Problems Involving Rate

Solving rate problems of more than one step: a rate with a change of units, two taps or two workers together, and rates given as a ratio.

Before you start

A rate is an amount of one quantity per unit of another, such as 12 L/min, $15 per hour or $8 per m². By the unitary method, find the amount for 1 unit first, then multiply: if a printer prints 90 pages in 2 min, it prints $90 \div 2 = 45$ pages per minute, so in 7 min it prints $45 \times 7 = 315$ pages.

Units of volume and time: 1 L = 1000 mL and 1 mL = 1 cm³; 1 h = 60 min and 1 min = 60 s. A part of a minute is changed to seconds by multiplying by 60: 0.2 min is 12 s.

In a ratio problem, quantities in the ratio 5 : 2 are drawn as 5 units and 2 units; once 1 unit is known, every quantity is known.

Working in Steps

The problems in this lesson take more than one step. None of the steps is new: each one is a rate, a unit change or a ratio you already know. The skill is in seeing which steps are needed and in what order, and in keeping the units right at every step.

How to solve a rate problem

1. Write down what is given, with its units, and what is to be found.

2. Draw it if it helps: a bar model for quantities in a ratio, a table for amounts over time.

3. Put every quantity into units that agree before you multiply or divide (all in litres and minutes, or all in millilitres and seconds).

4. Work one step at a time. Keep exact values and round only the final answer.

5. Give the answer in a sentence with its unit, and check that it is sensible.

Example 1: Filling a tank (a rate with a change of units)

An empty fish tank is a cuboid 80 cm long, 50 cm wide and 45 cm high. A tap pours water into it at 125 mL per second. How many minutes does it take to fill the tank?

Solution

Step 1: the volume of the tank. $80 \times 50 \times 45 = 180\,000$ cm³.

Step 2: put the volume in the unit of the rate. 1 cm³ = 1 mL, so the tank holds 180 000 mL.

Step 3: the time. Each second 125 mL flows in, so the time is $180\,000 \div 125 = 1440$ s.

Step 4: change to minutes. $1440 \div 60 = 24$ min.

Check another way: 125 mL/s is $125 \times 60 = 7500$ mL, or 7.5 L, each minute. The tank holds 180 L, and $180 \div 7.5 = 24$ min.

It takes 24 minutes to fill the tank.

An exchange rate is a rate too: it says how much of one currency you get for one unit of another. "S$1 = ¥108" means that each Singapore dollar buys 108 Japanese yen, so to change dollars into yen you multiply by 108, and to change yen into dollars you divide by 108.

Example 2: Cheaper in Tokyo or in Singapore?

Before a trip to Tokyo, Hui Min changes S$500 into Japanese yen (¥) at S$1 = ¥108. In Tokyo she sees a pair of running shoes for ¥16 200. The same shoes cost S$169 in Singapore.
(a) Where are the shoes cheaper, and by how much in Singapore dollars?
(b) If she buys the shoes in Tokyo, how many yen does she have left?

Solution

(a) Change the Tokyo price into Singapore dollars. Each S$1 buys ¥108, so ¥16 200 is worth $16\,200 \div 108 = 150$ Singapore dollars.

In Singapore the shoes cost S$169, and $169 - 150 = 19$ .

(b) Hui Min receives $500 \times 108 = 54\,000$ yen.

After the shoes she has $54\,000 - 16\,200 = 37\,800$ yen.

Check (a) the other way: S$169 is worth $169 \times 108 = 18\,252$ yen, which is more than ¥16 200.

(a) The shoes are cheaper in Tokyo, by S$19. (b) She has ¥37 800 left.

Working Together

When two taps fill the same tank, each pours in its own amount every minute, so together they pour in the sum of the two amounts. The rates add. Times do not add, and they do not average either: two taps together are faster than either tap alone.

Rule: rates working together

Find each rate as an amount per unit of time (litres per minute, pages per hour).

Working together on the same job, the combined rate = the sum of the rates. Something working against them, such as an open drain, takes its rate away.

Time taken together = total amount ÷ combined rate.

