Maths · 1 Number

1.11 Rates

Before you start

Multiplying and dividing decimals, with and without a calculator.
There are $60$ seconds in a minute and $60$ minutes in an hour, and times on the 24-hour clock are written like 07 50.
Metric conversions: $1$ km $= 1000$ m, $1$ litre $= 1000$ cm³ and $1$ m³ $= 1000$ litres.
Rounding to 3 significant figures, and to 2 decimal places for money.

Syllabus objectives covered (2)

Use common measures of rate.
Solve problems involving average speed.

Part 1 of 7

What a rate is, and how its unit tells you what to do

A rate compares two different kinds of quantity. It tells you how much of one goes with one unit of the other. A wage of $12 per hour means $12 for each hour worked. A speed of $60$ km/h means $60$ km for each hour of travel.
The word 'per' and the slash in a unit both mean 'for each'. They also tell you how the rate was made: the first quantity divided by the second. km/h is kilometres ÷ hours. m/s is metres ÷ seconds. g/cm³ is grams ÷ cubic centimetres.
So every rate links three quantities: the rate and the two quantities it compares. If you know two of the three, you can find the third.
$rate = \frac{first quantity}{second quantity} first quantity = rate \times second quantity$
To use a rate, multiply it by the number of units. At $60$ km/h for $3$ hours you travel $60 \times 3 = 180$ km. Notice that 'km per hour' times 'hours' leaves just km. Checking the units like this tells you whether to multiply or divide.
The same reading works for any unit, including ones from science. A metal block of $2.7$ g/cm³ has a mass of $2.7$ g for every cm³, so $10$ cm³ of it has a mass of $2.7 \times 10 = 27$ g.
Before you calculate, ask whether the answer should be bigger or smaller than the number you start with. That one question catches a multiply-or-divide slip before it costs you the mark.

Part 2 of 7

Hourly rates of pay

An hourly rate of pay is money per hour. Pay = hourly rate × hours worked. At $12.50 per hour, $36$ hours earns $12.50 \times 36 = 450$ , so $450.
To find the rate from the pay, divide by the hours, because the rate is the pay for one hour. Someone paid $552 for $40$ hours earns $552 \div 40 = 13.80$ , so $13.80 per hour.
Overtime is time worked beyond the normal hours, and it is paid at a higher rate. 'Time and a half' means $1.5$ times the normal rate. 'Double time' means $2$ times the normal rate. At $12.50 per hour, time and a half is $12.50 \times 1.5 = 18.75$ , so $18.75 per hour.
Work out the pay for the normal hours and the pay for the overtime hours separately, then add them. The trap is multiplying all the hours by $1.5$ : only the extra hours get the higher rate.
Write money to the cent: $18.75, not $18.8. $18.75 is exact, and an exact answer should not be rounded.

Part 3 of 7

Exchange rates

An exchange rate tells you how much of one currency you get for one unit of another. For example, a bank might offer £1 = $1.25. Rates change from day to day, so always use the rate the question gives you.
Going from the currency with the 1, multiply. £80 is $80 \times 1.25 = 100$ , so $100. Each pound is worth $1.25$ dollars, so $80$ pounds is $80$ lots of $1.25$ .
Going to the currency with the 1, divide. $150 is $150 \div 1.25 = 120$ , so £120. You are asking how many lots of $1.25 fit into $150.
Check the size. At this rate one pound is worth more than one dollar, so you get more dollars than the pounds you hand over, and fewer pounds than the dollars you hand over. If your answer goes the wrong way, you multiplied when you should have divided, or the other way round.
To compare prices in two countries, change both prices into the same currency first, then compare. Comparing $40 with £35 directly tells you nothing, because the units are different.

Part 4 of 7

Flow rates and fuel consumption

A flow rate is volume per unit of time: litres per minute, cm³ per second, m³ per hour. It works like any other rate.
$volume = flow rate \times time time = \frac{volume}{flow rate}$
A tap fills a $300$ -litre tank at $12$ litres per minute. The time is $300 \div 12 = 25$ minutes. You divide because you are asking how many lots of $12$ litres make $300$ litres.
Make the volume units match before you divide. $1$ litre $= 1000$ cm³ and $1$ m³ $= 1000$ litres, so a tank of $1.5$ m³ holds $1.5 \times 1000 = 1500$ litres. Dividing m³ by litres per minute gives a meaningless number.
Make the time units match as well. $0.5$ litres per second is $0.5 \times 60 = 30$ litres per minute, and $30 \times 60 = 1800$ litres per hour.
Fuel consumption says how much fuel a vehicle uses for the distance it travels. It is written in two different ways, so read the unit to see which one you have.
In kilometres per litre (km/l), the number is the distance for one litre. At $14$ km per litre, a $280$ km trip needs $280 \div 14 = 20$ litres. A bigger number means the vehicle uses less fuel.
In litres per 100 km, the number is the fuel for every $100$ km. At $5.5$ litres per $100$ km, a $360$ km trip is $3.6$ lots of $100$ km, so it needs $3.6 \times 5.5 = 19.8$ litres. Here a smaller number means the vehicle uses less fuel.
To change from one form to the other, use $100$ km as the link. $5$ litres per $100$ km means $100$ km on $5$ litres, which is $100 \div 5 = 20$ km per litre.

