SAT ratios, rates, percentages and units: review and practice

• Use a ratio or proportion to find a missing amount.
• Find and interpret a unit rate, like miles per gallon or dollars per pound.
• Convert between units, including square and cubic units.
• Find the percent, the part or the whole when you know the other two.
• Work with percent change, including two changes in a row.

• Using the new value as the base. A price that goes from $80 to $100 rose 25% (20 ÷ 80), not 20% (20 ÷ 100).
• Assuming +20% and −20% cancel. $100 up 20% is $120, and 20% off that is $96. Multiply the factors: 1.2 × 0.8 = 0.96.
• Reading "150% greater than" as × 1.5. A number 150% greater than a is 2.5a. "150% of a" is 1.5a.
• Converting square units with the length factor. 1 foot = 12 inches, so 1 square foot = 144 square inches, not 12.
• Flipping a ratio. If boys to girls is 3:5 in a class of 40, there are 15 boys and 25 girls. Reading the ratio backward picks the other number.
• Giving a percent when the question wants an amount. If 30% of 250 students walk to school, the answer is 75 students, not 30.

1. Question 1 · Easy

   The ratio 84 : m is equivalent to the ratio 6 : 15. What is the value of m?

   • A14
   • B15
   • C93
   • D210
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   D. The common multiplier is 84 ÷ 6 = 14, so m = 15 × 14 = 210. The value 14 stops at the multiplier, 15 leaves the second term unchanged, and 93 preserves the difference between the terms.

2. Question 2 · Easy

   A storage bin has a volume of 0.81 cubic yard. What is the volume, in cubic feet, of the bin? (1 yard = 3 feet)

   • A0.03
   • B2.43
   • C7.29
   • D21.87
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   D. Since 1 cubic yard = 3³ cubic feet = 27 cubic feet, multiplying cubic yards by 27 gives cubic feet. The volume is 0.81(27) = 21.87 cubic feet. The other values divide by 27, multiply by only 3, or multiply by only 9.

3. Question 3 · Medium

   For positive numbers u, v, and w, u is 275% of v and 85% of w. The number w is p% of v. Which of the following is closest to the value of p?

   • A234
   • B275
   • C324
   • D360
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   C. The equations u = 2.75v and u = 0.85w give w/v = 2.75/0.85, so p is approximately 323.53 and the closest choice is 324. The value 234 comes from multiplying the two multipliers and rounding, 275 ignores the second relationship, and 360 adds percentages with different reference quantities.

4. Question 4 · Medium

   For positive numbers a, b, and c, b is 188% greater than a, and c is 84% less than b. The number c is how many times the number a?

   • A0.3008
   • B0.4608
   • C1.5792
   • D2.4192
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   B. Since b = 2.88a and c = 0.16b, the ratio is c/a = (2.88)(0.16) = 0.4608. The value 0.3008 omits the original amount in the first factor, 1.5792 makes that error and uses 0.84 as the remaining fraction, and 2.4192 makes only the remaining-fraction error.

5. Question 5 · Hard

   The mass of object L is 125% of the mass of object M and 0.096% of the mass of object N. The mass of object M is p% of the mass of object N. What is the value of p?

   • A0.0768
   • B0.096
   • C0.12
   • D0.384
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   A. Letting m and n denote the masses of M and N, the shared mass gives 1.25m = 0.00096n, so p = 100(0.00096/1.25) = 0.0768. The value 0.096 ignores the 125% relationship, 0.12 multiplies by 1.25 instead of dividing, and 0.384 replaces the multiplier 1.25 with 0.25.

6. Question 6 · Hard

   A collection of coins increased in value during its first year. During its second year, the collection decreased in value by 15%. The amount lost during the second year was 40% of the amount gained during the first year. Over the two years, by what percent did the coin collection’s value increase?

   • A25%
   • B34%
   • C36%
   • D45%
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   C. Let G be the first gain and M the value after the first year; the loss relation gives 0.15M = 0.40G, hence M = (8/3)G and the initial value is (5/3)G. The net gain is G − 0.40G = 0.60G, so its percentage of the initial value is [0.60G/((5/3)G)](100) = 36%. The value 25% subtracts rates with different meanings, 34% treats 40% as the first growth rate and reduces only the gain, and 45% subtracts 15 from the recovered first growth rate of 60%.

Type the whole calculation on one line with each percent written as a multiplier, like 80 × 1.25 × 0.9, so every factor is visible. For a proportion such as 12/m = 3/8, type it as is and Desmos draws a vertical line at the solution; click it to read m. Keep unit conversions in one expression, like 5 × 12 × 2.54 for 5 feet in centimeters.

Ashore's SAT course starts with a 15-minute check that finds the question types costing you the most points, then gives you lessons and practice for each.

• Linear Equations and Inequalities
• Linear Functions and Systems
• One- and Two-Variable Data
• Probability and Statistical Inference
• Equivalent Expressions and Quadratics
• Nonlinear Functions and Exponentials
• Lines, Angles, and Triangles
• Right Triangles and Trigonometry
• Circles, Area, and Volume
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