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SAT Digital Math · SL AA

Problem-Solving and Data Analysis: Ratios, Rates, and Unit Conversions

Targeted Practice Sheet 1

Worked Examples: SAT Strategy

Example 1: Easy (Basic Unit Conversion)

Question: How many yards are equivalent to $1,116$ inches? (1 yard = 36 inches).
SAT Strategy: Use dimensional analysis. Write the given value as a fraction, and multiply by a conversion factor where the unit you want to cancel is on the opposite side (denominator).
  1. Start with the given value: $\frac{1116 \text{ inches}}{1}$.
  2. Multiply by the conversion factor: $\frac{1 \text{ yard}}{36 \text{ inches}}$.
  3. Notice that "inches" on top and "inches" on bottom cancel out.
  4. $\frac{1116}{1} \times \frac{1}{36} = \frac{1116}{36}$.
  5. $1116 \div 36 = 31$.
  6. Answer: $31$.

Example 2: Medium (Compound Rate Conversion)

Question: A cheetah can run at a top speed of $60$ miles per hour. What is the cheetah's top speed in feet per second? (1 mile = 5,280 feet, 1 hour = 3,600 seconds).
SAT Strategy: Set up a chain of fractions. Cancel the units one by one until you are left with feet in the numerator and seconds in the denominator.
  1. Start with the given rate: $\frac{60 \text{ miles}}{1 \text{ hour}}$.
  2. Cancel miles: multiply by $\frac{5280 \text{ feet}}{1 \text{ mile}}$. (Miles cancel, leaving feet/hour).
  3. Cancel hours: multiply by $\frac{1 \text{ hour}}{3600 \text{ seconds}}$. (Hours cancel, leaving feet/second).
  4. Multiply the numerators and denominators: $\frac{60 \times 5280 \times 1}{1 \times 1 \times 3600}$.
  5. Simplify: $\frac{316800}{3600} = 88$.
  6. Answer: $88$.

Example 3: Hard (Area and Volume Scaling)

Question: A rectangular carpet has an area of $12$ square yards. What is the area of this carpet in square feet? (1 yard = 3 feet).
SAT Strategy: Do NOT just multiply by 3! When dealing with area, the conversion factor must be squared. When dealing with volume, it must be cubed.
  1. The linear conversion is $1 \text{ yd} = 3 \text{ ft}$.
  2. Therefore, the area conversion is $(1 \text{ yd})^2 = (3 \text{ ft})^2 \implies 1 \text{ sq yd} = 9 \text{ sq ft}$.
  3. Multiply the area by the squared conversion factor: $12 \text{ sq yd} \times \frac{9 \text{ sq ft}}{1 \text{ sq yd}}$.
  4. $12 \times 9 = 108$.
  5. Answer: $108$.

Practice Questions

Level 1: Easy (Foundational SAT)

  1. A machine produces 45 parts every 3 minutes. At this rate, how many parts will the machine produce in 20 minutes? (Grid-in)
  2. A local theater has a ratio of children's tickets sold to adult tickets sold of $3 : 5$. If a total of 120 tickets were sold, how many children's tickets were sold?
    • A) $30$
    • B) $45$
    • C) $75$
    • D) $200$
  3. A hose puts exactly 88 ounces of water into a bucket in $x$ seconds. Which expression represents the number of ounces of water the hose puts into the bucket in 1 second?
    • A) $88x$
    • B) $\frac{x}{88}$
    • C) $\frac{88}{x}$
    • D) $88 - x$
  4. A recipe calls for 2 cups of sugar for every 3 cups of flour. If a baker uses 15 cups of flour to make a large batch, how many cups of sugar are needed? (Grid-in)
  5. How many centimeters are equivalent to 4.5 meters? (1 meter = 100 centimeters). (Grid-in)

Level 2: Medium (Standard SAT)

