Comparing Rational Numbers - ISEE Upper Level Quantitative Reasoning

1. Which is greater: $0.2\bar{7}$ or $\frac{5}{18}$?

   They are equal. Dividing $5$ by $18$ yields $0.2\bar{7}$, matching the repeating decimal.

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

   They are equal. The repeating decimal $0.\bar{6} = 0.666\ldots$ equals $\frac{2}{3}$ exactly.

3. Which is greater: $1.2$ or $\frac{6}{5}$?

   They are equal. Dividing $6$ by $5$ gives exactly $1.2$, confirming equivalence.

4. Which is greater: $0.\bar{1}$ or $\frac{1}{9}$?

   They are equal. The repeating decimal $0.\bar{1} = 0.111\ldots$ equals $\frac{1}{9}$ exactly.

5. Which is greater: $0.\bar{3}$ or $\frac{1}{3}$?

   They are equal. The repeating decimal $0.\bar{3} = 0.333\ldots$ equals $\frac{1}{3}$ exactly.

6. Which is greater: $\frac{17}{50}$ or $0.34$?

   They are equal. Dividing $17$ by $50$ yields exactly $0.34$, confirming equivalence.

7. Which is greater: $0.58$ or 58\text{%}?

   They are equal. $58\%$ converts to $0.58$, confirming both represent the same value.

8. Which is greater: $0.04$ or $\frac{1}{20}$?

   $\frac{1}{20}$ is greater. Converting $\frac{1}{20}$ to $0.05$ shows it exceeds $0.04$.

9. Which is greater: $0.\bar{27}$ or $\frac{3}{11}$?

   They are equal. Dividing $3$ by $11$ yields $0.\bar{27}$, matching the repeating decimal.

10. Which is greater: $0.125$ or $\frac{1}{8}$?

    They are equal. Dividing $1$ by $8$ yields exactly $0.125$, matching the decimal value.

11. Which is greater: 125\text{%} or $\frac{5}{4}$?

    They are equal. Converting $125\%$ to $1.25$ matches $\frac{5}{4} = 1.25$.

12. Which is greater: $-\frac{5}{4}$ or $-1.1$?

    $-1.1$ is greater. Converting $-\frac{5}{4} = -1.25$ shows $-1.1 > -1.25$ on the number line.

13. Which is greater: $\frac{5}{12}$ or $\frac{2}{5}$?

    $\frac{5}{12}$ is greater. Cross-multiplying gives $5 \times 5 = 25$ and $2 \times 12 = 24$, so $25 > 24$ implies $\frac{5}{12} > \frac{2}{5}$.

14. What must you do to an inequality when multiplying both sides by a negative number?

    Reverse the inequality sign. This preserves the inequality's direction, as negative multipliers reverse the order of numbers on the number line.

15. Which is greater: -40\text{%} or $-\frac{3}{8}$?

    $-\frac{3}{8}$ is greater. Converting $-40\%$ to $-0.4$ and $-\frac{3}{8} = -0.375$ shows $-0.375 > -0.4$.

16. Which is greater: $\frac{9}{4}$ or $2.2$?

    $\frac{9}{4}$ is greater. Dividing $9$ by $4$ yields $2.25$, which is larger than $2.2$.

17. Which is greater: $0.6$ or $\frac{3}{5}$?

    They are equal. Converting $\frac{3}{5}$ to a decimal yields $0.6$, confirming both values are identical.

18. Which is greater: $-\frac{2}{3}$ or $-0.7$?

    $-\frac{2}{3}$ is greater. On the number line, $-\frac{2}{3} \approx -0.666$ is to the right of $-0.7$, making it larger.

19. Which is greater: $\frac{11}{20}$ or $\frac{3}{5}$?

    $\frac{3}{5}$ is greater. Converting to decimals shows $\frac{11}{20} = 0.55$ and $\frac{3}{5} = 0.6$, confirming $0.6 > 0.55$.

20. Which is greater: $\frac{1}{6}$ or $0.16$?

    $\frac{1}{6}$ is greater. Converting $\frac{1}{6}$ to approximately $0.1667$ shows it exceeds $0.16$.

21. Which is greater: $\frac{13}{25}$ or $0.53$?

    $0.53$ is greater. Converting $\frac{13}{25}$ to $0.52$ shows $0.53 > 0.52$.

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