Ratios and Comparisons - ISEE Upper Level Quantitative Reasoning

1. What is the missing number $x$ in $\frac{7}{x}=\frac{21}{30}$?

   $10$. Cross-multiply to get $7 \times 30 = 21x$, then solve for $x$.

2. What is the scale factor from $2:5$ to $6:15$?

   $3$. Each term in the first ratio is multiplied by 3 to obtain the second ratio.

3. What does the ratio $a:b$ mean in fraction form?

   $\frac{a}{b}$. The ratio $a:b$ expresses the comparison of $a$ to $b$ as the fraction $\frac{a}{b}$.

4. What is the unit rate (miles per hour) for $150$ miles in $3$ hours?

   $50\ \text{mph}$. Divide total miles by total hours to find the rate per hour.

5. What is the ratio of $3$ feet to $18$ inches in simplest form?

   $2:1$. Convert 3 feet to 36 inches, then simplify 36:18 by dividing both by 18.

6. If the ratio of cats to dogs is $3:5$ and there are $40$ dogs, how many cats are there?

   $24$. Dogs represent 5 parts equaling 40, so 1 part is 8, and cats are 3 parts.

7. What is the missing number $x$ in the proportion $5:8=x:40$?

   $25$. Solve by cross-multiplying: $5 \times 40 = 8x$, so $x = \frac{200}{8}$.

8. What is the ratio of $\frac{3}{4}$ to $\frac{5}{8}$ in simplest form?

   $6:5$. Divide $\frac{3}{4}$ by $\frac{5}{8}$ to get $\frac{6}{5}$, or find a common denominator to compare.

9. What is the ratio of shaded to total if $9$ of $12$ squares are shaded, simplified?

   $3:4$. Shaded is 9 and total is 12, simplify by dividing both by 3.

10. Which is larger: $\frac{3}{5}$ or $\frac{5}{9}$?

    $\frac{3}{5}$. Compare by cross-multiplying: $3 \times 9 = 27$ and $5 \times 5 = 25$, since 27 > 25.

11. What is the unit price (dollars per pound) for $9 for $3$ lb?

    $\$3\ \text{per lb}$. Divide total cost by total pounds to find the price per pound.

12. What is the cross-multiplication condition for $\frac{a}{b}=\frac{c}{d}$?

    $ad=bc$. Cross-multiplication checks equality by verifying if the products of numerator and opposite denominator are equal.

13. What is the missing number $x$ in the proportion $\frac{3}{4}=\frac{x}{20}$?

    $15$. Solve by cross-multiplying or multiplying both sides by 20 and then by $\frac{3}{4}$.

14. Find and correct the ratio error: simplifying $18:24$ as $9:10$.

    Correct: $18:24=3:4$. Simplify 18:24 by dividing both by 6, not incorrectly by 2 to get 9:12 or other errors.

15. If the ratio is $2:3$ and the first quantity is $14$, what is the second quantity?

    $21$. The first quantity 14 is 7 times 2, so multiply 3 by 7 to find the second.

16. A recipe uses flour to sugar in a $5:2$ ratio. If flour is $20$ cups, how much sugar is needed?

    $8\ \text{cups}$. Flour is 5 parts equaling 20 cups, so 1 part is 4 cups, and sugar is 2 parts.

17. If the ratio is $4:7$ and the total is $33$, what is the first quantity?

    $12$. Total parts are 11, divide 33 by 11 to get 3 per part, then multiply by 4.

18. What is the ratio of girls to boys if there are $12$ boys and $8$ girls, simplified?

    $2:3$. Girls are 8 and boys are 12, so simplify 8:12 by dividing both by 4.

19. What is the ratio of boys to total students if there are $12$ boys and $8$ girls?

    $3:5$. Total students are 20, so simplify 12:20 by dividing both by 4.

20. What is the ratio of $0.6$ to $0.15$ in simplest form?

    $4:1$. Divide 0.6 by 0.15 to get 4, or multiply both by 100 and simplify 60:15 by dividing by 15.

21. Identify the equivalent ratio to $4:9$ among $8:18$, $6:16$, and $12:20$.

    $8:18$. Check equivalents: 8:18 simplifies to 4:9 by dividing by 2, while others do not.

22. What is the ratio of $45$ minutes to $2$ hours in simplest form?

    $3:8$. Convert 2 hours to 120 minutes, then simplify 45:120 by dividing both by 15.

23. What is the ratio of $12$ to $18$ in simplest form?

    $2:3$. Simplify by dividing both 12 and 18 by their greatest common divisor of 6.

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