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Apply Constant Rates - Praxis Core Math

  1. A city bus arrives at a certain stop every 25 minutes. If a bus just departed at 2:10 PM, at what time will the fourth bus to arrive after the 2:10 PM departure get to the stop?
    Answer: 3:50 PM The problem asks for the arrival time of the fourth bus after the 2:10 PM departure. This means we need to account for four 25-minute intervals. The total time that will pass is 4 × 25 minutes = 100 minutes. To find the arrival time, add 100 minutes to 2:10 PM. 100 minutes is equal to 1 hour and 40 minutes. Adding 1 hour to 2:10 PM gives 3:10 PM. Adding the remaining 40 minutes to 3:10 PM gives 3:50 PM.
  2. A factory machine produces widgets at a constant rate of 90 widgets every 5 minutes. How many hours will it take for this machine to produce 2,700 widgets?
    Answer: 2.5 hours First, calculate the total time in minutes required to produce 2,700 widgets. We can set up a proportion: (90 widgets / 5 minutes) = (2,700 widgets / x minutes). Solving for x gives x = (2,700 * 5) / 90 = 13,500 / 90 = 150 minutes. The question asks for the time in hours. To convert minutes to hours, divide by 60: 150 minutes / 60 minutes/hour = 2.5 hours.
  3. A pool containing 12,000 gallons of water is being drained at a constant rate of 40 gallons per minute. However, a hose is simultaneously adding water to the pool at a constant rate of 15 gallons per minute. How many hours will it take for the pool to be completely empty?
    Answer: 8.0 hours First, determine the net rate at which water is leaving the pool. The net drain rate is the drain rate minus the fill rate: 40 gal/min - 15 gal/min = 25 gal/min. This is the effective rate at which the volume is decreasing. Next, calculate the total time in minutes to empty the 12,000-gallon pool: Time (minutes) = 12,000 gallons / 25 gal/min = 480 minutes. The question asks for the time in hours, so convert minutes to hours: 480 minutes / 60 minutes/hour = 8.0 hours.
  4. The water level in a reservoir is decreasing at a constant rate of 4 inches per day. If the current water level is 60 feet, what will the water level be, in feet, after 15 days? (1 foot = 12 inches)
    Answer: 55 feet First, calculate the total decrease in water level in inches over the 15-day period. Total decrease = 4 inches/day × 15 days = 60 inches. Next, convert this decrease from inches to feet. Decrease in feet = 60 inches / 12 inches/foot = 5 feet. Finally, subtract this decrease from the initial water level to find the new level. New level = 60 feet - 5 feet = 55 feet.
  5. Maya and Ben are painting a fence. Working alone, Maya can paint the entire fence in 6 hours. Working alone, Ben can paint the same fence in 4 hours. If they start working together, what fraction of the fence will be painted after 1 hour?
    Answer: 512\frac{5}{12} First, determine the rate of work for each person as a fraction of the fence painted per hour. Maya's rate is 16\frac{1}{6} of the fence per hour. Ben's rate is 14\frac{1}{4} of the fence per hour. When they work together, their rates add. Their combined rate is 16+14\frac{1}{6} + \frac{1}{4}. To add these fractions, find a common denominator, which is 12. 212+312=512\frac{2}{12} + \frac{3}{12} = \frac{5}{12}. This means that together they can paint 512\frac{5}{12} of the fence in one hour.
  6. A train travels the 210 miles between City A and City B in 3.5 hours. At the same constant speed, how long would it take the train to travel 330 miles?
    Answer: 5.5 hours First, calculate the constant speed (rate) of the train. Speed = Distance / Time = 210 miles / 3.5 hours = 60 miles per hour. Now, use this speed to find the time it would take to travel 330 miles. Time = Distance / Speed = 330 miles / 60 mph = 5.5 hours.
  7. Using the equation from the passage, what is the total cost to produce 300 items?
    Answer: $4,100.00 The equation C = 12n + 500 represents a constant rate of $12 per item (the variable cost) plus a fixed cost of $500. To find the total cost to produce 300 items, substitute n = 300 into the equation. C = 12(300) + 500. C = 3600 + 500. C = $4,100.00.
  8. Two cyclists start at the same point and travel in opposite directions along a straight road. Cyclist A travels at a constant speed of 14 miles per hour, and Cyclist B travels at a constant speed of 18 miles per hour. How much time will have passed when they are 88 miles apart?