Example 3: Two taps filling a tank

Tap A alone fills a 360 L tank in 12 minutes. Tap B alone fills it in 18 minutes. How long do the two taps take to fill the empty tank together? Give the answer in minutes and seconds.

Solution

Rate of tap A: $360 \div 12 = 30$ L/min. Rate of tap B: $360 \div 18 = 20$ L/min.

Together they pour in $30 + 20 = 50$ L each minute.

Time to fill 360 L: $360 \div 50 = 7.2$ min.

0.2 min is $0.2 \times 60 = 12$ s, so 7.2 min is 7 min 12 s.

Check: in 7.2 min tap A pours in $30 \times 7.2 = 216$ L and tap B $20 \times 7.2 = 144$ L, and $216 + 144 = 360$ L. The time is less than 12 minutes, the time of the faster tap alone, as it must be.

The two taps together fill the tank in 7 minutes 12 seconds.

	Tap A	Tap B	Both taps
Water in 1 min	30 L	20 L	50 L
Time to fill 360 L	12 min	18 min	7 min 12 s

Example 4: Two pumps, one twice as fast as the other

Two pumps drain a fish pond at a community garden. Pump A works twice as fast as pump B. Together they drain 900 L of water in 15 minutes. How long would each pump take to drain the 900 L alone?

Solution

Combined rate: $900 \div 15 = 60$ L/min.

Pump A drains twice as much as pump B each minute, so their rates are in the ratio 2 : 1. Draw pump A's rate as 2 units and pump B's as 1 unit.

3 units → 60 L/min, so 1 unit → $60 \div 3 = 20$ L/min.

Pump B drains 20 L/min and pump A drains $2 \times 20 = 40$ L/min.

Pump A alone: $900 \div 40 = 22.5$ min, which is 22 min 30 s. Pump B alone: $900 \div 20 = 45$ min.

Check: $40 + 20 = 60$ L/min, and $60 \times 15 = 900$ L.

Pump A alone would take 22 minutes 30 seconds, and pump B alone 45 minutes.

Rates in a Ratio

Sometimes two rates are not given, but how they compare is: "the office rent per m² is 2.5 times the warehouse rent per m²". Write the comparison as a ratio of whole numbers and use units, as in ratio problems. Each unit is now a rate (dollars per m²), so first multiply each rate by its area to find how many units each part of the rent stands for, add them, then use the total rent to find 1 unit.

Example 5: Office and warehouse rent

A company rents an office of 100 m² and a warehouse of 300 m². The rent per m² of the office is 2.5 times the rent per m² of the warehouse. The total rent is $22 000 a month. Find the monthly rent per m² of the warehouse and of the office.

Solution

Office rate : warehouse rate = 2.5 : 1 = 5 : 2. Let the office rate be 5 units per m² and the warehouse rate 2 units per m².

The bars show rates per m², not rents, so 7 units is not $22 000. The rent is 100 m² at the office rate and 300 m² at the warehouse rate.

Office rent: $100 \times 5 = 500$ units. Warehouse rent: $300 \times 2 = 600$ units. Total: $500 + 600 = 1100$ units.

1100 units → $22 000, so 1 unit → $22\,000 \div 1100 = 20$ dollars per m².

Warehouse rate: 2 units = $2 \times 20 = 40$ dollars per m². Office rate: 5 units = $5 \times 20 = 100$ dollars per m².

Another way: 1 m² of office costs as much as 2.5 m² of warehouse, so the 100 m² office costs as much as 250 m² of warehouse. The whole rent is that of $250 + 300 = 550$ m² of warehouse, so the warehouse rate is $22\,000 \div 550 = 40$ dollars per m².

Check: $100 \times 100 + 300 \times 40 = 10\,000 + 12\,000 = 22\,000$ .

The warehouse rent is $40 per m² a month and the office rent is $100 per m² a month.

