Real-World Scaling - ISEE Middle Level: Quantitative Reasoning
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What is the scale factor from a drawing to the real object if $1$ cm represents $4$ m?
What is the scale factor from a drawing to the real object if $1$ cm represents $4$ m?
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$400$ (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.
$400$ (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.
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What is the scale factor $k$ if volume decreases from $64$ to $8$ cubic units?
What is the scale factor $k$ if volume decreases from $64$ to $8$ cubic units?
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$k= rac{1}{2}$. The scale factor $k$ is the cube root of the ratio of new volume to original volume, since volume scales by $k^3$.
$k= rac{1}{2}$. The scale factor $k$ is the cube root of the ratio of new volume to original volume, since volume scales by $k^3$.
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What is the new length if an object is scaled by a factor of $1.5$ from $8$ units?
What is the new length if an object is scaled by a factor of $1.5$ from $8$ units?
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$12$ units. Linear dimensions are multiplied by the scale factor to find the new length.
$12$ units. Linear dimensions are multiplied by the scale factor to find the new length.
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What happens to perimeter when all linear dimensions are scaled by a factor of $k$?
What happens to perimeter when all linear dimensions are scaled by a factor of $k$?
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Perimeter is multiplied by $k$. Perimeter, being a linear measurement, scales directly with the linear scale factor $k$.
Perimeter is multiplied by $k$. Perimeter, being a linear measurement, scales directly with the linear scale factor $k$.
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What happens to area when all linear dimensions are scaled by a factor of $k$?
What happens to area when all linear dimensions are scaled by a factor of $k$?
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Area is multiplied by $k^2$. Area scales with the square of the linear scale factor due to its two-dimensional nature.
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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What happens to volume when all linear dimensions are scaled by a factor of $k$?
What happens to volume when all linear dimensions are scaled by a factor of $k$?
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Volume is multiplied by $k^3$. Volume scales with the cube of the linear scale factor because it is three-dimensional.
Volume is multiplied by $k^3$. Volume scales with the cube of the linear scale factor because it is three-dimensional.
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What is the new area of a $10$ cm by $6$ cm rectangle scaled by $k=2$?
What is the new area of a $10$ cm by $6$ cm rectangle scaled by $k=2$?
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$240$ cm$^2$. The original area of 60 cm$^2$ is multiplied by $k^2=4$ to account for the scaling in both dimensions.
$240$ cm$^2$. The original area of 60 cm$^2$ is multiplied by $k^2=4$ to account for the scaling in both dimensions.
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What is the scale factor $k$ if area increases from $50$ to $200$ square units?
What is the scale factor $k$ if area increases from $50$ to $200$ square units?
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$k=2$. The scale factor $k$ is the square root of the ratio of new area to original area, as area scales by $k^2$.
$k=2$. The scale factor $k$ is the square root of the ratio of new area to original area, as area scales by $k^2$.
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What is the new perimeter of a square with side $5$ cm scaled by $k= rac{3}{2}$?
What is the new perimeter of a square with side $5$ cm scaled by $k= rac{3}{2}$?
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$30$ cm. The original perimeter of 20 cm is multiplied by $k= rac{3}{2}$ because perimeter scales linearly.
$30$ cm. The original perimeter of 20 cm is multiplied by $k= rac{3}{2}$ because perimeter scales linearly.
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What is the real distance in km if a map uses $1$ cm $=$ $2$ km and the map distance is $7.5$ cm?
What is the real distance in km if a map uses $1$ cm $=$ $2$ km and the map distance is $7.5$ cm?
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$15$ km. Multiply the map distance by the given scale of 2 km per cm to find the real distance.
$15$ km. Multiply the map distance by the given scale of 2 km per cm to find the real distance.
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What is the new number of servings if a recipe makes $6$ servings and is scaled by $k= rac{4}{3}$?
What is the new number of servings if a recipe makes $6$ servings and is scaled by $k= rac{4}{3}$?
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$8$ servings. The number of servings scales directly with the recipe scale factor $k= rac{4}{3}$.
$8$ servings. The number of servings scales directly with the recipe scale factor $k= rac{4}{3}$.
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What is the new cost if a recipe is scaled by $k= rac{5}{2}$ and the original cost is $8$ dollars?
What is the new cost if a recipe is scaled by $k= rac{5}{2}$ and the original cost is $8$ dollars?
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$20$ dollars. Recipe costs scale linearly with the scale factor $k= rac{5}{2}$, assuming proportional ingredients.
$20$ dollars. Recipe costs scale linearly with the scale factor $k= rac{5}{2}$, assuming proportional ingredients.