Part 5 of 7

Speed, distance and time

Speed is a rate: distance per unit of time. You must remember this formula. The syllabus says knowledge of it is required, so do not expect to be given it.
$speed = \frac{distance}{time}$
Rearrange it like any rate: distance = speed × time, and time = distance ÷ speed. As a memory aid, picture a triangle with D on top and S and T side by side underneath. Cover the one you want: side by side means multiply, one above the other means divide.
$d = s \times t s = \frac{d}{t} t = \frac{d}{s}$
Average speed is the total distance divided by the total time. A car's speed goes up and down on a journey. The average speed is the steady speed that would cover the same distance in the same time.
The units must match. Distance in km and time in hours gives km/h. Distance in m and time in s gives m/s. If the time is in minutes, change it to hours before you divide, or your answer is in km per minute.
To change minutes into hours, divide by $60$ , because there are $60$ minutes in an hour. $45$ minutes is $45 \div 60 = 0.75$ hours, so $3$ hours $45$ minutes is $3.75$ hours.
The trap: $3$ hours $45$ minutes is not $3.45$ hours. $3.45$ hours is $3$ hours and $0.45 \times 60 = 27$ minutes. Time counts in sixties; decimals count in tens.
Some numbers of minutes give recurring decimals. $20$ minutes is $\frac{20}{60} = \frac{1}{3}$ hour $= 0.333 \dots$ hours. Keep it as $\frac{1}{3}$ , or keep the full calculator value. Rounding it to $0.33$ early makes the final answer drift.
To turn a decimal number of hours back into hours and minutes, multiply only the decimal part by $60$ . $2.3$ hours is $2$ hours and $0.3 \times 60 = 18$ minutes.
For a time between two 24-hour clock times, count up in steps. From 08 35 to 11 10: 08 35 to 10 35 is $2$ hours, and 10 35 to 11 10 is $35$ minutes, so $2$ hours $35$ minutes. Do not subtract the times as decimals: $11.10 - 8.35 = 2.75$ is not a time at all.

Part 6 of 7

Changing between km/h and m/s

$1$ km $= 1000$ m and $1$ hour $= 60 \times 60 = 3600$ seconds. So to change km/h into m/s, multiply by $1000$ to get metres, then divide by $3600$ to get per second. For $72$ km/h:
$\frac{72 \times 1000}{3600} = 20$
So $72$ km/h is $20$ m/s. Multiplying by $1000$ and dividing by $3600$ is the same as dividing by $3.6$ : $72 \div 3.6 = 20$ .
Going back, from m/s to km/h, multiply by $3.6$ . $15$ m/s is $15 \times 3.6 = 54$ km/h.
Check the direction with a size test. The number in m/s is always smaller than the number in km/h for the same speed, because $1$ m/s is the same as $3.6$ km/h. If your m/s answer is bigger, you multiplied when you should have divided.

Part 7 of 7

Average speed for a whole journey

When a journey has more than one stage, the average speed for the whole journey is the total distance divided by the total time.
$average speed = \frac{total distance}{total time}$
Do not average the speeds. Drive $60$ km at $60$ km/h, then $60$ km at $30$ km/h. The first stage takes $1$ hour and the second takes $2$ hours. The average speed is $120 \div 3 = 40$ km/h, not $\frac{60 + 30}{2} = 45$ km/h. The slow stage lasts longer, so it pulls the average down further.
Find the time for each stage with time = distance ÷ speed, then add the times. A small table, with a row for each stage and columns for distance, speed and time, keeps the working in order.
Stops count. If the journey includes a $30$ -minute rest, add $0.5$ hours to the total time. The clock keeps running while you are stopped, but the distance does not grow.
Read exactly what the question asks for. 'Average speed for the whole journey' includes any stops. If a question asks for the average speed only while moving, leave the stops out.

Worked examples

Example 1. A train travels $156$ km in $1$ hour $36$ minutes. (a) Calculate its average speed in km/h. (b) Write this speed in m/s, correct to 3 significant figures.