Common Mistakes

Adding or averaging times. Taps that take 12 min and 18 min alone do not take 30 min together, nor 15 min. Add the rates (30 L/min + 20 L/min) and divide the amount by the sum.
Treating units of a rate as parts of a total. When the units stand for dollars per m², 5 units + 2 units is not the total rent. Multiply each rate by its area first.
Mixing units. A rate in litres per minute with a volume in cm³ and a time in seconds gives nonsense. Change everything to matching units first.
Multiplying when you should divide. Ask: should the answer be larger or smaller? For example, at S$1 = ¥108, yen are changed into dollars by dividing by 108, so the number of dollars is smaller than the number of yen.
Rounding in the middle. Rounding a rate or a time part-way puts the final answer out. Keep the exact value and round once, at the end.

Practice

Try each question before opening its solution. The questions get harder as you go.

Question 1

A printer prints 45 pages per minute. How long does it take to print 30 copies of a 12-page worksheet?

ShowHide solution

Pages: $30 \times 12 = 360$ .

Time: $360 \div 45 = 8$ min.

It takes 8 minutes.

Question 2

A machine fills 18 bottles every 45 s. How many bottles does it fill in 1 h 15 min?

ShowHide solution

Rate: $18 \div 45 = 0.4$ bottles per second.

1 h 15 min = 75 min = $75 \times 60 = 4500$ s.

Bottles: $0.4 \times 4500 = 1800$ .

Check: 4500 s is $4500 \div 45 = 100$ lots of 45 s, and $100 \times 18 = 1800$ .

The machine fills 1800 bottles.

Question 3

An empty tank is a cuboid 60 cm long, 40 cm wide and 50 cm high. Water flows into it at 8 L/min. How long does it take to fill the tank?

ShowHide solution

Volume: $60 \times 40 \times 50 = 120\,000$ cm³ = 120 000 mL = 120 L.

Time: $120 \div 8 = 15$ min.

It takes 15 minutes to fill the tank.

Question 4

Mei Ling changes S$800 into Malaysian ringgit (RM) at S$1 = RM3.45. She spends RM2150 in Malaysia. Back in Singapore she changes the ringgit she has left into Singapore dollars at S$1 = RM3.50. How many Singapore dollars does she get?

ShowHide solution

Ringgit received: $800 \times 3.45 = 2760$ , so RM2760.

Ringgit left: $2760 - 2150 = 610$ , so RM610.

Each S$1 costs RM3.50, so RM610 buys $610 \div 3.50 = 174.285...$ Singapore dollars.

She gets S$174.29 (to the nearest cent).

Question 5

Working together, Ali and Hui Min pack 300 goodie bags for a school carnival in 2 hours. Ali alone packs 90 bags an hour. How long would Hui Min take to pack the 300 bags alone?

ShowHide solution

Together: $300 \div 2 = 150$ bags an hour.

Hui Min: $150 - 90 = 60$ bags an hour.

Alone: $300 \div 60 = 5$ h.

Hui Min would take 5 hours.

Question 6

Tap A alone fills a 240 L tank in 20 min, and tap B alone fills it in 30 min. Both taps are turned on to fill the empty tank, but the drain has been left open and lets out 4 L of water every minute. How long does the tank take to fill?

ShowHide solution

Tap A: $240 \div 20 = 12$ L/min. Tap B: $240 \div 30 = 8$ L/min.

Each minute the water in the tank goes up by $12 + 8 - 4 = 16$ L.

Time: $240 \div 16 = 15$ min.

The tank takes 15 minutes to fill.

Question 7

Tap A pours water into an empty 360 L tank at 30 L/min. After 4 minutes, tap B, which pours 20 L/min, is also turned on. How long does the tank take to fill altogether? Give the answer in minutes and seconds.

ShowHide solution

In the first 4 min tap A alone pours in $30 \times 4 = 120$ L.

Left to fill: $360 - 120 = 240$ L.

Both taps together: $30 + 20 = 50$ L/min, so the rest takes $240 \div 50 = 4.8$ min, which is 4 min 48 s.

Altogether: 4 min + 4 min 48 s = 8 min 48 s.

The tank takes 8 minutes 48 seconds to fill.

