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Rates - SAT Math

  1. A copy center charges $18 for 120 pages of printing. At this rate, how much will it cost to print 350 pages?
    Answer: $52.50 We need to find the cost for 350 pages when 120 pages cost $18. First, find the cost per page: $18 ÷ 120 pages = $0.15 per page. For 350 pages: $0.15/page × 350 pages = $52.50. To verify: 350/120 = 2.917, and $18 × 2.917 ≈ $52.50. A common error is rounding the per-page rate too early, leading to accumulated error. Keep extra decimal places in intermediate calculations.
  2. A student types 1,050 words in 7 minutes at a constant rate. At this rate, how many words per minute is the student typing?
    Answer: 150 words/min150\text{ words/min} We need to find the typing rate in words per minute when 1,050 words are typed in 7 minutes. The rate is 1,050 words ÷ 7 minutes = 150 words per minute. This is a straightforward division to find the unit rate. A common error is misplacing the decimal point or confusing which number to divide by which. Remember that 'per minute' means we divide the total quantity by the number of minutes. Quick mental check: 1,050 ÷ 7 = 150 exactly.
  3. A hiker walks 7.5 miles in 2.5 hours at a steady pace. At this same rate, how many miles will the hiker walk in 4 hours?
    Answer: 12 mi12\text{ mi} We need to find how many miles the hiker walks in 4 hours at a steady pace. The hiker's rate is 7.5 miles ÷ 2.5 hours = 3 miles per hour. To find distance in 4 hours, multiply rate by time: 3 miles/hour × 4 hours = 12 miles. The hours cancel, leaving miles as our unit. A common error is to set up a proportion incorrectly or divide instead of multiply. When you have rate and time, multiply to get distance.
  4. A jacket is discounted 25% from an original price of 80. What is the sale price?
    Answer: 60 25% of 80 is 20, so the sale price is 80 - 20 = 60. 20 is just the discount, 55 subtracts 25 dollars instead of 25%, and 75 reflects a 5% decrease, not 25%.
  5. In a class of 48 students, 18 are seniors. What percent of the class are seniors?
    Answer: 37.50% 18 ÷ 48 = 0.375 = 37.5%. The other options either use the wrong base or give the complement instead.
  6. A recipe uses 2.5 cups of flour to make 20 cookies. At this same rate, how many cups of flour are needed to make 36 cookies? Express your answer in cups.
    Answer: 4.5 cups4.5\text{ cups} We need to find how much flour is needed for 36 cookies when 2.5 cups make 20 cookies. The rate is 2.5 cups ÷ 20 cookies = 0.125 cups per cookie. For 36 cookies: 0.125 cups/cookie × 36 cookies = 4.5 cups. The calculation involves finding the flour per cookie ratio first, then scaling up. A common mistake is trying to use mental math with the ratio 20:36 without finding the unit rate, which can lead to arithmetic errors. Always find the per-unit rate when dealing with recipes or similar scaling problems.
  7. A moving walkway carries people at 1.2 meters per second. A person stands still on the walkway and travels 180 meters. At this rate, how long does the trip take?
    Answer: 150 s The question asks how long the trip takes in seconds for 180 meters at 1.2 meters per second, standing still on the walkway. The rate relationship is speed in meters per second. To find time, use time = distance / rate = 180 meters / 1.2 meters per second = 150 seconds, with units meters canceling to leave seconds. This setup emphasizes the inversion of speed (distance per time) to time per distance, but unit analysis confirms correctness. A common error is multiplying instead of dividing, like 180 * 1.2, which gives meters squared per second. Always check units to avoid such mistakes. As a strategy, write the formula with units to verify the output.
  8. A printer can produce 140 flyers in 8 minutes at a constant rate. At this rate, how many flyers can the printer produce in 30 minutes?
    Answer: 525 flyers525\text{ flyers} We need to find how many flyers can be produced in 30 minutes at a constant rate. The printer's rate is 140 flyers ÷ 8 minutes = 17.5 flyers per minute. For 30 minutes: 17.5 flyers/minute × 30 minutes = 525 flyers. Notice how the minutes cancel out, leaving just flyers. A common mistake is trying to use proportions without first finding the unit rate. When dealing with rates, always identify what per what you're calculating.
  9. A plumber charges a $45 service fee plus $30 per hour of labor. If the total bill was $165, how many hours of labor were billed?
    Answer: 4 hours We need to find hours of labor when the total bill was $165. The plumber charges $45 service fee plus $30 per hour, so: $45 + $30h = $165. Subtract the service fee: $30h = $120, then h = 4 hours. The key is recognizing this as a linear equation with fixed and variable costs. A common error is dividing 165 by 30 without subtracting the service fee first. Always account for fixed costs before calculating variable quantities.
  10. A gym membership fee increases from 80 to 92. What is the percent increase?
    Answer: 15 Percent increase is (92 - 80)/80 * 100 = 15. 12 is the absolute difference, 13 uses the new price as the base, and 1.15 is the multiplier, not the percent.
  11. A cyclist rides 18 miles in 1.5 hours on a flat trail, then continues at the same speed. At this rate, how long will it take the cyclist to ride a total of 45 miles? (Assume the speed stays constant for the entire ride.)
