerator $21$?

    $\frac{21}{24}$. To reach numerator 21, multiply numerator and denominator of $\frac{7}{8}$ by 3, producing an equivalent fraction.

13. Which fraction is equivalent to $\frac{7}{9}$: $\frac{21}{27}$ or $\frac{21}{28}$?

    $\frac{21}{27}$. Cross products verify $\frac{21}{27}$ equals $\frac{7}{9}$ as $7\times27=9\times21=189$, unlike $\frac{21}{28}$.

14. Which fraction is equivalent to $\frac{5}{6}$: $\frac{15}{18}$ or $\frac{15}{20}$?

    $\frac{15}{18}$. Cross products show $\frac{15}{18}$ equals $\frac{5}{6}$ as $5\times18=6\times15=90$, unlike $\frac{15}{20}$.

15. What is an equivalent fraction to $\frac{9}{14}$ with numerator $27$?

    $\frac{27}{42}$. Multiplying numerator and denominator of $\frac{9}{14}$ by 3 reaches numerator 27, maintaining the fraction's value.

16. What is the missing denominator $x$ in $\frac{3}{8}=\frac{15}{x}$?

    $x=40$. Cross-multiplying gives $3x=8\times15$, solving to $x=40$ to make the fractions equivalent.

17. What is the simplest form of $\frac{35}{49}$?

    $\frac{5}{7}$. Dividing numerator and denominator by their greatest common divisor of 7 reduces $\frac{35}{49}$ to its simplest form.

18. Identify whether $\frac{4}{9}$ and $\frac{12}{27}$ are equivalent.

    Yes, they are equivalent. Cross products $4\times27=9\times12=108$ confirm the fractions represent the same value.

19. What is the definition of equivalent fractions using cross products?

    $\frac{a}{b}$ and $\frac{c}{d}$ are equivalent if $ad=bc$. Two fractions are equivalent if their cross products are equal, as this confirms they represent the same rational number.

20. What is the simplest form of $\frac{64}{80}$?

    $\frac{4}{5}$. Dividing numerator and denominator by their greatest common divisor of 16 reduces $\frac{64}{80}$ to its simplest form.

21. What is the simplest form of $\frac{42}{56}$?

    $\frac{3}{4}$. Dividing numerator and denominator by their greatest common divisor of 14 simplifies $\frac{42}{56}$ to lowest terms.

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