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Rate, Total, Units

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Knowing any two of rate, total and number of units gives you the third

Three numbers in every rate problem

Mia saves $5 every week. After 1 week she has saved $5, after 2 weeks $10, and after 3 weeks $15. Each week adds the same $5, so 3 weeks are 3 lots of $5: 3 × 5 = 15.

Every problem like this has three numbers in it. The rate is the amount for each one, here $5 per week. The number of units is how many there are, here 3 weeks. The total is the amount altogether, here $15.

051015week 1week 2week 3

The savings after each week. Every bar is $5 taller than the one before it, and after 3 weeks the total is $15.

One multiplication and two divisions

The three numbers are joined by one rule: rate × units = total. For Mia, 5 × 3 = 15.

Because of this rule, any two of the numbers find the third. If the rate is the number you do not know, divide the total by the number of units: $15 saved in 3 weeks is 15 ÷ 3 = $5 per week. If the number of units is missing, divide the total by the rate: $15 saved at $5 per week takes 15 ÷ 5 = 3 weeks.

55515

The total of 15 is 3 equal parts of 5. Dividing 15 by 3 gives the size of a part, and dividing 15 by 5 gives the number of parts.

Rates with units

A car travels at a steady 50 km per hour. In 5 hours it travels 50 × 5 = 250 km, which multiplies the rate by the number of hours. A trip of 400 km takes 400 ÷ 50 = 8 hours, which divides the total by the rate. And a trip of 150 km that takes 3 hours goes at 150 ÷ 3 = 50 km per hour, which divides the total by the number of hours.

The units must match before you multiply or divide. The rate is in km per hour, so a time given in minutes must first be changed into hours. 30 minutes is hour, so in 30 minutes the car travels km. The units also check the answer: km per hour times hours gives km.

hourskm002100420063008400

At 50 km per hour, every 2 hours add 100 km. 400 km sits under 8 hours, so 400 ÷ 50 = 8 hours.

The usual mistakes

Dividing by the wrong number. When 15 is shared over 3 equal amounts, each amount is 15 ÷ 3 = 5. The answer 3 is how many amounts there are, not how much is in each one.

Adding instead of multiplying. $5 a week for 3 weeks is not 5 + 3 = $8. Each of the 3 weeks adds $5, so the total is 3 × 5 = $15.

Taking one amount away from the total. 15 − 5 = 10 is not the size of a share: sharing means dividing, so each share is 15 ÷ 3 = 5.

Rate and work · Two People Scan 600 Pages Together Until One of Them Leaves

QuestionAlice and Bob were asked to scan 600 pages of old records. Working alone, Alice can scan all 600 pages in 20 hours. Alice and Bob began working together. After 4 hours, Bob left for another job. Alice carried on alone and took another 10 hours to scan the pages that were left. (a) How many pages did Bob scan in his 4 hours? (b) How long would Bob take to scan all 600 pages working alone?

  1. 1.Alice scans all 600 pages in 20 hours, so she scans 600 ÷ 20 = 30 pages an hour.

    Alice303030303030303030303030303030 an hour
    Alice scans 600 ÷ 20 = 30 pages an hour.

  2. 2.Alice worked 4 hours beside Bob and then 10 hours alone, which is 4 + 10 = 14 hours. In that time she scanned 14 × 30 = 420 pages.

    Alice3030303030303030303030303030420 pageswith Bob 4 halone 10 h
    She worked 14 hours in all: 14 × 30 = 420 pages.

  3. 3.(a) Bob scanned the rest: 600 − 420 = 180 pages.

    Alice3030303030303030303030303030420 pageswith Bob 4 halone 10 hBob????600 − 420 = 180
    (a) The other 180 pages were Bob’s, scanned in his 4 hours.

  4. 4.Bob scanned those 180 pages in 4 hours, so he scans 180 ÷ 4 = 45 pages an hour.

    Alice3030303030303030303030303030420 pageswith Bob 4 halone 10 hBob45454545600 − 420 = 180
    Bob scans 180 ÷ 4 = 45 pages an hour.

  5. 5.Alone, Bob needs 600 ÷ 45 = 1313 hours for all 600 pages.

    Alice3030303030303030303030303030420 pageswith Bob 4 halone 10 hBob45454545600 − 420 = 180Bob alone600 ÷ 4513 1/3 h
    Alone, Bob needs 600 ÷ 45 = 1313 hours.

