Comparing Fractions - ISEE Lower Level Quantitative Reasoning

1. Which option is greatest: $frac{4}{5}$ or $frac{13}{15}$?

   $\frac{13}{15}$. Cross-multiplying shows $4 \times 15 = 60 < 13 \times 5 = 65$, so $\frac{13}{15}$ is greater.

2. Which option is greatest: $frac{5}{6}$ or $frac{6}{7}$?

   $\frac{6}{7}$. Cross-multiplying shows $5 \times 7 = 35 < 6 \times 6 = 36$, so $\frac{6}{7}$ is greater.

3. Which option is greatest: $frac{3}{7}$ or $frac{5}{7}$?

   $\frac{5}{7}$. With the same denominator, $\frac{5}{7}$ is greater because its numerator is larger, representing more sevenths.

4. Which option is greatest: $frac{2}{9}$ or $frac{1}{4}$?

   $\frac{1}{4}$. Cross-multiplying gives $2 \times 4 = 8 < 1 \times 9 = 9$, indicating $\frac{1}{4}$ is greater.

5. What method compares $frac{a}{b}$ and $frac{c}{d}$ by comparing $ad$ and $bc$ (with $b,d>0$)?

   Cross-multiply: compare $ad$ and $bc$. This method determines which fraction is larger by comparing the cross products $ad$ and $bc$, where if $ad > bc$ then $\frac{a}{b} > \frac{c}{d}$ for positive denominators.

6. Which option is greatest: $frac{9}{10}$ or $frac{17}{20}$?

   $\frac{9}{10}$. Converting to twentieths gives $\frac{9}{10} = \frac{18}{20} > \frac{17}{20}$, or cross-multiplying $9 \times 20 = 180 > 17 \times 10 = 170$.

7. Which option is greatest: $frac{2}{3}$ or $frac{3}{5}$?

   $\frac{2}{3}$. Cross-multiplying gives $2 \times 5 = 10 > 3 \times 3 = 9$, so $\frac{2}{3}$ is greater.

8. Which option is greatest: $frac{10}{21}$ or $frac{1}{2}$?

   $\frac{1}{2}$. Cross-multiplying yields $10 \times 2 = 20 < 1 \times 21 = 21$, so $\frac{1}{2}$ is greater.

9. What is the rule for comparing $frac{a}{b}$ and $frac{a}{d}$ when numerators are equal and $b,d>0$?

   Smaller denominator gives the greater fraction. When numerators are identical and denominators positive, the fraction with the smaller denominator is larger because the whole is divided into fewer parts.

10. Which option is greatest: $frac{8}{9}$ or $frac{15}{16}$?

    $\frac{15}{16}$. Cross-multiplying results in $8 \times 16 = 128 < 15 \times 9 = 135$, confirming $\frac{15}{16}$ is greater.

11. Which option is greatest: $frac{13}{18}$ or $frac{7}{9}$?

    $\frac{7}{9}$. Cross-multiplying gives $13 \times 9 = 117 < 7 \times 18 = 126$, so $\frac{7}{9}$ is greater.

12. Which option is greatest: $frac{14}{27}$ or $frac{1}{2}$?

    $\frac{14}{27}$. Cross-multiplying yields $14 \times 2 = 28 > 1 \times 27 = 27$, so $\frac{14}{27}$ is greater.

13. Which option is greatest: $frac{6}{11}$ or $frac{7}{13}$?

    $\frac{6}{11}$. Cross-multiplying shows $6 \times 13 = 78 > 7 \times 11 = 77$, indicating $\frac{6}{11}$ is greater.

14. Which option is greatest: $frac{4}{9}$ or $frac{4}{11}$?

    $\frac{4}{9}$. With the same numerator, $\frac{4}{9}$ is greater because its smaller denominator divides the whole into fewer parts, making each part larger.

15. Which option is greatest: $frac{5}{14}$ or $frac{3}{7}$?

    $\frac{3}{7}$. Converting to fourteenths gives $\frac{3}{7} = \frac{6}{14} > \frac{5}{14}$, or cross-multiplying $5 \times 7 = 35 < 3 \times 14 = 42$.

16. What is the rule for comparing $frac{a}{b}$ and $frac{c}{b}$ when denominators are equal and $b>0$?

    Greater numerator gives the greater fraction. When denominators are identical and positive, the fraction with the larger numerator represents a greater value since it takes a bigger portion of the same whole.

17. Which option is greatest: $frac{11}{15}$ or $frac{3}{4}$?

    $\frac{3}{4}$. Cross-multiplying results in $11 \times 4 = 44 < 3 \times 15 = 45$, indicating $\frac{3}{4}$ is greater.

18. Which option is greatest: $frac{12}{25}$ or $frac{5}{11}$?

    $\frac{12}{25}$. Cross-multiplying shows $12 \times 11 = 132 > 5 \times 25 = 125$, confirming $\frac{12}{25}$ is greater.

19. Which option is greatest: $frac{5}{12}$ or $frac{3}{8}$?

    $\frac{5}{12}$. Cross-multiplying yields $5 \times 8 = 40 > 3 \times 12 = 36$, confirming $\frac{5}{12}$ is greater.

20. Which option is greatest: $frac{7}{15}$ or $frac{9}{20}$?

    $\frac{7}{15}$. Cross-multiplying results in $7 \times 20 = 140 > 9 \times 15 = 135$, indicating $\frac{7}{15}$ is greater.

21. Which option is greatest: $frac{1}{2}$ or $frac{4}{9}$?

    $\frac{1}{2}$. Cross-multiplying yields $1 \times 9 = 9 > 4 \times 2 = 8$, so $\frac{1}{2}$ is greater.

22. Which option is greatest: $frac{3}{10}$ or $frac{7}{20}$?

    $\frac{7}{20}$. Converting to twentieths yields $\frac{3}{10} = \frac{6}{20} < \frac{7}{20}$, or cross-multiplying $3 \times 20 = 60 < 7 \times 10 = 70$.

23. Which option is greatest: $frac{7}{12}$ or $frac{5}{8}$?

    $\frac{5}{8}$. Cross-multiplying yields $7 \times 8 = 56 < 5 \times 12 = 60$, confirming $\frac{5}{8}$ is greater.

24. Which option is greatest: $frac{9}{14}$ or $frac{5}{7}$?

    $\frac{5}{7}$. Cross-multiplying gives $9 \times 7 = 63 < 5 \times 14 = 70$, confirming $\frac{5}{7}$ is greater.

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