    Answer: 3.75 hr We need to find the total time to ride 45 miles at a constant speed. First, calculate the cyclist's rate: 18 miles ÷ 1.5 hours = 12 miles per hour. To find the time for 45 miles, use time = distance ÷ rate: 45 miles ÷ 12 mph = 3.75 hours. A common error is calculating additional time needed (27 more miles) instead of total time. When dealing with rate problems, always identify whether you need total or additional values.
  12. A grocery store sells almonds for $7.20 per 12 ounces. What is the unit price, in dollars per ounce, for the almonds at this rate?
    Answer: $0.60 per oz We need to find the unit price in dollars per ounce. The rate relationship is $7.20 for 12 ounces, so we calculate: $7.20 ÷ 12 ounces = $0.60 per ounce. This gives us the cost for exactly one ounce of almonds. A common mistake is dividing in the wrong order (12 ÷ 7.20), which would give ounces per dollar instead. When finding unit rates, always divide the total cost by the total quantity to get cost per unit.
  13. A pump fills a tank at a constant rate of 15 gallons per minute. At this rate, how many minutes will it take to fill 210 gallons, starting from empty?
    Answer: 1414 min We need to find how many minutes it takes to fill 210 gallons at 15 gallons per minute. Using the rate formula: time = amount ÷ rate, we get 210 gallons ÷ 15 gallons/minute = 14 minutes. Notice how the gallons cancel out, leaving minutes as our answer unit. A common error is multiplying instead of dividing, which would give 3,150 gallon-minutes, a meaningless unit. For rate problems, remember that time equals distance (or amount) divided by rate.
  14. A recipe costs $9.60 to make 12 muffins. Assuming the cost is proportional to the number of muffins, how much will it cost to make 35 muffins at the same rate? Round to the nearest cent.
    Answer: $28.00 We need to find the cost for 35 muffins when 12 muffins cost $9.60. The rate is $9.60 ÷ 12 muffins = $0.80 per muffin. For 35 muffins: $0.80/muffin × 35 muffins = $28.00. The cost scales proportionally with the number of muffins. A common mistake is setting up the proportion incorrectly. Always find the unit rate first, then multiply by the new quantity.
  15. A package of 12 identical notebooks costs $9.60. At this rate, what is the cost of 35 notebooks? (Assume notebooks can be purchased individually at the same unit price.)
    Answer: $28.00 We need the cost of 35 notebooks given the package pricing. The unit price is $9.60 ÷ 12 notebooks = $0.80 per notebook. For 35 notebooks: 35 × $0.80 = $28.00. Avoid multiplying the package price by 35/12 without finding unit price first. Breaking down to unit cost ensures accuracy in scaling.
  16. A student types 1,080 words in 18 minutes at a constant rate. At this rate, how many words does the student type per minute?
    Answer: 60 words/min60\text{ words/min} Words per minute means words divided by minutes, so divide 1,080 words by 18 minutes to get 60 words per minute. The check works: 60 words per minute for 18 minutes yields 1,080 words. The answer given as 19.4 min/word is wrong in both value and units, since minutes per word would be 18 divided by 1,080, roughly 0.017, and the question asks for words per minute. The 50 words per minute figure would produce only 900 words in 18 minutes, well short of 1,080. The 90 words per minute figure would produce 1,620 words, far more than the student actually typed; it comes from dividing 1,080 by 12 rather than by 18. Whenever you compute a rate, multiply it back by the time to confirm you recover the original total.
  17. A package delivery service charges $12.50 for 5 pounds of shipping weight. At the same rate, what is the cost to ship a package that weighs 8 pounds?
    Answer: $20.00 We need to find the shipping cost for 8 pounds when 5 pounds costs $12.50. First, find the cost per pound: $12.50 ÷ 5 pounds = $2.50 per pound. For 8 pounds: $2.50/pound × 8 pounds = $20.00. The calculation is straightforward: 2.50 × 8 = 20. A common mistake is setting up a proportion without finding the unit rate first. For linear pricing, the cost per unit remains constant.
  18. A faucet fills a tank at a constant rate of 2.5 gallons per minute. The tank needs 38 gallons of water. At this rate, how many minutes will it take to fill the tank? (Round to the nearest tenth if needed.)
    Answer: 15.2 min We need to find how many minutes it takes to fill 38 gallons at 2.5 gallons per minute. Using time = volume ÷ rate: 38 gallons ÷ 2.5 gallons/minute = 15.2 minutes. Don't multiply rate by volume (95) or confuse the units. When the rate is given as 'per minute,' the answer will be in minutes.
  19. A cashier scans 84 items in 6 minutes at a constant rate. What is the cashier's scanning rate, in items per minute?
    Answer: 1414 items/min We need to find the scanning rate in items per minute. Calculate: 84 items ÷ 6 minutes = 14 items per minute. This tells us the cashier scans 14 items every minute at this constant rate. A common mistake is computing 6 ÷ 84 = 0.071, which gives minutes per item instead of items per minute. Always check that your final units match what the question asks for - here we want items/minute, not minutes/item.
  20. A website subscription costs $54 for 9 months. At the same monthly rate, how much will the subscription cost for 14 months?
    Answer: $84 We need the cost for 14 months at the same monthly rate. The monthly rate is $54÷9 months=$6\$54 \div 9 \text{ months} = \$6 per month. For 14 months: 14×$6=$8414 \times \$6 = \$84. Don't select the monthly rate ($6) as the answer—multiply it by the number of months. For subscription problems, find the per-month rate then scale appropriately.

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