Ratio and Proportion - SSAT Upper Level Quantitative

1. What is the value of $x$ if $x$ is $\frac{2}{5}$ of $60$?

   $x=24$. Multiply $60$ by the fraction $\frac{2}{5}$ to find the proportional amount.

2. What is $x$ if $\frac{4}{x}=\frac{10}{25}$?

   $x=10$. Cross-multiply the proportions: $4 \times 25 = 10 \times x$, then divide by $10$.

3. What is $x$ if $\frac{x}{14}=\frac{9}{21}$?

   $x=6$. Cross-multiply: $x \times 21 = 14 \times 9$, then divide by $21$ to isolate $x$.

4. What is the missing value $x$ if $4$ items cost $10 and $x$ items cost $25?

   $x=10$. Set up the proportion $\frac{4}{10} = \frac{x}{25}$ for items to cost, then cross-multiply to solve.

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

   $ad=bc$ (with $b\neq^0$, $d\neq^0$). For equal ratios, the product of the means equals the product of the extremes, ensuring proportionality.

6. What is the scale factor from $8$ to $20$ in a proportional relationship?

   $\frac{5}{2}$. Divide the new value by the original: $20 \div 8 = \frac{5}{2}$, representing the proportional increase.

7. What is each share if $48$ is divided in the ratio $1:3$ (give both parts)?

   $12$ and $36$. Total parts are $4$; divide $48$ by $4$ to get $12$ per part, then multiply by $1$ and $3$.

8. What is the simplest form of the ratio $18:24$?

   $3:4$. Simplify by dividing both terms by their greatest common divisor, which is $6$.

9. What is $x$ if $\frac{x-2}{6}=\frac{3}{4}$?

   $x=\frac{13}{2}$. Cross-multiply to get $4(x-2) = 18$, solve for $x-2$, then add $2$.

10. What does it mean to say two ratios are proportional?

    They are equal: $\frac{a}{b}=\frac{c}{d}$ (with $b\neq^0$, $d\neq^0$). Two ratios are proportional when they express the same relationship, meaning their values are equal under the given conditions.

11. What is the unit rate for a ratio written as $a:b$?

    The per-1 rate: $\frac{a}{b}$ (meaning $\frac{a}{b}$ per $1$). The unit rate converts the ratio to a fraction representing the amount per single unit.

12. What is the total number of parts in the ratio $3:5:2$?

    $10$ parts. Add the individual ratios $3 + 5 + 2$ to find the total divisions of the whole.

13. What is the time in hours for $240$ miles at $60$ mph (constant speed)?

    $4$ hours. Use the time formula: distance divided by speed, $240 \div 60$.

14. What is the missing number $x$ if $\frac{7}{x}=\frac{21}{60}$?

    $x=20$. Cross-multiply: $7 \times 60 = 21 \times x$, then divide by $21$ to solve.

15. What is the constant of proportionality $k$ if $y$ is proportional to $x$?

    $k=\frac{y}{x}$, so $y=kx$. When $y$ varies directly with $x$, their ratio is constant, defining the proportional relationship.

16. What is the larger share if $70$ is divided in the ratio $3:4$?

    $40$. Total parts are $7$; divide $70$ by $7$ to get $10$ per part, then multiply the larger ratio by $10$.

17. What is the speed in mph if $150$ miles are driven in $3$ hours at a constant rate?

    $50$ mph. Use the speed formula: distance divided by time, $150 \div 3$.

18. What is the new length if a $12$ cm segment is scaled by a factor of $\frac{3}{2}$?

    $18$ cm. Multiply the original length by the scale factor: $12 \times \frac{3}{2}$.

19. What is the percent equivalent of the ratio $3:20$?

    $15\%$. Convert the ratio to a fraction $\frac{3}{20}$, multiply by $100$ to get the percentage.

20. What value of $x$ makes the ratios proportional: $7:9=x:45$?

    $x=35$. Set up the proportion and cross-multiply: $7 \times 45 = 9 \times x$, then solve for $x$.

21. Which option is proportional to $6:15$: $2:5$, $4:9$, or $9:20$?

    $2:5$. Simplify $6:15$ to $2:5$; check which option matches this reduced form.

22. What is the missing term $x$ in the proportion $\frac{5}{8}=\frac{15}{x}$?

    $x=24$. Cross-multiply the proportion: $5 \times x = 8 \times 15$, then solve for $x$.

23. What is $x$ if $x:36=5:9$?

    $x=20$. Set up the proportion $\frac{x}{36} = \frac{5}{9}$ and solve by multiplying $36$ by $\frac{5}{9}$.

24. What value of $x$ makes the ratios proportional: $\frac{3}{5}=\frac{x}{20}$?

    $x=12$. Cross-multiply to solve the proportion: $3 \times 20 = 5 \times x$, then divide by $5$.

25. What is the ratio of boys to girls if there are $12$ boys and $18$ girls (simplest form)?

    $2:3$. Express as $12:18$ and divide both by $6$, their greatest common divisor, to simplify.

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