Question 8

Streaming a video on Siti's phone uses 4 MB of data every 10 s. Her phone plan has 18 GB of data left this month. For how many hours of video is that enough? (Take 1 GB = 1000 MB.)

ShowHide solution

Rate: $4 \div 10 = 0.4$ MB per second.

Data left: $18 \times 1000 = 18\,000$ MB.

Time: $18\,000 \div 0.4 = 45\,000$ s.

In hours: $45\,000 \div 3600 = 12.5$ h, which is 12 h 30 min.

It is enough for 12 hours 30 minutes of video.

Question 9

For a school hall, grade A tiles cost 3 times as much per m² as grade B tiles. The school uses 40 m² of grade A tiles and 90 m² of grade B tiles, and pays $5250 for them. Find the cost per m² of each grade.

ShowHide solution

Grade A rate : grade B rate = 3 : 1. Let grade A cost 3 units per m² and grade B 1 unit per m².

Grade A: $40 \times 3 = 120$ units. Grade B: $90 \times 1 = 90$ units. Total: 210 units.

210 units → $5250, so 1 unit → $5250 \div 210 = 25$ dollars per m².

Grade B: $25 per m². Grade A: $3 \times 25 = 75$ dollars per m².

Check: $40 \times 75 + 90 \times 25 = 3000 + 2250 = 5250$ .

Grade A tiles cost $75 per m² and grade B tiles $25 per m².

Question 10

Mrs Goh pays a cleaner and a gardener by the hour. The cleaner's rate per hour is 1.5 times the gardener's. In one month the cleaner works 20 hours and the gardener 12 hours, and Mrs Goh pays them $672 altogether. Find each one's rate per hour.

ShowHide solution

Cleaner's rate : gardener's rate = 1.5 : 1 = 3 : 2. Let the cleaner earn 3 units per hour and the gardener 2 units per hour.

Cleaner: $20 \times 3 = 60$ units. Gardener: $12 \times 2 = 24$ units. Total: 84 units.

84 units → $672, so 1 unit → $672 \div 84 = 8$ dollars per hour.

Gardener: $2 \times 8 = 16$ dollars per hour. Cleaner: $3 \times 8 = 24$ dollars per hour.

Check: $20 \times 24 + 12 \times 16 = 480 + 192 = 672$ .

The cleaner earns $24 per hour and the gardener $16 per hour.

Question 11

Printer A prints 3 pages for every 2 pages that printer B prints in the same time. Working together, they print 600 pages in 4 minutes. How long would each printer take to print the 600 pages alone?

ShowHide solution

Together: $600 \div 4 = 150$ pages per minute.

Rate of A : rate of B = 3 : 2. Draw A's rate as 3 units and B's as 2 units.

5 units → 150 pages per minute, so 1 unit → $150 \div 5 = 30$ pages per minute.

Printer A: $3 \times 30 = 90$ pages per minute. Printer B: $2 \times 30 = 60$ pages per minute.

A alone: $600 \div 90 = 6\frac{2}{3}$ min, and $\frac{2}{3}$ min is $\frac{2}{3} \times 60 = 40$ s, so 6 min 40 s. B alone: $600 \div 60 = 10$ min.

Check: $90 + 60 = 150$ pages per minute, and $150 \times 4 = 600$ .

Printer A alone would take 6 minutes 40 seconds and printer B alone 10 minutes.

Key Points

Solve a problem in steps: what is given and what is asked, a drawing if it helps, units that agree, one step at a time, an answer in a sentence with its unit.
A rate can be used both ways: multiply by it to go from one quantity to the other, divide by it to go back (at S$1 = ¥108, dollars × 108 gives yen, and yen ÷ 108 gives dollars).
Working together on one job, rates add; something working against them takes its rate away. Time = amount ÷ combined rate.
Never add or average the times of two workers or taps: add their rates.
When rates are given as a ratio, write the ratio in whole numbers and let each unit be a rate; multiply each rate by its quantity before using the total to find 1 unit.
Keep exact values to the end; give money to the nearest cent and other values that are not exact to 3 significant figures.