Scaling and Unit Rates - ISEE Upper Level Quantitative Reasoning

1. What is the cost for $8$ lb if the unit price is $2.75 per lb?

   $22. Multiply the unit price by the number of pounds to compute the total cost.

2. What is the unit rate in gallons per minute for $15$ gallons in $5$ minutes?

   $3$ gal/min. Divide the total gallons by the total minutes to find the unit rate in gallons per minute.

3. What is the scale factor if the volume changes from $27$ to $216$ for similar solids?

   $2$. The linear scale factor is the cube root of the ratio of new volume to original volume.

4. What is the scale factor from an original length of $12$ to a new length of $18$?

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

5. What is the actual distance if a map scale is $1$ in $=$ $20$ mi and the map shows $3.5$ in?

   $70$ mi. Multiply the map distance by the scale ratio to find the actual distance.

6. What is the volume scale factor when all linear dimensions are multiplied by $k$?

   $k^3$. Volumes scale with the cube of the linear dimensions, giving a factor of $k^3$.

7. What is the new length if a $24$ m distance is reduced by a scale factor of $\frac{1}{3}$?

   $8$ m. Multiply the original distance by the scale factor to determine the reduced length.

8. What is the unit rate in words per minute for $450$ words typed in $9$ minutes?

   $50$ wpm. Divide the total words by the total minutes to determine the unit rate in words per minute.

9. What is the unit rate in dollars per item for $18 for $6$ notebooks?

   $3 per notebook. Divide the total cost by the number of notebooks to obtain the unit rate in dollars per notebook.

10. What is the area scale factor when all linear dimensions are multiplied by $k$?

    $k^2$. Areas scale with the square of the linear dimensions, resulting in a factor of $k^2$.

11. What is the new area if a rectangle with area $48$ is scaled by a linear factor of $\frac{1}{2}$?

    $12$. Multiply the original area by the square of the linear scale factor to find the new area.

12. What is the new volume if a solid with volume $5$ is scaled by a factor of $2$?

    $40$. Multiply the original volume by the cube of the scale factor to determine the new volume.

13. What is the unit rate in dollars per pound for $7.50 for $3$ lb?

    $2.50 per lb. Divide the total cost by the number of pounds to find the unit rate in dollars per pound.

14. What is the scale factor if a model length is $9$ cm and the actual length is $36$ cm?

    $4$. Divide the actual length by the model length to find the scale factor from model to actual.

15. What is the new area if a figure with area $20$ is scaled by a factor of $3$?

    $180$. Multiply the original area by the square of the scale factor to find the new area.

16. What is the new length if a $10$ cm segment is enlarged by a scale factor of $1.4$?

    $14$ cm. Multiply the original length by the scale factor to find the enlarged length.

17. What is the new circumference if a circle with circumference $30$ is scaled by $\frac{2}{3}$?

    $20$. Multiply the original circumference by the scale factor, as circumference is a linear dimension.

18. What is the perimeter scale factor when all side lengths are multiplied by $k$?

    $k$. The perimeter scales linearly with the side lengths, so the scale factor is $k$.

19. What is the scale factor if the area changes from $16$ to $64$ for similar figures?

    $2$. The linear scale factor is the square root of the ratio of new area to original area.

20. What is the distance traveled in $2.5$ hours at a unit rate of $48$ mph?

    $120$ miles. Multiply the speed by the time to calculate the distance traveled.

21. What is the map distance if the scale is $1$ cm $=$ $5$ km and the actual distance is $40$ km?

    $8$ cm. Divide the actual distance by the scale ratio to determine the map distance.

22. What is the unit rate in dollars per hour for $84 earned in $7$ hours?

    $12 per hour. Divide the total earnings by the total hours to calculate the unit rate in dollars per hour.

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

    $50$ mph. Divide the total miles by the total hours to determine the unit rate in miles per hour.

24. What is the unit rate in minutes per mile for $24$ minutes to run $3$ miles?

    $8$ min/mi. Divide the total minutes by the number of miles to find the unit rate in minutes per mile.

25. What is the time per item if $45$ items are produced in $9$ minutes?

    $\frac{1}{5}$ min/item. Divide the total time by the number of items to find the unit rate in minutes per item.

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