  6. 6.(b) 13 of an hour is 60 ÷ 3 = 20 minutes, so Bob would take 13 hours 20 minutes. Check: in 13 hours he scans 13 × 45 = 585 pages, and in 20 more minutes 45 ÷ 3 = 15 pages, which makes 600.

    Alice3030303030303030303030303030420 pageswith Bob 4 halone 10 hBob45454545600 − 420 = 180Bob alone600 ÷ 4513 1/3 h
    (b) 13 of an hour is 20 minutes: 13 hours 20 minutes.

Answer: (a) 180 pages; (b) 1313 hours, which is 13 hours 20 minutes

Watch out for

  • Sharing Bob's 180 pages over 14 hours to find his rate. Bob left after 4 hours, so he scanned his 180 pages in 4 hours: 180 ÷ 4 = 45 pages an hour.
  • Counting only Alice's 10 hours alone: 10 × 30 = 300 pages, which leaves 300 for Bob. Alice also worked in the first 4 hours, beside Bob, so she worked 14 hours in all.

Water in and water out

In the next problem, water flows into a reservoir and out of it at the same time. Each minute, the amount that flows in and the amounts that flow out combine into one rate: how much the reservoir gains, or loses, in that minute. Then the rule gives the time: the amount of water to be gained, divided by that rate.

Rate and work · A Reservoir Filled by One Pipe While Water Leaves It Two Ways

QuestionA village reservoir holds 36000 liters when it is full, and it now holds 24000 liters. An intake pipe fills it at 100 liters a minute. At the same time, the reservoir sends 70 liters a minute to the village, and a crack leaks 10 liters a minute. (a) How many hours will it take to fill the reservoir? (b) The intake pipe is closed the moment the reservoir is full. The village supply and the leak carry on at the same rates. How long will it take for the reservoir to empty, in hours and minutes?

  1. 1.The reservoir needs 36000 − 24000 = 12000 liters more to be full.

    Reservoir24000 in12000 to fill36000
    The reservoir is 36000 − 24000 = 12000 liters short of full.

  2. 2.Each minute, 100 liters come in, and 70 + 10 = 80 liters go out to the village and through the crack.

    Reservoir24000 in12000 to fill36000Intake open?−70 village−10+100 in
    Each minute 100 liters come in and 70 + 10 = 80 liters go out.

  3. 3.So each minute the reservoir gains 100 − 80 = 20 liters.

    Reservoir24000 in12000 to fill36000Intake open+20 net−70 village−10+100 in
    So each minute the reservoir gains 100 − 80 = 20 liters.

  4. 4.(a) 12000 liters at 20 liters a minute take 12000 ÷ 20 = 600 minutes, which is 600 ÷ 60 = 10 hours.

    Reservoir24000 in+12000 in 600 min36000 fullIntake open+20 net−70 village−10+100 in
    (a) 12000 ÷ 20 = 600 minutes, which is 10 hours.

  5. 5.Once the intake is closed, nothing comes in, and 80 liters go out each minute from the full 36000 liters.

    Reservoir24000 in+12000 in 600 min36000 fullIntake open+20 net−70 village−10+100 inIntake shut−70−10−80 a minute
    With the intake shut, nothing comes in and 80 liters a minute go out of the full 36000.

  6. 6.(b) The reservoir empties in 36000 ÷ 80 = 450 minutes. 450 = 7 × 60 + 30, so that is 7 hours 30 minutes. Check: 450 × 80 = 36000 liters.

    Reservoir24000 in+12000 in 600 min36000 fullIntake open+20 net−70 village−10+100 inIntake shut−70−10−80 a minute
    (b) 36000 ÷ 80 = 450 minutes: 7 hours 30 minutes.

Answer: (a) 10 hours; (b) 7 hours 30 minutes

Watch out for

  • Dividing the whole 36000 liters by 20 to find the filling time: 1800 minutes. The reservoir already holds 24000 liters, so only the 12000 liters still missing have to come in.
  • Taking the leak away from the village supply: 70 − 10 = 60 liters a minute out. The water sent to the village and the water lost through the crack both leave the reservoir, so together 70 + 10 = 80 liters leave it each minute.

Your turn

1.6 each, 5 times over. What is the total? 

6 in each of 5
Practice Rate, Total, Units in the app