Proportional Reasoning with Rates — Challenge

A rate compares two different kinds of quantity. Turn it into a unit rate and you can compare anything.

What these problems cover

These questions use 1 idea from the course. Follow any of them for a definition, where it is taught, and what it leads to.

Problem 1

A printer produces 84 pages in 7 minutes, working at a steady rate. How many pages does it produce each minute?

  1. 7
  2. 10
  3. 12
  4. 14
  5. 18
Worked answer

12

A unit rate is the amount for one unit of time, so divide the pages by the minutes.

84 pages7 min = 12 pages per minute.

Problem 2

A car covers 180 kilometres in 2.5 hours. Travelling at the same steady rate, how many kilometres does it cover in 4 hours?

  1. 240
  2. 270
  3. 280
  4. 288
  5. 320
Worked answer

288

Find the rate first: 1802.5 = 72 kilometres per hour.

Then multiply by the new time: 72· 4 = 288 kilometres.

Problem 3

One tap fills a tank in 12 minutes and a second tap fills the same tank in 18 minutes. If both run together, how many minutes does the tank take to fill?

  1. 6
  2. 7.2
  3. 7.5
  4. 9
  5. 15
Worked answer

7.2

Add rates, not times. The first tap fills 112 of the tank each minute and the second fills 118.

Together they fill 112+118 = 336+236 = 536 of the tank per minute.

The time to fill one tank is the reciprocal: 365 = 7.2 minutes, which is sensibly less than either tap alone.

Problem 4

Three kilograms of apples cost $7.50. At the same price per kilogram, what do five kilograms cost?

  1. $10.00
  2. $12.50
  3. $13.75
  4. $15.00
  5. $18.75
Worked answer

$12.50

The price for one kilogram is 7.503 = 2.50 dollars.

Five kilograms therefore cost 5· 2.50 = 12.50 dollars.

Problem 5

A runner covers 5 kilometres in 25 minutes. What is the runner's speed in kilometres per hour?

  1. 6
  2. 10
  3. 12
  4. 15
  5. 20
Worked answer

12

The units asked for are kilometres per hour, so convert the time to hours: 25 minutes is 2560 = 512 of an hour.

Speed is distance divided by time: 5÷512 = 5·125 = 12 kilometres per hour.

Problem 6

A hiker walks up a hill at 3 kilometres per hour and returns down the same path at 6 kilometres per hour. What is the average speed for the whole walk, in kilometres per hour?

  1. 3.5
  2. 4
  3. 4.5
  4. 5
  5. 6
Worked answer

4

Average speed is total distance over total time, which is not the average of the two speeds.

Let the path be d kilometres each way. Going up takes d3 hours and coming down takes d6, a total of 2d6+d6 = 3d6 = d2 hours.

The total distance is 2d, so the average speed is 2dd/2 = 4 kilometres per hour. It is below 4.5 because more time is spent at the slower speed.

Problem 7

A recipe uses 2 cups of flour to make 18 muffins. How many cups of flour are needed for 45 muffins?

  1. 4
  2. 5
  3. 5.4
  4. 6
  5. 7.5
Worked answer

5

Each muffin needs 218 = 19 of a cup.

For 45 muffins that is 459 = 5 cups.

Problem 8

One machine prints 150 labels an hour and another prints 100 labels an hour. Running at the same time, how many hours do they take to print 1000 labels?

  1. 2.5
  2. 3
  3. 3.5
  4. 4
  5. 5
Worked answer

4

Working together the machines print 150+100 = 250 labels an hour.

The time needed is 1000250 = 4 hours.

Problem 9

A resting heart beats 207 times in 3 minutes. What is the rate in beats per minute?

  1. 62
  2. 66
  3. 69
  4. 70
  5. 72
Worked answer

69

Divide the number of beats by the number of minutes to get the rate for one minute.

2073 = 69 beats per minute.

Problem 10

A train leaves at 9:40 and arrives at 13:10 the same day, travelling at a steady 96 kilometres per hour. How far does it travel, in kilometres?

  1. 288
  2. 312
  3. 320
  4. 336
  5. 384
Worked answer

336

From 9:40 to 13:10 is 3 hours and 30 minutes, which is 3.5 hours.

Distance is speed multiplied by time: 96· 3.5 = 336 kilometres.

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