Scale and Rate Problems - SSAT Upper Level Quantitative

1. How long does it take to travel $150\text{ km}$ at $75\text{ km/h}$?

   $2\text{ h}$. Divide the distance by the speed to determine the time required.

2. What is the speed if a car travels $120\text{ miles}$ in $3\text{ hours}$?

   $40\text{ mph}$. Divide the total distance by the total time to find the constant speed.

3. State the scale factor formula when map distance is $d_m$ and actual distance is $d_a$.

   $k=\frac{d_a}{d_m}$. The scale factor is the ratio of actual distance to map distance.

4. What is the actual distance if the scale is $1:50\,000$ and the map shows $3\text{ cm}$?

   $150\,000\text{ cm}$. Multiply the map distance by the scale ratio to convert to actual distance.

5. What is the volume scale factor if the linear scale factor is $k$?

   $k^3$. Volumes of similar figures scale by the cube of the linear scale factor.

6. What is the map distance if the scale is $1:20$ and the actual length is $60\text{ m}$?

   $3\text{ m}$. Divide the actual distance by the scale ratio to find the map distance.

7. What is the density if a substance has mass $30\text{ g}$ and volume $10\text{ cm}^3$?

   $3\text{ g/cm}^3$. Divide the mass by the volume to compute the density.

8. Find the average speed for $60\text{ mi}$ at $30\text{ mph}$ then $60\text{ mi}$ at $60\text{ mph}$.

   $40\text{ mph}$. Compute total distance over total time for varying speeds to find average speed.

9. State the unit rate formula for speed using distance $d$ and time $t$.

   $v=\frac{d}{t}$. Speed is calculated as the distance traveled divided by the time taken.

10. What is the area scale factor if the linear scale factor is $k$?

    $k^2$. Areas of similar figures scale by the square of the linear scale factor.

11. If a model uses linear scale $1:5$, what is the ratio of model volume to actual volume?

    $1:125$. Cube the linear scale ratio to obtain the volume ratio of model to actual.

12. State the average speed formula for total distance $D$ and total time $T$.

    $\text{avg speed}=\frac{D}{T}$. Average speed equals total distance divided by total time elapsed.

13. State the formula for density using mass $m$ and volume $V$.

    $\rho=\frac{m}{V}$. Density is mass divided by the volume it occupies.

14. What is the combined rate if worker A does $\frac{1}{4}$ job/h and worker B does $\frac{1}{6}$ job/h?

    $\frac{5}{12}\text{ job/h}$. Sum the individual work rates to obtain the combined rate.

15. What is the scale factor from figure A to figure B if a side changes from $8$ to $12$?

    $\frac{3}{2}$. Divide the new side length by the original to find the scale factor.

16. If a model uses linear scale $1:10$, what is the ratio of model area to actual area?

    $1:100$. Square the linear scale ratio to obtain the area ratio of model to actual.

17. State the proportion to convert map distance $d_m$ using scale $1:n$ to actual distance $d_a$.

    $\frac{d_m}{d_a}=\frac{1}{n}$. The ratio of map distance to actual distance equals the reciprocal of the scale denominator.

18. If the linear scale factor from A to B is $\frac{2}{3}$, what is the area scale factor from A to B?

    $\frac{4}{9}$. Square the linear scale factor to determine the area scale factor.

19. What is the unit price if $12$ items cost $18?

    $\$1.50\text{ per item}$. Divide the total cost by the number of items to find the price per item.

20. Identify the scale $1:n$ if $2\text{ cm}$ on a map represents $10\text{ km}$.

    $1:500\,000$. Convert actual distance to cm and divide by map distance to determine $n$.

21. How far do you travel in $0.5\text{ h}$ at $60\text{ mph}$?

    $30\text{ miles}$. Multiply the speed by the time to calculate the distance traveled.

22. How long to finish a job if one person takes $6\text{ h}$ and another takes $3\text{ h}$ working together?

    $2\text{ h}$. Add individual rates and take the reciprocal to find combined time for one job.

23. State the formula for price per unit if total cost is $C$ for $q$ units.

    $\text{unit price}=\frac{C}{q}$. Unit price is the total cost divided by the quantity of units.

24. State the work rate formula if one job takes $t$ hours at constant rate.

    $\text{rate}=\frac{1}{t}$. Work rate is the reciprocal of the time to complete one full job.

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