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.
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.
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.
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.
Worked example: Joint Work with Staggered Departures
Question Alice and Bob were assigned to digitize 600 pages of historical archives. Working alone, Alice can digitize all 600 pages in 20 hours. Alice and Bob began working together. After 4 hours of joint work, Bob had to leave for an urgent assignment. Alice continued alone and took another 10 hours to finish digitizing the remaining pages. (a) How many pages did Bob digitize during the 4 hours? (b) How many hours would Bob take to digitize all 600 pages working completely alone?
1.Alice produces 600 ÷ 20 = 30 pages every hour.
Alice does 30 pages an hour. 2.Alice was present for 14 hours in total: 14 × 30 = 420 pages.
She worked 14 hours in all: 420 pages. 3.The remaining pages were completed by Bob in his 4 hours: 600 − 420 = 180 pages.
(a) The other 180 pages were Bob’s, in his 4 hours. 4.(a) Bob digitized 180 pages.
(a) The other 180 pages were Bob’s, in his 4 hours. 5.Bob's rate: 180 ÷ 4 = 45 pages/hr.
6.Time for Bob alone: 600 ÷ 45 = 1313 hours (13 hr 20 min).
Bob’s rate: 45 pages an hour.
Answer: (a) 180 pages; (b) 1313 hours, that is 13 h 20 min
Common mistakes
- Assuming Bob worked for 14 hours as well, rather than only during the first 4 hours.
- Dividing 180 pages by 14 hours instead of Bob's actual working duration of 4 hours.
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.
Worked example: Rate with Continuous Leakage / Depletion
Question A community water reservoir with a total capacity of 36000 liters currently holds 24000 liters of water. An intake pipe replenishes the reservoir at a rate of 100 liters/min. At the same time, the reservoir supplies water to a village at 70 liters/min, and an undetected crack leaks water at 10 liters/min. (a) How many hours will it take to fill the reservoir completely to its capacity? (b) If the intake pipe is closed the moment the reservoir is completely full, how many hours and minutes will it take for the reservoir to empty completely while supplying the village and leaking at the same rates?
1.Capacity deficit = 36000 − 24000 = 12000 liters.
12000 liters short of full. 2.Every minute: +100 ℓ in, −70 ℓ to village, −10 ℓ leak ⟹ +20 ℓ/min net gain.
Each minute the reservoir gains 100 and loses 80: net +20. 3.Fill time: 12000 ÷ 20 = 600 min = 10 hours.
(a) 12000 ÷ 20 = 600 minutes: 10 hours. 4.(a) 10 hours.
5.When tap shuts: loss is 70 + 10 = 80 ℓ/min.
With the intake shut it loses 80 a minute. 6.Empty time from full (36000 ℓ): 36000 ÷ 80 = 450 minutes = 7 hr 30 min.
(b) 36000 ÷ 80 = 450 minutes: 7 hours 30 minutes.
Answer: (a) 10 hours; (b) 7 hours 30 minutes
Common mistakes
- Using the full 36,000 liters instead of the 12,000 liter deficit to find the filling time.
- Subtracting the 10 liter leak from the 70 liter supply instead of adding them together to find total outflow.