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What is the scale factor $k$ if a length changes from $12$ to $9$ units?
What is the scale factor $k$ if a length changes from $12$ to $9$ units?
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$k= rac{3}{4}$. The scale factor $k$ is the ratio of the new length to the original length.
$k= rac{3}{4}$. The scale factor $k$ is the ratio of the new length to the original length.
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What is the new area if a circle is scaled by $k=3$ from area $10\pi$ cm$^2$?
What is the new area if a circle is scaled by $k=3$ from area $10\pi$ cm$^2$?
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$90\pi$ cm$^2$. The area of a circle scales by $k^2=9$ when linear dimensions are scaled by $k=3$.
$90\pi$ cm$^2$. The area of a circle scales by $k^2=9$ when linear dimensions are scaled by $k=3$.
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What is the new circumference if a circle is scaled by $k=3$ from circumference $8\pi$ cm?
What is the new circumference if a circle is scaled by $k=3$ from circumference $8\pi$ cm?
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$24\pi$ cm. Circumference, a linear measurement, is multiplied by the scale factor $k=3$.
$24\pi$ cm. Circumference, a linear measurement, is multiplied by the scale factor $k=3$.
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What is the new volume if a cube with volume $27$ cm$^3$ is scaled by $k=2$?
What is the new volume if a cube with volume $27$ cm$^3$ is scaled by $k=2$?
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$216$ cm$^3$. The new volume is the original volume multiplied by $2^3=8$, increasing 27 cm$^3$ to 216 cm$^3$.
$216$ cm$^3$. The new volume is the original volume multiplied by $2^3=8$, increasing 27 cm$^3$ to 216 cm$^3$.
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What is the new area if a figure is scaled down by $k= rac{1}{3}$ from $81$ cm$^2$?
What is the new area if a figure is scaled down by $k= rac{1}{3}$ from $81$ cm$^2$?
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$9$ cm$^2$. The new area is the original area multiplied by $left( rac{1}{3}
ight)^2= rac{1}{9}$, reducing 81 cm$^2$ to 9 cm$^2$.
$9$ cm$^2$. The new area is the original area multiplied by $left( rac{1}{3} ight)^2= rac{1}{9}$, reducing 81 cm$^2$ to 9 cm$^2$.
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What is the real volume if a $1:10$ model aquarium holds $2$ liters of water when full?
What is the real volume if a $1:10$ model aquarium holds $2$ liters of water when full?
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$2{,}000$ liters. The real volume is the model volume multiplied by $10^3=1{,}000$, as volume scales with the cube of the linear factor.
$2{,}000$ liters. The real volume is the model volume multiplied by $10^3=1{,}000$, as volume scales with the cube of the linear factor.
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What is the model length if a $1:200$ scale model represents a real length of $30$ m?
What is the model length if a $1:200$ scale model represents a real length of $30$ m?
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$15$ cm. The model length is the real length divided by 200, converting 30 m to 3000 cm before division.
$15$ cm. The model length is the real length divided by 200, converting 30 m to 3000 cm before division.
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What is the real length if a map scale is $1:50{,}000$ and the map distance is $3$ cm?
What is the real length if a map scale is $1:50{,}000$ and the map distance is $3$ cm?
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$1{,}500$ m. Multiply the map distance by the scale ratio to find the real distance in centimeters, then convert to meters by dividing by 100.
$1{,}500$ m. Multiply the map distance by the scale ratio to find the real distance in centimeters, then convert to meters by dividing by 100.
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What is the map distance if the scale is $1:25{,}000$ and the real distance is $2.5$ km?
What is the map distance if the scale is $1:25{,}000$ and the real distance is $2.5$ km?
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$10$ cm. Divide the real distance, converted to centimeters, by the scale ratio to obtain the map distance in centimeters.
$10$ cm. Divide the real distance, converted to centimeters, by the scale ratio to obtain the map distance in centimeters.
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What is the scale ratio if $2$ inches on a blueprint represent $5$ feet in real life?
What is the scale ratio if $2$ inches on a blueprint represent $5$ feet in real life?
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$2:60$ 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.
$2:60$ 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.
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What is the new radius if a circle is scaled by $k=0.6$ from radius $10$ cm?
What is the new radius if a circle is scaled by $k=0.6$ from radius $10$ cm?
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$6$ cm. Linear dimensions such as radius are multiplied by the scale factor $k=0.6$.
$6$ cm. Linear dimensions such as radius are multiplied by the scale factor $k=0.6$.
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