Comparing Rational Forms - SSAT Upper Level Quantitative

1. What is the quickest method to compare $\frac{a}{b}$ and $\frac{c}{d}$ for positive $b,d$?

   Compare $ad$ and $bc$ (cross-multiply). Cross-multiplication determines the relationship without converting to decimals or finding common denominators, as $ad > bc$ implies $\frac{a}{b} > \frac{c}{d}$.

2. Which is greater: $\frac{-7}{8}$ or $-0.9$?

   $-\frac{7}{8}$. Approximating $-\frac{7}{8} = -0.875$, which has a smaller magnitude than $-0.9$, placing it higher on the number line.

3. Which is greater: $\frac{5}{6}$ or $83\%$?

   $\frac{5}{6}$. Converting $83\%$ to $0.83$ and $\frac{5}{6} \approx 0.8333$ shows the fraction exceeds it slightly.

4. Which is greater: $0.\overline{6}$ or $\frac{2}{3}$?

   They are equal. Converting $0.\overline{6}$ using the formula for repeating decimals yields exactly $\frac{2}{3}$.

5. Which is smaller: $\frac{1}{4}$ or $25\%$?

   They are equal. Converting $25\%$ to $\frac{25}{100} = \frac{1}{4}$ shows both are identical.

6. What is the rule to convert a terminating decimal to a fraction?

   Write over $10^n$ and simplify. The number of decimal places determines the power of $10$ in the denominator, allowing simplification to the equivalent fraction.

7. What is the comparison result: $\frac{2}{5}$ $\square$ $40\%$?

   $=$. Both $\frac{2}{5} = 0.4$ and $40\% = 0.4$ represent the same value.

8. Which is greater: $\frac{3}{4}$ or $0.7$?

   $\frac{3}{4}$. Converting the fraction to $0.75$ shows it exceeds $0.7$ on the number line.

9. Which is greater: $-\frac{3}{5}$ or $-0.55$?

   $-0.55$. Since $|-0.55| < |-0.6|$, $-0.55$ is positioned to the right of $-\frac{3}{5} \approx -0.6$ on the number line.

10. Which is less: $-\frac{5}{6}$ or $-0.8$?

    $-\frac{5}{6}$. Approximating $-\frac{5}{6} \approx -0.833$, which is less than $-0.8$ since it is further left on the number line.

11. Which is less: $\frac{9}{20}$ or $0.46$?

    $\frac{9}{20}$. Converting $\frac{9}{20} = 0.45$ reveals it is smaller than $0.46$ by $0.01$.

12. What is the sign rule when comparing two negative rational numbers?

    The number with the smaller absolute value is greater. For negative numbers, the one closer to zero (smaller magnitude) is larger on the number line.

13. What is $0.\overline{3}$ written as a fraction in simplest form?

    $\frac{1}{3}$. The repeating decimal $0.\overline{3}$ represents an infinite series summing to $\frac{1}{3}$.

14. Which is greater: $\frac{4}{9}$ or $0.44\overline{4}$?

    They are equal. Dividing $4$ by $9$ produces the repeating decimal $0.\overline{4}$, matching the given form.

15. Which is greater: $0.375$ or $\frac{3}{10}$?

    $0.375$. Expressing $\frac{3}{10} = 0.3$ shows $0.375$ is larger due to the higher tenths and hundredths digits.

16. Which is greater: $\frac{11}{12}$ or $0.91$?

    $\frac{11}{12}$. Converting yields $\frac{11}{12} \approx 0.9167 > 0.91$, due to the higher hundredths place.

17. Which is greater: $\frac{13}{15}$ or $0.86\overline{6}$?

    They are equal. Dividing $13$ by $15$ yields $0.8\overline{6}$, exactly matching the given decimal.

18. Which is greater: $0.125$ or $\frac{1}{9}$?

    $0.125$. Comparing decimals, $0.125 > 0.\overline{1} \approx 0.111$, as the former has a higher tenths digit.

19. Identify the larger number: $\frac{5}{8}$ or $0.62$.

    $\frac{5}{8}$. Dividing $5$ by $8$ yields $0.625$, which is greater than $0.62$.

20. Which is greater: $37.5\%$ or $\frac{3}{8}$?

    They are equal. Expressing $37.5\%$ as $\frac{37.5}{100} = \frac{3}{8}$ confirms their equivalence.

21. What fraction is equal to $0.\overline{12}$ in simplest form?

    $\frac{4}{33}$. Applying the conversion method for two-digit repeating decimals results in the simplified fraction $\frac{4}{33}$.

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