Worked Examples: SAT Strategy
Example 1: Easy (Basic Proportions)
Question: A local restaurant makes its own hot sauce using a ratio of 3 parts chili peppers to 2 parts vinegar. If the restaurant uses 15 cups of chili peppers, how many cups of vinegar are needed?
SAT Strategy: Set up a simple fraction proportion and cross-multiply. Keep your units aligned (chili on top, vinegar on bottom).
- Setup: $\frac{\text{3 parts chili}}{\text{2 parts vinegar}} = \frac{\text{15 cups chili}}{x \text{ cups vinegar}}$.
- Cross-multiply: $3x = 2(15) \implies 3x = 30$.
- Divide by 3: $x = 10$.
- Answer: 10.
Example 2: Medium (Multi-Step Unit Conversion)
Question: A car is traveling at a constant speed of 60 miles per hour. Which of the following is closest to the car's speed in feet per second? (1 mile = 5,280 feet)
SAT Strategy: Use dimensional analysis (the "train track" method) to cancel out units you don't want and leave the units you do want.
- Start with the given rate: $\frac{60 \text{ miles}}{1 \text{ hour}}$.
- Convert miles to feet: $\frac{60 \text{ miles}}{1 \text{ hour}} \times \frac{5280 \text{ feet}}{1 \text{ mile}} = \frac{316,800 \text{ feet}}{1 \text{ hour}}$. (Miles cancel out).
- Convert hours to minutes, then seconds: $\frac{316,800 \text{ feet}}{1 \text{ hour}} \times \frac{1 \text{ hour}}{60 \text{ mins}} \times \frac{1 \text{ min}}{60 \text{ secs}}$.
- The math becomes: $\frac{316,800}{3,600} = 88$.
- Answer: 88 feet per second.
Example 3: Hard (Abstract Constants in Rates)
Question: A machine can print $p$ pages every $m$ minutes. At this rate, how many hours will it take the machine to print $500$ pages?
SAT Strategy: Treat the abstract variables exactly like numbers. Find the unit rate (hours per page) and multiply by the total number of pages.
- First, find the rate in minutes per page: $\frac{m \text{ minutes}}{p \text{ pages}}$.
- Convert this rate to hours per page by dividing by 60: $\frac{m}{60p} \text{ hours per page}$.
- Multiply this rate by the total number of pages (500): $\frac{m}{60p} \times 500 = \frac{500m}{60p}$.
- Simplify the fraction: $\frac{50m}{6p} = \frac{25m}{3p}$.
- Answer: $\frac{25m}{3p}$.
Practice Questions
Level 1: Easy (Foundational SAT)
- A recipe for cookies calls for 2 cups of sugar for every 3 cups of flour. If a baker uses 9 cups of flour, how many cups of sugar are needed? (Grid-in)
- What is $15%$ of $300$? (Grid-in)
- If $40%$ of a number is $80$, what is $100%$ of the number?
- A) $120$
- B) $160$
- C) $200$
- D) $320$
- A runner completes a 5-kilometer race in 25 minutes. What is the runner's average speed in kilometers per hour? (Grid-in)
- The ratio of boys to girls in a school club is 4 to 5. If there are 36 students in the club, how many are girls?
- A) $16$
- B) $20$
- C) $25$
- D) $30$
Level 2: Medium (Standard SAT)
- A store buys a jacket for $\$40$ and sells it for $\$60$. What is the percentage markup on the cost of the jacket?
- A) $20%$
- B) $33.3%$
- C) $50%$
- D) $66.7%$
- An architect is using a scale of 1 inch = 5 feet for a blueprint. If a room is 15 feet by 20 feet in reality, what is the area of the room on the blueprint, in square inches? (Grid-in)
- A liquid is leaking from a tank at a rate of 12 milliliters per second. How many liters of liquid leak from the tank in 1 hour? (1 liter = 1,000 milliliters)
- A) $0.72$
- B) $7.2$
- C) $43.2$
- D) $720$
- The price of an item was decreased by $20%$. By what percentage must the new price be increased to return it to its original price?
- A) $20%$
- B) $25%$
- C) $40%$
- D) $80%$
- Machine A can produce 120 widgets in 3 hours. Machine B can produce 150 widgets in 5 hours. If both machines run continuously, how many total widgets will they produce in 8 hours? (Grid-in)
Level 3: Hard (Advanced SAT)
- A painter can cover $x$ square meters of wall in $y$ minutes. At this rate, how many hours will it take the painter to cover $z$ square meters of wall?
- A) $\frac{yz}{60x}$
- B) $\frac{60yz}{x}$
- C) $\frac{xz}{60y}$
- D) $\frac{60xz}{y}$
- A population of bacteria increases by $40%$ every 3 hours. If the initial population is 500, what is the best estimate of the population after 9 hours?
- A) $1,100$
- B) $1,372$
- C) $1,500$
- D) $1,960$
- In a certain chemical process, $p$ grams of substance A react with $q$ grams of substance B to produce $r$ grams of substance C. If the ratio of $p$ to $q$ is strictly 5 to 2, and no material is lost in the reaction ($p+q=r$), how many grams of substance A are needed to produce 140 grams of substance C? (Grid-in)
- A rectangular plot of land has a length that is $25%$ greater than its width. If the width is decreased by $20%$ and the length is increased by $20%$, what is the overall percentage change in the area of the plot?
- A) It decreases by $4%$
- B) It decreases by $5%$
- C) It remains the same
- D) It increases by $4%$
- The density of a certain metal is 8.5 grams per cubic centimeter. A solid cube of this metal has a mass of 544 grams. What is the side length of the cube in centimeters? (Grid-in)
Level 4: Very Hard (Slightly beyond SAT)
- A water tank can be filled by Pipe X in 4 hours and drained by Pipe Y in 6 hours. If both pipes are opened simultaneously when the tank is empty, how many hours will it take to completely fill the tank? (Grid-in)
- On a map, 1 centimeter represents $k$ kilometers. A region has an area of 50 square centimeters on the map. If the actual area of the region is 1,800 square kilometers, what is the value of $k$? (Grid-in)
- The currency in Country A is the Alpha, and the currency in Country B is the Beta. The exchange rate is 1 Alpha = 1.25 Betas. A trader exchanges $x$ Alphas for Betas, spends $20%$ of the Betas, and then exchanges the remaining Betas back into Alphas at a new exchange rate of 1 Alpha = 1.20 Betas. What fraction of the original $x$ Alphas does the trader have left?
- A) $\frac{5}{6}$
- B) $\frac{19}{24}$
- C) $\frac{1}{1.2}$
- D) $\frac{4}{5}$
- A car travels from City A to City B at an average speed of 40 miles per hour, and immediately returns from City B to City A along the same route at an average speed of 60 miles per hour. What is the car's average speed for the entire round trip?
- A) $48$ mph
- B) $50$ mph
- C) $52$ mph
- D) $54$ mph
- The population of a city grew by $r%$ in 2021 and then grew by $s%$ in 2022. If the population at the end of 2022 was exactly $21%$ greater than the population at the beginning of 2021, and $r$ and $s$ are positive integers, what is the value of $r \times s$? (Grid-in)
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