Proportional Scaling - SSAT Middle Level Quantitative

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

   $40$. Perimeters of similar figures scale linearly by the scale factor.

2. What is the rule for how perimeter changes under a scale factor $k$?

   Perimeter multiplies by $k$. Since perimeters are sums of linear dimensions, they scale by the factor $k$.

3. What is the original volume if the new volume is $216$ and the linear scale factor is $3$?

   $8$. Divide the new volume by the cube of the scale factor to find the original volume.

4. What is the original length if the new length is $21$ and the scale factor is $\frac{3}{2}$?

   $14$. Divide the new length by the scale factor to recover the original length.

5. What is the new volume if a volume of $12$ is scaled by a factor of $2$ in all dimensions?

   $96$. Multiply the original volume by the cube of the scale factor for similar solids.

6. What is the scale factor $k$ if a model is $\frac{1}{25}$ of the actual object in every length?

   $k = \frac{1}{25}$. The scale factor is the ratio of model length to actual length when the model is smaller.

7. What drawing length (in inches) represents $18$ feet if the scale is $1$ inch $=$ $6$ feet?

   $3$ inches. Divide the actual length by the scale conversion factor to find the drawing length.

8. What is the scale factor if a length changes from $8$ to $14$?

   $\frac{7}{4}$. Divide the new length by the original length to compute the scale factor.

9. What is the new area if a square with side $7$ is scaled by a factor of $\frac{2}{7}$?

   $4$. Square the scale factor and multiply by the original area of $49$ to get the scaled area.

10. What actual length (in feet) does $7$ inches represent if the scale is $1$ inch $=$ $3$ feet?

    $21$ feet. Multiply the drawing length by the scale conversion factor to obtain the actual length.

11. What is the constant of proportionality $k$ if $y$ is proportional to $x$ and $y=15$ when $x=6$?

    $k = \frac{5}{2}$. Divide the given $y$ by the corresponding $x$ to find the constant of proportionality.

12. What is the scale factor from an original length $L$ to a new length $L'$?

    $k = \frac{L'}{L}$. The scale factor is defined as the ratio of the new length to the original length in proportional scaling.

13. What is the scale factor if a drawing uses $1$ inch to represent $4$ feet?

    $1:48$. Convert feet to inches to express the ratio of drawing length to actual length as 1:48.

14. What is the scale factor from $15$ to $10$, expressed as a simplified fraction?

    $\frac{2}{3}$. Divide the new value by the original to express the scale factor as a simplified fraction.

15. What is $y$ if $y$ is proportional to $x$, $y=10$ when $x=4$, and $x=14$?

    $35$. Determine $k$ from known values and multiply by the new $x$ to find the corresponding $y$.

16. What is the missing value $x$ in the proportion $\frac{3}{5}=\frac{x}{20}$?

    $12$. Cross-multiply the proportion and solve for the unknown value $x$.

17. What is the new length if an original length is $12$ and the scale factor is $\frac{3}{2}$?

    $18$. Multiply the original length by the scale factor to determine the scaled length.

18. What is the rule for how volume changes under a scale factor $k$ in all dimensions?

    Volume multiplies by $k^3$. Volumes of similar solids scale by the cube of the linear scale factor.

19. What is the new area if an area of $50$ is scaled by a factor of $3$?

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

20. What is the new width if a rectangle has width $9$ and is reduced by a scale factor of $\frac{2}{3}$?

    $6$. Apply the scale factor to the original width by multiplication to get the reduced width.

21. What is the missing value $x$ in the proportion $\frac{7}{x}=\frac{21}{30}$?

    $10$. Cross-multiply the proportion to equate products and solve for $x$.

22. What is the scale factor $k$ if area changes from $36$ to $81$ for similar figures?

    $\frac{3}{2}$. Take the square root of the ratio of new area to original area for similar figures.

23. What is the new length if $5$ cm on a map represents $40$ km, and the map length is $8$ cm?

    $64$ km. Use the unit rate from the given scale to multiply by the new map length for actual distance.

24. What is the rule for how area changes under a scale factor $k$?

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

25. What is the scale factor $k$ if volume changes from $8$ to $27$ for similar solids?

    $\frac{3}{2}$. Take the cube root of the ratio of new volume to original volume for similar solids.

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