    Answer: 2.75 hours When two objects move in opposite directions, their relative speed is the sum of their individual speeds. The rate at which the distance between them is increasing is 14 mph + 18 mph = 32 mph. To find the time it takes for them to be 88 miles apart, use the formula Time = Distance / Rate. Time = 88 miles / 32 mph = 2.75 hours.
  9. At a fruit stand, apples are sold at a rate of 3 for $2.00. Pears are sold at a rate of 5 for $4.00. What is the total cost of purchasing 12 apples and 10 pears?
    Answer: $16.00 Calculate the cost of the apples first. The rate is 3 apples for $2.00. To buy 12 apples, the customer is buying 12/3 = 4 sets of apples. The cost for apples is 4 × $2.00 = $8.00. Next, calculate the cost of the pears. The rate is 5 pears for $4.00. To buy 10 pears, the customer is buying 10/5 = 2 sets of pears. The cost for pears is 2 × $4.00 = $8.00. The total cost is the sum of the cost of apples and pears: $8.00 + $8.00 = $16.00.
  10. It takes 4 identical pipes 90 minutes to fill a swimming pool. How many minutes would it take 6 of the same pipes to fill the same pool, assuming they all work at the same constant rate?
    Answer: 60 minutes This is an inverse proportion problem. The total work can be measured in 'pipe-minutes'. Total work = 4 pipes × 90 minutes = 360 pipe-minutes. This represents the amount of work needed to fill the pool. To find the time it would take for 6 pipes, divide the total work by the number of pipes: Time = 360 pipe-minutes / 6 pipes = 60 minutes.
  11. A 20-gallon container has a leak and loses water at a constant rate of 2.5 fluid ounces per minute. If the container is initially full, approximately how many hours will it take for the container to be one-quarter full? (1 gallon = 128 fluid ounces)
    Answer: 12.8 hours First, determine the volume of water that must be lost for the container to be one-quarter full. This is three-quarters of the total volume: 3/4 × 20 gallons = 15 gallons. Next, convert this volume to fluid ounces: 15 gallons × 128 ounces/gallon = 1,920 ounces. Now, calculate the time in minutes it will take to lose this amount of water: Time (minutes) = 1,920 ounces ÷ 2.5 ounces/minute = 768 minutes. Finally, convert the time from minutes to hours: 768 minutes ÷ 60 minutes/hour = 12.8 hours.
  12. The value of a certain stock increased at a constant rate over a period. On Monday, its value was $32.50. By Wednesday of the same week, its value was $33.70. If the stock continues to increase at the same rate, what will its value be on Friday of that week?
    Answer: $34.90 First, find the rate of increase per day. The time from Monday to Wednesday is 2 days. The increase in value was $33.70 - $32.50 = $1.20. The rate is $1.20 / 2 days = $0.60 per day. The time from Wednesday to Friday is another 2 days. The additional increase will be 2 days × $0.60/day = $1.20. The value on Friday will be the value on Wednesday plus this additional increase: $33.70 + $1.20 = $34.90.
  13. A delivery truck travels at a constant speed of 50 miles per hour. The driver makes a 45-minute stop for a delivery. If the total distance of the trip is 125 miles, what is the total time elapsed from the beginning of the drive to the completion of the stop?
    Answer: 3 hours, 15 minutes First, calculate the driving time. Time = Distance / Speed. Driving time = 125 miles / 50 mph = 2.5 hours. Convert 0.5 hours to minutes: 0.5 hours * 60 minutes/hour = 30 minutes. So, the driving time is 2 hours and 30 minutes. The driver also makes a 45-minute stop. The total elapsed time is the sum of the driving time and the stop time. Total time = (2 hours 30 minutes) + 45 minutes = 2 hours 75 minutes. Since 60 minutes equals 1 hour, 75 minutes is 1 hour and 15 minutes. So, the total time is 2 hours + 1 hour 15 minutes = 3 hours 15 minutes.
  14. A tank is filled by an inlet pipe at a constant rate of 12 gallons per minute. At the same time, water is drained from the tank by an outlet pipe at a constant rate of 4 gallons per minute. If the tank is empty and has a capacity of 300 gallons, how long will it take to fill the tank completely?