Show the working

Change the time into hours, because the answer is wanted in km per hour. $36$ minutes is $36 \div 60 = 0.6$ hours, so the time is $1.6$ hours, not $1.36$ hours.
(a) Speed is distance divided by time.
$speed = \frac{156}{1.6} = 97.5$
Check it another way. $1$ hour $36$ minutes is $96$ minutes. $156 \div 96 = 1.625$ km per minute, and $1.625 \times 60 = 97.5$ km per hour. The same answer.
(b) To change km/h to m/s, multiply by $1000$ and divide by $3600$ , which is the same as dividing by $3.6$ .
$\frac{97.5 \times 1000}{3600} = 27.083 \dots$
Round only at the end: $27.083 \dots$ is $27.1$ to 3 significant figures. The m/s number is smaller than the km/h number, as it must be.

Answer: (a) $97.5$ km/h (b) $27.1$ m/s

Example 2. The exchange rate is €1 = $1.08. (a) Change €350 into dollars. (b) Change $500 into euros, giving your answer to the nearest cent.

Show the working

Decide the direction first. The rate starts from €1, so euros to dollars is a multiply, and dollars to euros is a divide.
(a) Each euro is worth $1.08, so €350 is $350$ lots of $1.08$ .
$350 \times 1.08 = 378$
Size check: at this rate one euro is worth more than one dollar, so the number of dollars ( $378$ ) should be bigger than the number of euros ( $350$ ). It is.
(b) Now you want to know how many lots of $1.08 fit into $500, so divide.
$500 \div 1.08 = 462.962 \dots$
To the nearest cent means 2 decimal places: €462.96. Size check: fewer euros than dollars, which is the right way round.
Check by going back: $462.96 \times 1.08 = 499.9968$ , which is $500 to the nearest cent.

Answer: (a) $378 (b) €462.96

Example 3. A car uses $6.5$ litres of fuel per $100$ km. (a) Calculate how much fuel it uses on a journey of $380$ km. (b) Fuel costs $1.74 per litre. Calculate the cost of the fuel for this journey, to the nearest cent. (c) Write the car's fuel consumption in kilometres per litre, correct to 3 significant figures.

Show the working

(a) The rate is per $100$ km, so count the hundreds in the journey: $380 \div 100 = 3.8$ .
Each $100$ km uses $6.5$ litres, so multiply.
$3.8 \times 6.5 = 24.7$
(b) Cost is the price per litre times the number of litres.
$24.7 \times 1.74 = 42.978$
To the nearest cent that is $42.98. Here $24.7$ from (a) was exact; whenever an earlier answer is not exact, carry its full value into the next part.
(c) $6.5$ litres per $100$ km means $100$ km on $6.5$ litres. So one litre takes the car $100 \div 6.5$ km.
$\frac{100}{6.5} = 15.384 \dots$
To 3 significant figures, that is $15.4$ km per litre. Check: $15.4 \times 6.5 = 100.1$ , which is close to $100$ .

Answer: (a) $24.7$ litres (b) $42.98 (c) $15.4$ km per litre

Example 4. Leila drives $120$ km at an average speed of $80$ km/h. She stops for $30$ minutes. She then drives a further $90$ km, which takes $1$ hour $15$ minutes. She sets off at 08 40. (a) Find the time she arrives. (b) Calculate her average speed for the whole journey.

Show the working

Find the time for each part, in hours. Stage 1: time = distance ÷ speed.
$t_{1} = \frac{120}{80} = 1.5$
The stop is $30$ minutes, which is $0.5$ hours. Stage 2 is $1$ hour $15$ minutes, which is $1.25$ hours, because $15 \div 60 = 0.25$ .
Add the times, including the stop. The total is $3.25$ hours, which is $3$ hours $15$ minutes.
$1.5 + 0.5 + 1.25 = 3.25$
(a) 08 40 plus $3$ hours is 11 40. Plus $15$ minutes is 11 55.
(b) The total distance is $120 + 90 = 210$ km. Divide by the total time.
$average speed = \frac{210}{3.25} = 64.615 \dots$
To 3 significant figures that is $64.6$ km/h. It is lower than the $80$ km/h of the first stage, because the slower second stage ( $90 \div 1.25 = 72$ km/h) and the stop both pull the average down.

Answer: (a) 11 55 (b) $64.6$ km/h

Where people go wrong

Mistake: Writing $3$ hours $45$ minutes as $3.45$ hours in a speed calculation.

Instead: Divide the minutes by $60$ : $45 \div 60 = 0.75$ , so the time is $3.75$ hours.

