0580 · 1.12Rates

Understanding rates and wages

0:00A job paying twelve pounds per hour and a tap delivering six litres per minute describe different situations. But their numbers do the same mathematical job. Each tells us how much of one quantity corresponds to one unit of another. By understanding that connection, you can calculate wages, exchange currencies, compare fuel consumption and solve average speed problems. You need multiplication, division and a little unit conversion for this topic.

0:25Multiplication builds up from one unit to several units. Division can find the amount for one unit, or how many units fit into a total. Before calculating, make sure the units in the question match the units in the rate. A rate compares quantities measured in different units. The word per means for each, so twelve pounds per hour means twelve pounds for each hour worked. Dividing forty-eight pounds by four hours gives twelve pounds per hour.

0:49We have shared the total pay across the hours to find the amount corresponding to one hour. The units tell you what the calculation means. Pounds divided by hours gives pounds per hour, while litres divided by minutes gives litres per minute. On the page, a slash is often used to mean per. Always read the entire unit, because swapping its order gives a different rate. Consider a question asking you to calculate pay for three hours and thirty minutes at twelve pounds per hour.

1:1330 ÷ 60 gives zero point five hours. The working time is 3 + 0.5, giving three point five hours. Pay is 12 × 3.5, which equals forty-two pounds. We multiply because each hour contributes another twelve pounds. If instead the question gives a total payment of sixty pounds, we divide sixty by twelve. That gives five hours of work. Choosing multiplication or division becomes easier when you explain what each number represents, rather than searching for a rule to memorise.

Exchange, flow and fuel rates

1:43An exchange rate tells us the value of one currency in another. Suppose one pound exchanges for one point two five dollars, with no fees. To convert eighty pounds into dollars, multiply eighty by the rate of one point two five. The result is one hundred dollars, because each pound contributes one point two five dollars. To convert seventy-five dollars back into pounds, calculate 75 ÷ 1.25. Multiplying both numbers in that division by one hundred gives 7500 ÷ 125.

2:13The result is sixty pounds. Going back the other way needs the opposite operation, so we divide rather than multiply. A flow rate describes the volume passing through something per unit time. A tap fills a container at six litres per minute, and the container needs ninety litres. The filling time is 90 ÷ 6, giving fifteen minutes. At the same constant rate, four minutes supplies 6 × 4, giving twenty-four litres.

2:37These calculations assume the stated rate stays constant throughout the interval. If a rate changes, one multiplication may not describe the whole situation. You would need to calculate the amounts for separate stages and combine them. Also, a rate stated per minute must be used with minutes, unless you first convert the units. Fuel consumption needs particular care because it can be expressed in different ways. Suppose a car uses six litres per one hundred kilometres and travels two hundred and fifty kilometres.

3:03250 ÷ 100 gives two point five lots of one hundred kilometres. Fuel used is 2.5 × 6, giving fifteen litres. For the same journey, distance per litre is 250 ÷ 15. This gives sixteen point six recurring kilometres per litre, or sixteen point seven to three significant figures. Lower litres per one hundred kilometres means less fuel used for the same distance. Higher kilometres per litre means more distance covered using the same amount of fuel.

Density, population density and pressure

3:33Not every rate compares an amount with time. Density compares mass with volume, while population density compares population with area. Pressure compares force with area. In these questions, the required formulas will be given, so concentrate on interpreting the quantities and substituting values with matching units. Density means mass per unit volume. Using the given formula, take an object with mass one hundred and eighty grams and volume sixty cubic centimetres.

3:59Its density is 180 ÷ 60. The answer is three grams per cubic centimetre. For uniform material, a volume of twenty-five cubic centimetres has mass 3 × 25, giving seventy-five grams. Population density means the number of people per unit area. Suppose a region contains forty-eight thousand people across one hundred and twenty square kilometres. Its population density is 48000 ÷ 120, giving four hundred people per square kilometre. This is an average across the region, not a claim that every square kilometre contains exactly four hundred people.

4:34Pressure means force per unit area. If the given force is six hundred newtons over zero point zero three square metres, divide six hundred by zero point zero three. The pressure is twenty thousand newtons per square metre. For the same force, a smaller area gives a greater pressure, because the push is more concentrated.

Average speed and journey times

4:51Speed is distance travelled per unit time. You must know how speed, distance and time are related. Dividing distance by time finds how far you go in one unit of time. Multiplying speed by time therefore gives distance, while dividing distance by speed gives time. Average speed uses the total distance and total time for the entire journey being considered. A cyclist travels forty-five kilometres in three hours and forty-five minutes.

5:15Calculate the average speed in kilometres per hour by first converting the minutes into hours. 45 ÷ 60 gives zero point seven five hours. The total time is 3 + 0.75, giving three point seven five hours. Average speed is 45 ÷ 3.75. Multiplying both numbers in this division by four gives 180 ÷ 15. The answer is twelve kilometres per hour. This average describes the complete journey, even if the cyclist’s speed varied along the way.

5:46Writing the time as three point four five hours would be incorrect. Minutes are parts of sixty, not parts of one hundred. Also, kilometres divided by hours cannot produce metres per second without conversion. To obtain metres per second directly, express the distance in metres and the time in seconds before dividing. An exam question describes eighteen kilometres travelled in one hour, then twelve kilometres in half an hour, followed by a stop of half an hour.

6:10Calculate the average speed for the whole journey. Total distance is 18 + 12, giving thirty kilometres. Total time is 1 + 0.5 + 0.5, giving two hours. Average speed is 30 ÷ 2, giving fifteen kilometres per hour. The stop contributes time but no distance, so it reduces the overall average. Simply averaging the two travelling speeds would ignore their different durations and the stop. Adding all distances and all times is the reliable method.

6:38These methods must work without a calculator on Paper two; Paper four allows a calculator. Show unit conversions and substitutions clearly, rather than writing only an answer. Unless instructed otherwise, give answers that are not exact to three significant figures, and money to the nearest cent or penny. Keep full values during working and round only the final answer. A rate tells you how much of one quantity corresponds to another.

7:01Read its units, identify what is given, and decide whether you are finding a rate, an amount or a duration. Use the supplied formulas for pressure, density and population density. For average speed, remember total distance divided by total time, including stops within the journey.