Proportional Scaling - ISEE Lower Level Quantitative Reasoning

1. What is the volume scale factor when the linear scale factor is $\frac{2}{3}$?

   Volume factor $=\left(\frac{2}{3}\right)^3=\frac{8}{27}$. The volume scale factor is the cube of the linear scale factor for similar solids.

2. What is the time needed to travel $150$ miles if you travel $100$ miles in $2$ hours?

   $3$ hours. Use constant speed to find time by dividing new distance by speed.

3. What is the new volume if the original volume is $27$ and the linear scale factor is $\frac{1}{3}$?

   $1$. Volumes scale by the cube of the linear factor, reducing the original volume accordingly.

4. 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 given scale factor to determine the corresponding new length.

5. What is the area scale factor when the linear scale factor is $5$?

   Area factor $=5^2=25$. The area scale factor is the square of the linear scale factor for similar figures.

6. What real distance corresponds to $3.5$ in on a map with scale $1$ in : $8$ mi?

   $28$ miles. Multiply the map distance by the scale ratio to find the actual distance.

7. What map distance corresponds to $30$ mi on a map with scale $1$ in : $6$ mi?

   $5$ in. Divide the actual distance by the scale ratio to find the map distance.

8. What is the missing value $x$ in the proportion $\frac{x}{9}=\frac{14}{21}$?

   $6$. Cross-multiply in the proportion and solve for the missing variable.

9. What is the cost for $9$ pounds if $6$ pounds cost $15 and cost is proportional to weight?

   $22.50. Use the proportional relationship by multiplying the unit cost by the new weight.

10. What is the scale factor from a model height of $9$ cm to an actual height of $1.8$ m?

    Scale factor $=\frac{180}{9}=20$. Convert units and divide actual height by model height for the scale factor.

11. What is the new width if a $10$ by $6$ rectangle is scaled so the length becomes $25$?

    New width $=15$. Determine the scale factor from lengths and apply it to the width for similarity.

12. What is the new area if a circle with area $36\pi$ is scaled by linear factor $\frac{1}{3}$?

    $4\pi$. Circle areas scale by the square of the linear factor, reducing accordingly.

13. What is the linear scale factor if volume changes from $8$ to $64$ for similar solids?

    Linear factor $=\sqrt[3]{\frac{64}{8}}=2$. The linear scale factor is the cube root of the ratio of new volume to original volume.

14. What is the original length if the new length is $24$ and the scale factor is $3$?

    $8$. Divide the new length by the scale factor to find the corresponding original length.

15. What is the linear scale factor if area changes from $18$ to $72$ for similar figures?

    Linear factor $=\sqrt{\frac{72}{18}}=2$. The linear scale factor is the square root of the ratio of new area to original area.

16. What is the scale factor from $9$ inches to $6$ inches?

    Scale factor $=\frac{6}{9}=\frac{2}{3}$. The scale factor is the ratio of the new measurement to the original, simplified for a reduction.

17. What is the distance for $5$ hours if you travel $180$ miles in $3$ hours at constant speed?

    $300$ miles. Determine constant speed and multiply by the new time to find the distance.

18. What is the new circumference if a circle with circumference $14\pi$ is scaled by factor $\frac{3}{2}$?

    $21\pi$. Circumferences scale linearly, so multiply original by the scale factor.

19. What is the missing value $x$ in the proportion $\frac{4}{7}=\frac{x}{21}$?

    $12$. Solve the proportion by cross-multiplying and dividing to find the missing value.

20. What is the scale factor from a $1$ in map distance to a $5$ mi real distance?

    Scale $=1$ in : $5$ mi. The scale expresses the ratio of map distance to actual distance.

21. What is the new area if a figure with area $16$ is scaled by linear factor $\frac{1}{2}$?

    $4$. Areas scale by the square of the linear factor, reducing the original area accordingly.

22. What is the new area if the original area is $50$ and the linear scale factor is $3$?

    $450$. Multiply the original area by the square of the linear scale factor to find the new area.

23. What is the scale factor from an original length of $6$ to a new length of $15$?

    Scale factor $=\frac{15}{6}=\frac{5}{2}$. The scale factor is calculated as the ratio of the new length to the original length, simplified to lowest terms.

24. What is the new perimeter if a figure with perimeter $30$ is scaled by factor $\frac{4}{3}$?

    $40$. Perimeters scale linearly, so multiply the original perimeter by the linear scale factor.

Study these flashcards