There are $60$ minutes in an hour, not $100$ . $3.45$ hours is really $3$ hours $27$ minutes, so every answer that uses it is wrong.

Mistake: Finding the average speed of a two-stage journey by adding the two speeds and halving.

Instead: Divide the total distance by the total time. Find each stage's time with time = distance ÷ speed first.

When the stages are the same distance at different speeds, the slower stage takes longer, so it counts for more of the time than a simple average of the speeds allows.

Mistake: Multiplying by the exchange rate when you should divide, e.g. changing $150 into pounds at £1 = $1.25 as $150 \times 1.25 = 187.5$ .

Instead: Multiply when you go from the currency with the 1; divide when you go to it. $150 is $150 \div 1.25 = 120$ , so £120.

Both operations give a number, so a size check is the quickest way to catch the slip. One pound buys more than one dollar here, so you must get fewer pounds than the dollars you started with.

Mistake: Changing km/h into m/s by multiplying by $3.6$ .

Instead: Divide by $3.6$ to go from km/h to m/s, and multiply by $3.6$ to go from m/s to km/h. $72$ km/h is $20$ m/s.

$1$ m/s is the same as $3.6$ km/h, so the number in m/s is always the smaller one.

Mistake: Leaving a stop out of the total time when the question asks for the average speed for the whole journey.

Instead: Add the time of every stop to the total time. Leave stops out only if the question asks for the average speed while moving.

The whole journey lasts from setting off to arriving, and the clock keeps running during a stop even though no distance is covered.

Check yourself

Write $2$ hours $24$ minutes in hours, as a decimal.
Answer
$24 \div 60 = 0.4$ , so $2.4$ hours.
A car travels $252$ km in $3$ hours $30$ minutes. Calculate its average speed.
Answer
$252 \div 3.5 = 72$ km/h.
Change $90$ km/h into m/s.
Answer
$90 \times 1000 \div 3600 = 90 \div 3.6 = 25$ m/s.
Raj is paid $391 for $34$ hours of work. Work out his hourly rate of pay.
Answer
$391 \div 34 = 11.5$ , so $11.50 per hour.
The exchange rate is £1 = €1.16. Change €290 into pounds.
Answer
Going to the currency with the 1, so divide: $290 \div 1.16 = 250$ , so £250.
An empty tank holds $0.6$ m³. Water flows in at $15$ litres per minute. How many minutes does it take to fill the tank?
Answer
$0.6$ m³ $= 600$ litres. $600 \div 15 = 40$ minutes.
A van uses $7.5$ litres of fuel per $100$ km. How much fuel does it use on a $340$ km journey?
Answer
$340 \div 100 = 3.4$ , and $3.4 \times 7.5 = 25.5$ litres.
Tom walks $6$ km at $4$ km/h, then runs another $6$ km at $12$ km/h. Calculate his average speed for the $12$ km.
Answer
Times: $6 \div 4 = 1.5$ hours and $6 \div 12 = 0.5$ hours, so $2$ hours in total. Average speed $= 12 \div 2 = 6$ km/h, not $8$ km/h.

Exam questions

Try each part first. Open the mark scheme after you have written your answer.

Question 1. A bus leaves a town at 07 50 and arrives at a city at 10 14. The distance it travels is $186$ km.

(a) Write down the time the journey takes, in hours and minutes. 1 mark

Mark scheme

B1 2 h 24 min cao

(b) Calculate the average speed of the bus, in km/h. 2 marks

Mark scheme

M1 186 ÷ 2.4 oe, e.g. 186 ÷ 144 × 60; FT their (a) correctly changed to hours
A1 77.5 cao

(c) Write your answer to part (b) in metres per second. 2 marks

Mark scheme

M1 their 77.5 × 1000 ÷ 3600 oe, e.g. their 77.5 ÷ 3.6
A1 21.5 (21.527…) awrt 21.5; FT their (b)

(d) At the city the bus stops for $20$ minutes. It then travels a further $64$ km at an average speed of $48$ km/h. Calculate the average speed of the bus for the whole journey, from 07 50 until the end of the $64$ km, including the stop. 4 marks

Mark scheme

M1 64 ÷ 48 (= 1.33… hours or 80 minutes) soi
M1 total time 2.4 + 20/60 + their 64/48 (= 4.066… hours or 244 minutes); FT their (a)
M1 (186 + 64) ÷ their total time in hours, or 250 ÷ their total minutes × 60
A1 61.5 (61.47…) awrt 61.5

Question 2. Ines is paid €15.20 per hour for the first $37.5$ hours she works in a week. Any extra hours are overtime, paid at time and a half.

