Solutions

  1. 1 minute
    R
    (ft/sec)    		 × T
    (sec)    		 = D
    (ft)
    5 ×  t = 300

    This is a simple application of the RT = D equation, involving one unit conversion. First convert the rate, 60 inches/second, into 5 feet/second (given that 12 inches = 1 foot). Substitute this value for R. Substitute the distance, 300 feet, for D. Then solve:

  2. 48 hours
    R
    (ft3/hr)    		 × T
    (hr)    		 = W
    (ft)
    4 ×  t = 192

    The capacity of the tank is 6 × 4 × 8, or 192 cubic feet. Use the RT = W equation, substituting the rate, 4 ft3/hour, for R, and the capacity, 192 cubic feet, for W:

  3. 9 years

    Organize the information given in a population chart. Notice that because the population is increasing exponentially, it does not take very long for the population to top 1,000,000.

    Time Elapsed Population
    NOW 2,000
    1 year 4,000
    2 years 8,000
    3 years 16,000
    4 years 32,000
    5 years 64,000
    6 years 128,000
    7 years 256,000
    8 years 512,000
    9 years 1,024,000
  4. minutes
    R
    (bkt/min)    		 × T
    (min)    		 = W
    (bucket)
    r ×  6 = 7/10

    Use the RT = W equation to solve for the rate, with t = 6 minutes and w = 7/10:

    R
    Regular    		 × T
    Regular    		 = W
    Regular
    7/60 ×  t = 3/10

    Then, substitute this rate into the equation again, using 3/10 for w (the remaining work to be done):

  5. 1 × 107

    Organize the information given in a population chart.

    Time Elapsed Population
    4 years ago 0.1 × 108
    2 years ago 0.2 × 108
    NOW 0.4 × 108
    in 2 years 0.8 × 108
    in 4 years 1.6 × 108

    Then, convert:

    0.1 × 108 = 10,000,000 = 1 × 107 bees

  6. 12 songs

    This is a “working together” problem, so add the individual rates: 5 + 5 = 10 songs per hour.

    The two machines together can produce 10 bad songs in 1 hour. Convert the given time into hours:

    R
    (songs/hr)    		 × T
    (hr)    		 = W
    (songs)
    10 ×  1.2 = w

    Then, use the RT = W equation to find the total work done:

       (10)(1.2 hours) = w
    w = 12 bad songs

  7. 3 boxes per hour

    The average rate is equal to the total work done divided by the time in which the work was done. Remember that you cannot simply average the rates. You must find the total work and total time. The total time is 4 hours. To find the total work, add up the boxes Jack put together in each hour: 3 + 2 + 2 + 5 = 12. Therefore, the average rate is , or 3 boxes per hour. The completed chart looks like this:

      R
    (box/hr)    		 × T
    (hr)    		 = W
    (box)
    Phase 1 3 × 1 = 3
    Phase 2 2 × 2 = 4
    Phase 3 5 × 1 = 5
    Total 3
    = 12/4
    		4
    Sum
    		12
    Sum
  8. (B)
      R
    (mi/hr)    		 × T
    (hr)    		 = D
    (mi)
    Train K to T 240 × t + 1/6 = 240(t + 1/6)
    Train T to K 160 × t = 160t
    Total —   —   300

    Solve this problem by filling in the RTD chart. Note that the train going from Kyoto to Tokyo leaves first, so its time is longer by 10 minutes, which is 1/6 hour.

    Next, write the expressions for the distance that each train travels, in terms of t. Now, sum those distances and set that total equal to 300 miles:

    The first train leaves at 12 noon. The second train leaves at 12:10pm. Thirty-nine minutes after the second train has left, at 12:49pm, the trains pass each other.

  9. Approximately 74.5 mph

    Use a Multiple RTD chart to solve this problem. Start by selecting a Smart Number for d, such as 720 miles. (This is a common multiple of the 3 rates in the problem.) Then work backward to find the time for each trip and the total time:
      R
    (mi/hr)    		 × T
    (hr)    		 = D
    (mi)
    A to B 60 × tA = 720
    B to A 80 × tB = 720
    A to B 90 × tC = 720
    Total —   t   2,160

  10. miles per hour

    Use a Multiple RTD chart to solve this problem. Start by selecting a Smart Number for d, such as 200 miles. (This is a common multiple of the two rates in the problem.) Then work backward to find the time for each trip and the total time:

    Do NOT simply average 40 miles per hour and 50 miles per hour to get 45 miles per hour. The fact that the right answer is very close to this wrong result makes this error especially pernicious: avoid it!

  11. If Hose 1 can fill the pool in 6 hours, its rate is 1/6 pool per hour, or the fraction of the job it can do in 1 hour. Likewise, if Hose 2 can fill the pool in 4 hours, its rate is 1/4 pool per hour. Therefore, the combined rate is 5/12 pool per hour (1/4 + 1/6 = 5/12).

      R
    (pool/hr)    		 × T
    (hr)    		 = W
    (pool)
      5/12 × t = 2/3
  12. 5 hours

    Working together, Aimee and Brianna pack boxes per minute. Next use a proportion:

  13. (A)

    This problem can be solved using an RTW chart or by a proportion. There are 20 minutes between 11:40am and noon, and 40 minutes between noon and 12:40pm. Hector’s work rate is different for the two time periods. For the work period before noon, this is the proportion:

    Let b represent the number of problems Hector solves before noon:

    Let a represent the number of problems Hector solves after noon:

    Rewrite the quantities:

    Quantity A Quantity B
    The number of word problems Hector can solve between 11:40am and noon = 5 The number of word problems Hector can solve between noon and 12:40pm = 4

    Therefore, Quantity A is greater.

  14. (A)

    Set up a Population chart, letting X denote the number of users one year ago:

    Time Number of Users
    12 months ago X
    8 months ago 2X
    4 months ago 4X
    NOW 8X

    Ten times the number of users one year ago is 10X, while the number of users today is 8X. Rewrite the quantities:

    Quantity A Quantity B
    Ten times the number of users one year ago = 10X The number of users today = 8X

    Therefore, 10X is greater than 8X because X must be a positive number. Thus, Quantity A is greater.

  15. (B)

    You can use the rate equation to solve for the time it will take the train to cover the distance. Your answer will be in hours because the given rate is in kilometers per hour. Let t stand for the total time of the trip:

    (Note that you can omit the units in your calculation if you verify ahead of time that you are dealing with a consistent system of units.) Finally, convert the time from hours into minutes:

    Rewrite the quantities:

    Quantity A Quantity B
    The number of minutes it will take the train to travel from Xenia to York 110

    An efficient way to solve this problem is to use the value in Quantity B to “cheat.” Assume the train traveled for 110 minutes. Convert 110 minutes to hours:

    Now multiply the time ( ) by the rate (240 kilometers per hour) to calculate the distance:

    The train can travel 440 kilometers in 110 minutes, but the distance between the cities is 420 kilometers. Therefore, the train must have traveled less than 110 minutes to reach its destination. Thus, Quantity B is greater.