Rates

What a rate is

A rate tells you how much of one quantity there is for each unit of another. £12 per hour, 60 km per hour, 8 litres per minute and 2.7 grams per cubic centimetre are all rates. The word per means 'for each', and in a calculation it means divide: a speed in km per hour is a distance in km divided by a time in hours.

Because a rate links two quantities, every rate question can be answered in one of three ways:

• multiply the rate by an amount to find the total (12 hours at £11 per hour is $12\times 11=132$, so £132);
• divide a total by an amount to find the rate (£132 for 12 hours is $132÷12=11$, so £11 per hour);
• divide a total by the rate to find the amount (£132 at £11 per hour is $132÷11=12$ hours).

The units tell you which. If you want hours, and you have pounds and pounds per hour, then pounds divided by pounds per hour leaves hours. Always write the units on your answer: they are your check that you divided the right way round.

Pay, exchange rates, flow and fuel

Hourly rates of pay. Pay = hours worked $\times$ rate per hour. Overtime is often paid at a higher rate: time and a half means 1.5 times the normal rate, and double time means 2 times. Work out the normal hours and the overtime hours separately, then add.

Exchange rates. An exchange rate such as £1 = €1.17 says how many euros you get for each pound. To change pounds into euros, multiply by 1.17. To change euros into pounds, divide by 1.17. Check the direction with a quick sense test: one pound buys more than one euro here, so a number of euros should come out bigger than the same value in pounds. Money answers are given to 2 decimal places, the nearest cent or penny.

Flow rates. A tap that delivers 12 litres per minute fills a 180-litre tank in $180÷12=15$ minutes. When two taps run together, add their rates, not their times: 12 and 8 litres per minute together give 20 litres per minute.

Fuel consumption is written in two ways, and you must read which. 'km per litre' says how far one litre takes you, so a bigger number is better. 'litres per 100 km' says how much fuel 100 km uses, so a smaller number is better. To compare them, convert one into the other: 8 litres per 100 km is $100÷8=12.5$ km per litre.

Speed, distance and time

You must know the speed formula; it is not given on the paper:

$$\text{speed}=\frac{\text{distance}}{\text{time}}$$

Rearranged, distance = speed $\times$ time, and time = distance $÷$ speed. If the units match, the formula works: km and hours give km/h; metres and seconds give m/s.

Times must be in the right unit. A time of 3 hours 45 minutes is not 3.45 hours. Minutes are sixtieths of an hour, so 45 minutes is $\frac{45}{60}=0.75$ hours, and 3 hours 45 minutes is 3.75 hours. The same care is needed in reverse: 2.4 hours is 2 hours and $0.4\times 60=24$ minutes.

Average speed for a whole journey is the total distance divided by the total time:

$$\text{average speed}=\frac{\text{total distance}}{\text{total time}}$$

It is not the average of the speeds for each stage. A driver who spends longer on the slow stage spends most of the journey going slowly, so the average speed is pulled towards the slow speed. Find the time for each stage, add the times, add the distances, then divide.

Converting units of speed

Speeds are written with a slash for 'per': km/h is kilometres per hour, and m/s is metres per second. To change km/h into m/s you change both the distance and the time:

• 1 km is 1000 m, so multiply by 1000;
• 1 hour is $60\times 60=3600$ seconds, so divide by 3600.

So 72 km/h is $72\times 1000÷3600=20$ m/s. The two steps together are the same as dividing by 3.6. To go back from m/s to km/h, multiply by 3.6: 25 m/s is $25\times 3.6=90$ km/h.

Do not just memorise which way 3.6 goes; check it. A car at 72 km/h covers 20 metres in each second, and 20 is a smaller number than 72. The number in m/s is always smaller than the number in km/h, so if your m/s answer is bigger, you multiplied by 3.6 when you should have divided.

Density, pressure and population density

These rates compare a quantity with a volume or an area. The syllabus says the formula you need will be given in the question, so the skill being tested is using it correctly, including rearranging it.

• Density $=\frac{\text{mass}}{\text{volume}}$, measured for example in ${\text{g/cm}}^3$ (grams per cubic centimetre). Water has a density of about $1{\text{g/cm}}^3$; gold is about $19.3{\text{g/cm}}^3$.
• Pressure $=\frac{\text{force}}{\text{area}}$, measured in newtons per square metre, ${\text{N/m}}^2$. The same force on a smaller area gives a bigger pressure, which is why a sharp knife cuts.
• Population density $=\frac{\text{population}}{\text{area}}$, measured in people per ${\text{km}}^2$.

Each formula has the same shape as speed = distance $÷$ time, so it rearranges the same way: mass = density $\times$ volume, and volume = mass $÷$ density.

Watch the units. If the density is in ${\text{g/cm}}^3$, the volume must be in ${\text{cm}}^3$ and the mass comes out in grams. A question that asks for kilograms needs one more step: divide by 1000.