Real-World Scaling - ISEE Middle Level Quantitative Reasoning

1. What is the scale ratio if $2$ inches on a blueprint represent $5$ feet in real life?

   $2:60$ or simplified $1:30$. Convert 5 feet to 60 inches to form the ratio of blueprint inches to real inches, then simplify by dividing both by 2.

2. What is the new radius if a circle is scaled by $k=0.6$ from radius $10$ cm?

   $6$ cm. Linear dimensions such as radius are multiplied by the scale factor $k=0.6$.

3. What is the new area of a $10$ cm by $6$ cm rectangle scaled by $k=2$?

   $240$ cm$^2$. The original area of 60 cm$^2$ is multiplied by $k^2=4$ to account for the scaling in both dimensions.

4. What is the real volume if a $1:10$ model aquarium holds $2$ liters of water when full?

   $2{,}000$ liters. The real volume is the model volume multiplied by $10^3=1{,}000$, as volume scales with the cube of the linear factor.

5. What is the new number of servings if a recipe makes $6$ servings and is scaled by $k=\frac{4}{3}$?

   $8$ servings. The number of servings scales directly with the recipe scale factor $k=\frac{4}{3}$.

6. What is the map distance if the scale is $1:25{,}000$ and the real distance is $2.5$ km?

   $10$ cm. Divide the real distance, converted to centimeters, by the scale ratio to obtain the map distance in centimeters.

7. What is the new perimeter of a square with side $5$ cm scaled by $k=\frac{3}{2}$?

   $30$ cm. The original perimeter of 20 cm is multiplied by $k=\frac{3}{2}$ because perimeter scales linearly.

8. What happens to perimeter when all linear dimensions are scaled by a factor of $k$?

   Perimeter is multiplied by $k$. Perimeter, being a linear measurement, scales directly with the linear scale factor $k$.

9. What is the real distance in km if a map uses $1$ cm $=$ $2$ km and the map distance is $7.5$ cm?

   $15$ km. Multiply the map distance by the given scale of 2 km per cm to find the real distance.

10. What happens to volume when all linear dimensions are scaled by a factor of $k$?

    Volume is multiplied by $k^3$. Volume scales with the cube of the linear scale factor because it is three-dimensional.

11. What is the new circumference if a circle is scaled by $k=3$ from circumference $8\pi$ cm?

    $24\pi$ cm. Circumference, a linear measurement, is multiplied by the scale factor $k=3$.

12. What is the scale factor $k$ if area increases from $50$ to $200$ square units?

    $k=2$. The scale factor $k$ is the square root of the ratio of new area to original area, as area scales by $k^2$.

13. What is the scale factor from a drawing to the real object if $1$ cm represents $4$ m?

    $400$ (real units per drawing unit). The scale factor is the ratio of real dimensions to drawing dimensions, converting 4 m to 400 cm for consistency with the 1 cm drawing unit.

14. What is the new cost if a recipe is scaled by $k=\frac{5}{2}$ and the original cost is $8$ dollars?

    $20$ dollars. Recipe costs scale linearly with the scale factor $k=\frac{5}{2}$, assuming proportional ingredients.

15. What is the real length if a map scale is $1:50{,}000$ and the map distance is $3$ cm?

    $1{,}500$ m. Multiply the map distance by the scale ratio to find the real distance in centimeters, then convert to meters by dividing by 100.

16. What is the new length if an object is scaled by a factor of $1.5$ from $8$ units?

    $12$ units. Linear dimensions are multiplied by the scale factor to find the new length.

17. What is the new area if a figure is scaled down by $k=\frac{1}{3}$ from $81$ cm$^2$?

    $9$ cm$^2$. The new area is the original area multiplied by left(\frac{1}{3}\right)^2=\frac{1}{9}, reducing 81 cm$^2$ to 9 cm$^2$.

18. What is the new area if a circle is scaled by $k=3$ from area $10\pi$ cm$^2$?

    $90\pi$ cm$^2$. The area of a circle scales by $k^2=9$ when linear dimensions are scaled by $k=3$.

19. What is the scale factor $k$ if volume decreases from $64$ to $8$ cubic units?

    $k=\frac{1}{2}$. The scale factor $k$ is the cube root of the ratio of new volume to original volume, since volume scales by $k^3$.

20. What is the scale factor $k$ if a length changes from $12$ to $9$ units?

    $k=\frac{3}{4}$. The scale factor $k$ is the ratio of the new length to the original length.

21. What is the new volume if a cube with volume $27$ cm$^3$ is scaled by $k=2$?

    $216$ cm$^3$. The new volume is the original volume multiplied by $2^3=8$, increasing 27 cm$^3$ to 216 cm$^3$.

22. What is the model length if a $1:200$ scale model represents a real length of $30$ m?

    $15$ cm. The model length is the real length divided by 200, converting 30 m to 3000 cm before division.

23. What happens to area when all linear dimensions are scaled by a factor of $k$?

    Area is multiplied by $k^2$. Area scales with the square of the linear scale factor due to its two-dimensional nature.

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