Ratios & Proportions - SAT Math

A model car is built at a scale of $1:18$ (model:actual). If the actual car is 162 inches long, how long is the model car?
Answer: 9 in. With a model car scale of 1:18 (model:actual) and an actual car length of 162 inches, we need the model length. The scale 1:18 means the model is 1/18 the size of the actual car, so model length = 162 ÷ 18 = 9 inches. We can verify by setting up the proportion: 1/18 = x/162, which gives x = 162/18 = 9 inches. A common mistake is to multiply by the scale factor instead of dividing. For scale models, divide the actual measurement by the scale factor to get the model measurement.
A blueprint uses a scale of $3$ cm to $4.5$ m. A hallway is shown as $8.2$ cm long on the blueprint. What is the actual length of the hallway, in meters?
Answer: 12.3 m The blueprint scale is 3 cm : 4.5 m, and the hallway measures 8.2 cm on the blueprint. Setting up the proportion: 3 cm/4.5 m = 8.2 cm/x m. Cross-multiplying: 3x = 8.2 × 4.5 = 36.9, so x = 12.3 m. The actual hallway length is 12.3 meters. A common error is inverting the ratio or using the wrong units. When working with scales, always set up the proportion with consistent units and check that your answer is reasonable - a hallway of 12.3 m is plausible.
If the ratio of boys to girls in a class is 2:3 and there are 18 girls, how many boys are in the class?
Answer: 12 boys This problem asks us to find the number of boys when the ratio of boys to girls is 2:3 and there are 18 girls. We set up the proportion: boys/girls = 2/3, so x/18 = 2/3. Cross-multiplying gives us 3x = 2 × 18 = 36, therefore x = 12 boys.
A similar-figures diagram shows two similar right triangles. The smaller triangle has hypotenuse 13 and a leg 5. The larger triangle's corresponding leg is 15. What is the length of the larger triangle's hypotenuse?
Answer: 39 In similar right triangles, the smaller has hypotenuse 13 and leg 5, while the larger has corresponding leg 15. The scale factor is 15/5 = 3, meaning the larger triangle is 3 times the size. Therefore, the larger triangle's hypotenuse = 13 × 3 = 39. We can verify using the Pythagorean theorem: in the smaller triangle, the other leg = √(13² - 5²) = √(169 - 25) = 12, so in the larger triangle it's 36, and the hypotenuse = √(15² + 36²) = √(225 + 1296) = √1521 = 39. Always use the ratio of corresponding parts consistently.
A café makes a house blend by mixing dark roast beans and light roast beans in a ratio of $5:8$ (dark:light). If the café makes $39$ kilograms of the blend, how many kilograms of light roast beans are used?
Answer: 24 kg This problem asks for the amount of light roast beans in a 39 kg blend with a dark:light ratio of 5:8. Set up the proportion where the total ratio parts are 5 + 8 = 13, so light roast represents 8/13 of the total blend. To find the amount of light roast: (8/13) × 39 = 312/13 = 24 kg. A common error is using 5/8 instead of 8/13, which would give an incorrect answer. When dealing with part-to-part ratios, always find the total parts first to determine what fraction each component represents.
A photocopier enlarges images so that every length is multiplied by $1.25$ . A logo's original area is $64\text{ cm}^2$ . What is the area of the enlarged logo in square centimeters?
Answer: 100 When a photocopier enlarges with scale factor 1.25 for linear dimensions, areas are multiplied by (1.25)². The original area is 64 cm², so the enlarged area is 64 × (1.25)² = 64 × 1.5625 = 100 cm². A critical concept: when linear dimensions are scaled by factor k, areas scale by k². Common errors include multiplying area by 1.25 instead of 1.25² (giving 80), or adding 25% to 64 (giving 80). Remember that area scaling involves squaring the linear scale factor, and volume scaling involves cubing it.
A map uses a scale of $1$ inch to $12$ miles. Two towns are $3.75$ inches apart on the map. What is the actual distance between the towns, in miles?
Answer: 45 miles This map scale problem requires converting map distance to actual distance using the scale 1 inch : 12 miles. We set up the proportion: 1 inch/12 miles = 3.75 inches/x miles. Cross-multiplying gives 1 × x = 12 × 3.75, so x = 45 miles. The actual distance between the towns is 45 miles. A common error is dividing instead of multiplying, which would give 0.3125 miles. Remember that map scales show how real distances are reduced, so actual distances are larger than map distances.