(a) One week Ines works $42$ hours. Calculate her pay for that week. 3 marks

Mark scheme

M1 37.5 × 15.20 (= 570)
M1 4.5 × 15.20 × 1.5 oe (= 102.60), e.g. 4.5 × 22.80
A1 €672.60 (accept 672.6) cao. Alternative method: M1 for 42 × 15.20 (= 638.40), M1 for 4.5 × 0.5 × 15.20 oe (= 34.20), A1 for 672.60

(b) On holiday, Ines changes €450 into Swiss francs. The exchange rate is €1 = $0.94$ francs. Calculate the number of francs she receives. 1 mark

Mark scheme

B1 423 cao

(c) A watch costs $260$ francs in Switzerland and €285 in Ines's home country. Using the exchange rate in part (b), determine where the watch is cheaper, and by how many euros. 2 marks

Mark scheme

M1 260 ÷ 0.94 (= 276.59…), or 285 × 0.94 (= 267.9) to compare in francs
A1 Switzerland, cheaper by €8.40 (8.404…) awrt 8.40

Question 3. An empty storage tank holds $2.4$ m³ of water when it is full. Water is pumped into the tank at a constant rate of $16$ litres per minute.

(a) Work out how long it takes to fill the tank. Give your answer in hours and minutes. 3 marks No calculator

Mark scheme

B1 2400 (litres) seen
M1 their 2400 ÷ 16 (= 150 minutes)
A1 2 hours 30 minutes cao

(b) Show that a rate of $16$ litres per minute is $267$ cm³ per second, correct to 3 significant figures. 2 marks No calculator

Mark scheme

M1 16 × 1000 ÷ 60 oe, e.g. 16 000 ÷ 60
A1 266.6… or 266.7 seen before the final 267; nfww

How this is examined

Finding one average speed for a journey with several stages and a stop: A later part of a journey question: a second stage and a stop are added, and you find the average speed from setting off to arriving The time for each stage from time = distance ÷ speed (e.g. $64 \div 48$ hours), a total time that includes the stop, then total distance ÷ total time. Each of those steps is a separate method mark; the answer to 3 significant figures is the accuracy mark. Averaging the speeds earns nothing.
Working out a week's pay when some hours are overtime: A Calculate part with an hourly rate, a number of normal hours, and overtime at time and a half or double time Normal hours × normal rate seen, overtime hours × $1.5$ × normal rate seen, then the two added. Show both products, so a slip in one still leaves the method mark for the other.
Average speed when the time is given in hours and minutes or as two clock times: A Write down part for the time taken, then a Calculate part for the average speed in km/h The elapsed time counted correctly (07 50 to 10 14 is $2$ hours $24$ minutes), the minutes changed with ÷ $60$ ( $2.4$ hours, never $2.24$ ), then distance ÷ time with the unit km/h.
Using a flow rate when the volume is in different units: A tank or pool given in m³, filled at a rate in litres per minute, with the time asked for in hours and minutes The volume changed to litres (× $1000$ ) before dividing, time = volume ÷ flow rate, and the minutes turned into hours and minutes ( $150$ minutes $= 2$ hours $30$ minutes).
Changing money between currencies and comparing two prices: A one-mark conversion, then a short part asking where something is cheaper and by how much Multiply going from the currency with the 1, divide going to it. To compare, both prices in the same currency, then a statement of which is cheaper and the difference with its currency sign.
Changing a speed between km/h and m/s: A short part asking you to write an earlier speed answer in the other unit × $1000$ ÷ $3600$ (or ÷ $3.6$ ) seen for km/h to m/s, using the unrounded speed from the earlier part, and the answer to 3 significant figures.

Words to know

Rate: How much of one quantity goes with one unit of another, e.g. $12 per hour or $60$ km/h.
Per: 'For each'. In a unit it is written as a slash: km/h means kilometres per hour.
Hourly rate of pay: The money earned for one hour of work. Pay = hourly rate × hours worked.
Overtime: Hours worked beyond the normal hours, paid at a higher rate.
Time and a half: An overtime rate of $1.5$ times the normal hourly rate. Double time is $2$ times.
Exchange rate: How much of one currency you get for one unit of another, e.g. €1 = $1.08.
Flow rate: The volume that passes in one unit of time, e.g. litres per minute.
Fuel consumption: How much fuel a vehicle uses for the distance it travels, given in km per litre or in litres per 100 km.
Speed: Distance travelled per unit of time. Speed = distance ÷ time.
Average speed: Total distance divided by total time, including any stops unless the question says otherwise.

All Maths lessons Next: 1.12 Percentages