    Answer: 37.5 minutes First, find the net filling rate. Since water is entering and leaving simultaneously, subtract the outlet rate from the inlet rate. Net rate = 12 gallons/minute - 4 gallons/minute = 8 gallons/minute. Now, use this net rate to find the time to fill the 300-gallon tank. Time = Total Volume / Net Rate. Time = 300 gallons / 8 gallons/minute = 37.5 minutes.
  15. The temperature of a chemical solution is decreasing at a constant rate. Over a 25-minute period, it drops from 80°C to 45°C. What will the temperature of the solution be after the first 10 minutes of this period?
    Answer: 66°C First, find the total temperature change: 80°C - 45°C = 35°C. This change occurred over 25 minutes. The rate of cooling is 35°C / 25 minutes = 1.4°C per minute. Now, calculate the temperature drop after the first 10 minutes: 10 minutes × 1.4°C/minute = 14°C. Finally, subtract this drop from the initial temperature: 80°C - 14°C = 66°C.
  16. A snail moves at a constant rate of 2 centimeters per minute. An ant starts at the same point 5 minutes after the snail and moves in the same direction at a constant rate of 10 centimeters per minute. How far from the starting point will the ant catch up to the snail?
    Answer: 12.5 cm First, find the head start distance of the snail. In 5 minutes, the snail travels 5 min × 2 cm/min = 10 cm. The ant gains on the snail at a rate of 10 cm/min - 2 cm/min = 8 cm/min. The time for the ant to catch up is Time = Head Start Distance / Relative Speed = 10 cm / 8 cm/min = 1.25 minutes. The question asks for the distance from the starting point where they meet. This is the distance the ant travels in that time. Distance = Ant's Speed × Time = 10 cm/min × 1.25 min = 12.5 cm.
  17. A candle is 25 centimeters tall and burns at a constant rate of 0.4 centimeters per minute. How tall will the candle be after 35 minutes of burning?
    Answer: 11 cm First, calculate the total length of the candle that has burned after 35 minutes. Amount burned = Rate × Time = 0.4 cm/minute × 35 minutes = 14 cm. This is the amount that is gone. To find the remaining height, subtract the burned amount from the original height. Remaining height = 25 cm - 14 cm = 11 cm.
  18. A long-distance runner maintains a constant pace of 8 minutes per mile. The runner plans to take two 5-minute breaks during a marathon of 26.2 miles. What is the runner's total time from start to finish, including breaks?
    Answer: 3 hours, 39.6 minutes First, calculate the total running time. Running time = Pace × Distance = 8 minutes/mile × 26.2 miles = 209.6 minutes. Next, add the total break time. Total breaks = 2 breaks × 5 minutes/break = 10 minutes. The total time from start to finish is the sum of the running time and the break time: Total time = 209.6 minutes + 10 minutes = 219.6 minutes. Finally, convert this to hours and minutes. 219.6 minutes / 60 minutes/hour = 3 with a remainder. 3 hours = 3 × 60 = 180 minutes. The remaining minutes are 219.6 - 180 = 39.6 minutes. So the total time is 3 hours, 39.6 minutes.
  19. A leaky faucet drips water at a constant rate, filling a 200-milliliter beaker in 40 minutes. At this rate, how many liters of water will drip from the faucet in a 24-hour period? (1 liter = 1,000 milliliters)
    Answer: 7.2 liters First, find the drip rate in milliliters per minute: 200 mL / 40 min = 5 mL/min. Next, calculate the total number of minutes in a 24-hour period: 24 hours × 60 minutes/hour = 1,440 minutes. Now, calculate the total volume of water dripped in 24 hours: 5 mL/min × 1,440 min = 7,200 mL. Finally, convert this volume from milliliters to liters by dividing by 1,000: 7,200 mL / 1,000 mL/L = 7.2 liters.
  20. An employee earns $24.00 per hour for a standard 40-hour work week. For any hours worked beyond 40, the employee earns 1.5 times the standard hourly rate. How much does this employee earn for working 52 hours in one week?
    Answer: $1392.00 This problem involves two rates. First, calculate the earnings for the standard 40 hours: 40 hours × $24.00/hour = $960.00. Next, determine the number of overtime hours: 52 hours - 40 hours = 12 overtime hours. Calculate the overtime rate: 1.5 × $24.00/hour = $36.00/hour. Calculate the overtime earnings: 12 hours × $36.00/hour = $432.00. Finally, add the standard and overtime earnings: $960.00 + $432.00 = $1392.00.

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