If 5 kg of apples cost $15, what is the cost of 8 kg of apples assuming the cost is proportional?
Answer: $24 This is a direct proportionality problem asking for the cost of a different quantity of apples. We set up the proportion 5 kg : $15 = 8 kg : $x, where x is the cost we're finding. Cross-multiplying gives us 5x = 15 × 8 = 120, so x = 120 ÷ 5 = $24. Alternatively, we could find the unit rate: $15 ÷ 5 kg = $3 per kg, then multiply by 8 kg to get $24. Common mistakes include setting up the proportion with mismatched units or forgetting to include the dollar sign in the final answer.
A chemistry solution is $35\%$ acid by volume. How many milliliters of acid are in $480$ mL of the solution?
Answer: 168 mL To find the amount of acid in 480 mL of a 35% acid solution, calculate 35% of 480 mL. Convert percentage to decimal: 35% = 0.35, then multiply: 0.35 × 480 = 168 mL of acid. You can also set up the proportion: 35/100 = x/480, then cross-multiply to get 100x = 16,800, so x = 168 mL. A common error is forgetting to convert the percentage or misplacing the decimal. Remember that finding a percentage of a quantity means multiplying by the decimal form.
A store advertises that the number of reward points earned is proportional to dollars spent. A customer earns $1{,}260$ points for spending $84$ dollars. How many points would the customer earn for spending $135$ dollars at the same rate?
Answer: 2,025 The problem states that reward points are proportional to dollars spent, with 1,260 points for $84. First find the rate: 1,260 points ÷ $84 = 15 points per dollar. For $135, multiply: 135 × 15 = 2,025 points. A common error is setting up a complex proportion when simple unit rate multiplication works perfectly. Another mistake is inverting the ratio and getting dollars per point instead of points per dollar. When dealing with proportional relationships, finding the unit rate (points per dollar) often simplifies calculations and reduces errors.
A solution is $35\%$ salt by mass. How many grams of salt are in $260$ grams of this solution?
Answer: 91 g To find the amount of salt in a 35% salt solution, we multiply the total mass by the percentage as a decimal. Salt amount = 260 g × 0.35 = 91 g. We can verify: if 91 g is salt, then 91/260 = 0.35 = 35%. A common error is using 35 instead of 0.35, which would give 9,100 g. When working with percentages in proportion problems, always convert to decimal form before multiplying.
How many students are in class A?
Answer: 36 This question asks us to find the number of students in class A when the ratio of class A to class B is 4:5 and class B has 45 students. We can set up the proportion: class A/class B = 4/5, which gives us x/45 = 4/5. Cross-multiplying, we get 5x = 180, so x = 36. This means class A has 36 students. We can verify this by checking that 36:45 simplifies to 4:5 (dividing both by 9 gives us 4:5). A common mistake would be confusing which class corresponds to which part of the ratio or making an arithmetic error during cross-multiplication. Always double-check your arithmetic and verify that your answer creates the correct ratio when paired with the given information.
Two similar right triangles are shown. Triangle $A$ has legs $9$ cm and $12$ cm. Triangle $B$ has the leg corresponding to $9$ cm labeled $15$ cm. What is the length of the leg in triangle $B$ that corresponds to $12$ cm?
Answer: 20 cm This question asks for the length of the leg in triangle B that corresponds to the 12 cm leg in similar triangle A, where triangle A has legs 9 cm and 12 cm, and triangle B's corresponding leg to 9 cm is 15 cm. Since the triangles are similar, the ratio of corresponding sides is constant, so set up the proportion 9/15 = 12/x, or use the scale factor 15/9 = 5/3. Multiply the smaller leg by the scale factor: x = 12 * (5/3) = 20 cm. Alternatively, cross-multiply in the proportion: 9x = 180, so x = 20 cm, matching units in cm. A key error is mismatching corresponding sides, such as using 12/15 directly. Another mistake could be adding ratios instead of multiplying. In similar figures, identify correspondences carefully and verify units to avoid setup errors.
A smoothie recipe uses yogurt, berries, and juice in the ratio $3:5:2$ by volume. If a batch uses $420$ mL total, how many milliliters of juice are in the batch?