  1. A car travels at a constant speed of 75 miles per hour. How many miles does the car travel in 40 minutes?
    • A) $30$
    • B) $40$
    • C) $50$
    • D) $112.5$
  2. On a map, 1 inch represents an actual distance of 15 miles. If two cities are 4.2 inches apart on the map, what is the actual distance between the two cities, in miles? (Grid-in)
  3. Machine A can print 400 pages in 2 hours. Machine B can print 600 pages in 3 hours. If both machines work together at their respective constant rates, how many total pages will they print in 5 hours?
    • A) $1000$
    • B) $1500$
    • C) $2000$
    • D) $2500$
  4. A traveler is converting US Dollars (USD) to Euros. The exchange rate is 1 USD = 0.85 Euros. If the traveler buys a jacket that costs 170 Euros, what is the cost of the jacket in USD? (Grid-in)
  5. A swimming pool has a volume of 15 cubic meters. What is the volume of the swimming pool in cubic centimeters? (1 meter = 100 centimeters).
    • A) $1,500$
    • B) $15,000$
    • C) $1,500,000$
    • D) $15,000,000$

Level 3: Hard (Advanced SAT)

  1. A liquid flows through a pipe at a rate of 12 gallons per minute. What is the rate of flow in quarts per second? (1 gallon = 4 quarts, 1 minute = 60 seconds).
    • A) $0.8$
    • B) $1.25$
    • C) $3$
    • D) $48$
  2. Pump A can fill a water tank in 3 hours working alone. Pump B can fill the exact same tank in 6 hours working alone. If both pumps are turned on at the same time, how many hours will it take them to fill the tank together? (Grid-in)
  3. A gear system consists of Gear X, which has 12 teeth, and Gear Y, which has 36 teeth. The gears are interlocked so that when Gear X turns, it turns Gear Y. If Gear X completes 15 full revolutions, how many full revolutions does Gear Y complete? (Grid-in)
  4. A car travels $m$ miles in $h$ hours. Which of the following expressions represents the number of feet the car travels in $x$ minutes? (1 mile = 5,280 feet, 1 hour = 60 minutes).
    • A) $\frac{88mx}{h}$
    • B) $\frac{5280m}{60hx}$
    • C) $\frac{88hx}{m}$
    • D) $\frac{mx}{88h}$
  5. A chemist has 50 liters of a solution that is $20%$ acid and $80%$ water. How many liters of pure water must the chemist add to the solution to dilute it so that the new solution is exactly $10%$ acid? (Grid-in)

Level 4: Very Hard (Slightly beyond SAT)

  1. A student mixes 200 milliliters (mL) of a juice drink that is $30%$ sugar with 300 mL of a juice drink that is $80%$ sugar. What is the sugar percentage of the final 500 mL mixture?
    • A) $45%$
    • B) $50%$
    • C) $55%$
    • D) $60%$
  2. A construction crew of 5 workers can build 3 identical brick walls in 4 days, assuming each worker works at the same constant rate. How many workers are needed to build 6 identical brick walls in exactly 2 days? (Grid-in)
  3. A certain metal alloy has a density of 2.5 grams per cubic centimeter ($\text{g/cm}^3$). A solid block of this alloy has a volume of 0.04 cubic meters ($\text{m}^3$). What is the mass of the block, in kilograms (kg)? (1 meter = 100 cm, 1 kg = 1,000 g). (Grid-in)
  4. A driver travels from City A to City B at an average speed of 40 miles per hour. The journey takes 2 hours. The driver then travels from City B to City C at an average speed of 60 miles per hour, which takes 3 hours. What is the driver's average speed for the entire journey from City A to City C, in miles per hour?
    • A) $48$
    • B) $50$
    • C) $52$
    • D) $54$
  5. A factory produces $p$ widgets for every $q$ dollars spent on manufacturing. Which of the following expressions represents the cost, in cents, to manufacture $r$ widgets? (1 dollar = 100 cents).
    • A) $\frac{100pr}{q}$
    • B) $\frac{100qr}{p}$
    • C) $\frac{qr}{100p}$
    • D) $\frac{pr}{100q}$

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