Ratios, Rates, and Proportions - Quantitative Reasoning

1. What is the ratio of areas of similar figures if the side-length ratio is $3:5$?

   $9:25$. Areas of similar figures scale with the square of the linear dimensions ratio.

2. What is the definition of a rate?

   A ratio comparing quantities with different units. A rate is a special ratio that compares two quantities measured in different units, such as speed or density.

3. What is the definition of the ratio $a:b$ in fraction form?

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

4. What is the value of $x$ if $\frac{x}{15}=\frac{2}{5}$?

   $6$. Solve the proportion by cross-multiplying: $x \cdot 5 = 15 \cdot 2$, so $x=30/5=6$.

5. What is the part-to-whole fraction for $b$ if $a:b=2:5$?

   $\frac{5}{7}$. The whole is $2+5=7$ parts, so $b$ represents $5$ out of $7$.

6. What is the ratio of volumes of similar solids if the side-length ratio is $2:7$?

   $8:343$. Volumes of similar solids scale with the cube of the linear dimensions ratio.

7. State the cross-multiplication rule for $\frac{a}{b}=\frac{c}{d}$.

   $ad=bc$. Cross-multiplying the proportion $\frac{a}{b}=\frac{c}{d}$ yields $ad=bc$ to verify equality or solve for unknowns.

8. What is the larger part if two parts are in ratio $3:5$ and the total is $64$?

   $40$. Total parts are 8; each part is $64/8=8$, so larger is $5 \times 8$.

9. State the general equation for inverse variation between $x$ and $y$ with constant $k$.

   $y=\frac{k}{x}$. Inverse variation means $y$ decreases as $x$ increases, with product $xy$ constant.

10. What is the total if two parts are in ratio $3:5$ and the smaller part is $21$?

    $56$. The smaller part (3 units) is 21, so each unit is 7; total is $8 \times 7$.

11. What is the mixture ratio of water to concentrate if $2$ liters of concentrate are added to $8$ liters of water?

    $4:1$. The ratio water:concentrate is 8:2, simplified by dividing by 2.

12. 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.

13. What is the part-to-whole fraction for $a$ if $a:b=2:5$?

    $\frac{2}{7}$. The whole is $2+5=7$ parts, so $a$ represents $2$ out of $7$.

14. Identify the constant of proportionality $k$ if $y$ is proportional to $x$ and $y=18$ when $x=6$.

    $k=3$. In direct proportion $y=kx$, substitute values to solve for $k=18/6$.

15. What is the percent of the whole represented by $3$ parts out of $8$ parts?

    $37.5\%$. Convert the fraction $3/8$ to a percentage: $(3/8) \times 100$.

16. What is $y$ if $y$ is directly proportional to $x$, $y=10$ when $x=4$, and $x=12$?

    $30$. Find $k=10/4=2.5$, then $y=2.5 \times 12$ for the new $x$.

17. What is the unit rate if $180$ miles are traveled in $3$ hours?

    $60$ miles per hour. Divide total distance by total time to find the rate per unit time.

18. What is $a:c$ in simplest integer form if $a:b=3:7$ and $b:c=14:5$?

    $6:5$. Simplify the ratio $a:c=6:5$ after combining and reducing to integers.

19. What is the ratio after scaling both terms of $4:7$ by $3$?

    $12:21$. Scaling a ratio by a constant multiplies each term by that factor, preserving the ratio.

20. What is the combined ratio of $a:b=2:3$ and $b:c=4:5$ as $a:b:c$?

    $8:12:15$. Align the common term $b$ by scaling to make it equal (12), then combine into a single ratio.

21. What is the value of $x$ if $x:8=3:4$?

    $6$. Convert the ratio to a proportion $x/8=3/4$ and solve by cross-multiplying.

22. What is the definition of a proportion?

    An equation stating that two ratios are equal. A proportion sets two ratios equal, indicating that the relationships between the pairs are equivalent.

23. What is the speed in miles per hour for $150$ miles in $2.5$ hours?

    $60$ miles per hour. Compute the rate by dividing distance by time: $150 / 2.5$.

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