Answer: 84 mL We need to find the amount of juice in a smoothie with ingredients in the ratio 3:5:2 (yogurt:berries:juice) totaling 420 mL. The ratio tells us there are 3 + 5 + 2 = 10 total parts, with juice making up 2 parts. To find the juice amount: (2/10) × 420 = 84 mL. Students often make the mistake of using the wrong part of the ratio or forgetting to add all parts for the denominator. Remember that in a three-part ratio, each number represents that ingredient's share of the total.
Triangle $ABC$ is similar to triangle $DEF$ . The ratio of a side in $ABC$ to its corresponding side in $DEF$ is $7:9$ . If the area of triangle $ABC$ is $98\text{ cm}^2$ , what is the area of triangle $DEF$ ?
Answer: $162\text{ cm}^2$ Similar triangles ABC and DEF have corresponding sides in ratio 7:9, with ABC having area 98 cm². Since areas of similar figures are proportional to the square of the linear ratio, the area ratio is (7/9)² = 49/81. Set up: 49/81 = 98/x, where x is DEF's area. Cross-multiplying: 49x = 81 × 98 = 7,938, so x = 7,938/49 = 162 cm². A critical error is using the linear ratio instead of squaring it. Always square the linear ratio when dealing with areas of similar figures.
A scale model of a building uses the scale $1\text{ cm} : 2.5\text{ m}$ . In the model, a hallway is $14.8$ cm long. What is the actual hallway length, in meters?
Answer: 37.0 m We need to find the actual hallway length when a model uses scale 1 cm : 2.5 m and the model hallway is 14.8 cm. The scale tells us each centimeter in the model represents 2.5 meters in reality. To find the actual length: 14.8 cm × 2.5 m/cm = 37.0 meters. Students often forget to convert units or mix up which measurement to multiply. When using scales, multiply the model measurement by the real-world value per model unit.
How many oranges are there?
Answer: 14 This question asks us to find the number of oranges when the ratio of apples to oranges is 3:2 and there are 21 apples. We can set up the proportion: apples/oranges = 3/2, which gives us 21/x = 3/2. Cross-multiplying, we get 3x = 42, so x = 14. This means there are 14 oranges in the basket. We can also solve this by recognizing that 21 ÷ 3 = 7, so we multiply the orange portion by 7: 2 × 7 = 14 oranges. A common error would be setting up the proportion incorrectly or confusing which fruit corresponds to which part of the ratio. When checking your work, verify that your final answer maintains the original ratio: 21:14 should simplify to 3:2.
How many blue marbles are there?
Answer: 16 This question asks us to find the number of blue marbles when we know the ratio of red to blue marbles is 3:4 and there are 12 red marbles. We can set up the proportion: red marbles/blue marbles = 3/4, which gives us 12/x = 3/4. Cross-multiplying, we get 3x = 48, so x = 16. This means there are 16 blue marbles. A common error would be confusing which part of the ratio corresponds to which color or setting up the proportion incorrectly. Always double-check that your ratio setup matches the given information exactly.
In a school orchestra, the ratio of violins to cellos to flutes is $7:3:5$ . If there are $45$ total musicians in the orchestra, how many cellos are there?
Answer: 9 This question asks for the number of cellos in a school orchestra with a total of 45 musicians, where the ratio of violins to cellos to flutes is 7:3:5. The total parts in the ratio are 7 + 3 + 5 = 15, so set up the proportion for cellos as 3/15 of the total musicians. Calculate one part by dividing 45 by 15, yielding 3 musicians per part, then multiply by 3 for cellos: 3 3 = 9. Alternatively, directly compute (3/15) 45 = 9 cellos, ensuring the total adds up correctly. A common error is forgetting to sum the ratio parts, leading to incorrect fractions like 3/7. Another mistake might be distributing unevenly without proportioning. For multi-part ratios, always sum parts first and check if the total matches by adding individual counts.
A paint color is made by mixing blue and yellow in the ratio $7:9$ (blue:yellow). A painter wants $64$ ounces of this paint. How many ounces of blue paint are needed?
Answer: 28 oz To make 64 ounces of paint with a blue:yellow ratio of 7:9, find the amount of blue paint needed. The total ratio parts are 7 + 9 = 16, so blue represents 7/16 of the total. Calculate: (7/16) × 64 = 448/16 = 28 ounces of blue paint. A common error is using the ratio 7/9 instead of 7/16, which would give 49.8 ounces. Remember to find the fraction of the whole that each color represents, not the fraction of one color relative to the other.

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