Ratio, speed, proportion and variation

A Ratio tells you the relative size of the portions into which a quantity is to be divided

With a question about ratios, always find out what one part represents. Usually you'll need to work out the Total number of parts the quantity is to be divided into and then divide the quantity by that total. Then, once you know the size of one part, you can work out the size of each portion.

Ratios can be treated like fractions - it's OK to multiply or divide all parts by the same number to make them simpler to handle. If the ratio given is not in whole numbers, multiply it through to convert to whole numbers.

Ben and Bob's auntie gives them £20 for Christmas but tells them to share in the ratio of their ages. If Ben is 6 and Bob is 2, how much does each one receive?

The ratio they're dividing it up into is 6 : 2 which cancels down to 3 : 1. Now find the total number of parts which is 3 + 1 = 4. So, the size of one part is £20/4 = £5 and the size of three parts is £15 So, Ben gets £15 and Bob gets £5. You can do a final check - add up the portions and see if it equals the original total. £15 + £5 = £20

A 5m plank is divided into two parts, one 3.5m long, the other 1.5m long. Express the ratio of the parts in its simplest form.

The ratio here is 3.5 : 1.5. We're told to put it in its simplest form. First multiply by 10 to get rid of the decimal points So it's now 35 : 15. We can divide through by 5 to get 7 : 3 We can't divide any further so that's our answer.

Ratios and fractions are basically the same thing. They're both ways of dividing a quantity into different sized parts.

As long as you remember the golden rule of ratios - always find what one part represents. Normally this will involve working out the total number of parts and dividing that number into the total quantity.

Make sure you write any ratios in the correct order - if you're asked to give the ratio of red : blue, make sure you don't give the ratio of blue : red

For a fruit punch, the quantity of orange juice to apple juice to pineapple juice is 4 : 3 : 2. What fraction of the punch is apple juice?

First, work out the total number of parts: 4 + 3 + 2 = 9 Now, 3 of those 9 parts are apple juice so, as a fraction, 3/9 of the punch is apple juice. Cancel through by 3 for the answer, 1/3

In this type of question, you're given a ratio to apply to a practical situation.

In some of these questions, we don't actually know the total amount so we might have to work out the value of one part from the information we're given. Just use your common sense!

A can of purple paint is mixed using red and blue paint in the ratio 5 : 12. If 9 litres of blue paint were used, how many litres of red paint were used?

We know that 12 parts of the paint is blue and this represents 9 litres. So, 1 part of the paint will be 9/12 litres ie. 0.75 l The red paint makes up 5 parts so is 5 x 0.75 l = 3.75 l.

These are basically common sense problems that require you to apply your knowledge of ratios and fractions.

Read the question carefully and find out what it's asking. Then decide what information you need to answer it.

As ever, you'll almost certainly need to work out what 1 part represents.

Finally, check that the answer you give 'makes sense'.

If it takes 7 gardeners 12 hours to mow Lord Chumley's vast lawn, how long will it take 3 gardeners?

Here the total number of parts is the 7 gardeners. We're told they take 12 hours to mow the lawn. Next step is to convert that information for 1 part ie. 1 gardener. Here's where common sense comes in. Will 1 gardener take more or less time than 7? The answer is obviously more so we multiply. So, 1 gardener will take 7 x 12 hours. Next bit of common sense - will 3 gardeners take more or less time than 1? The answer is less so we divide. 3 gardeners will take 84/3 hours = 28 hours. That's one big lawn!

A Best Buy is simply the best deal per unit on an item.

These are very straightforward - you probably do them already without even thinking when you go shopping.

All you need to do is find a unit price for each offer and then decide which is cheapest.

Milky Tray chocolate bars come in three sizes: the 500g bar costs £4.90, the 400g bar costs £3.80, the 200g bar costs £2.10. Which bar is the best value for money?

We need to find a unit price. As 100g divides into each bar size, let's use that. Consider each bar in turn: 500g costs £4.90 so 100g costs £4.90/5 = £0.98 400g costs £3.80 so 100g costs £3.80/4 = £0.95 200g costs £2.10 so 100g costs £2.10/2 = £1.05 So, the best value bar is the 400g one.

Speed is a ratio of distance to time. It tells us how far we travel in a given time period.

There is one simple formula to learn namely:

From this you can derive and and with these three, you can answer pretty much any question you like on speed.

Make sure you use units consistently - if there's a combination of hours and minutes, convert to one or the other, don't use both. Also, make sure you answer the question in the unit specified.

If you're asked a sneaky question about average speed remember to use Total Distance and Total Time - don't take the average of two speeds, it'll only end in tears.

During a 5km race, Amar runs the first 4km at an average of 10km per hour and the last 1km at an average of 12km per hour. To 1 d.p., what's his average speed in km/h for the whole race?

Don't forget - use Total Distance and Total Time NOT an average of the two speeds. We know the total distance - 5km. So, we need to work out the total time. Using on the first 4km, time = 4/10 = 2/5 hours = 24 minutes For the last 1km, the time is 1/12 hours = 5 minutes So, he runs 5km in 24 + 5 = 29 minutes. First work out his speed in km/minute - this is 5/29. So, his speed in km/h will be 60 x this ie. 300/29 = 10.3km/h to 1 d.p.

The Density of an object is the ratio of its mass to its volume. If two objects have the same volume, then the denser one will weigh more because its density is higher, ie. its mass to volume ratio is higher

As with speed, there's one basic formula to learn and from this you can derive two other formulas and

The units of density can be a bit tricky - they're normally given in g / cm³ ie mass per volume. Make sure you convert all units to be consistent with each other ie. don't use kg if the density is given in g / cm³

Make sure you're familiar with converting cm³ to litres and vice versa. 1000cm³ = 1 litre

A container is filled to the top with a liquid that has density 2.5 g/cm³. If the weight of the liquid in the container is 7.5 kg, what is the volume of the container in litres?

First, convert the weight to grams, 7500g Now, use the formula So, Volume = 7500 / 2.5 = 3000 cm³. Now convert to litres for the answer, 3 litres.

If two quantities are in Direct Proportion to each other, it means they increase or decrease together.

There's a symbol that's used to represent Direct Proportion: y α x means that y is directly proportional to x. This relationship can also be written y = kx where k is a constant.

In fact, this method of expressing Direct Proportion is what you'll need to use when answering questions. Usually, you'll need to find out the value of k.

The relationships aren't always linear, y α x², y α x ³ can also crop up.

y α x² and when x = 5, y = 7.5. Find the value of y when x = 10

We can write y = kx² where k is a constant. Put the figures we're given into this equation to get 7.5 = 25k So, k = 0.3 So, when x = 10, y = 0.3(10²) = 30

When two quantities are Inversely Proportional, as one Increases, the other Decreases.

The short hand way of writing this is y α 1/xThis relationship can also be written y = k/x where k is a constant.

As with Direct Proportion questions, you'll usually need to find out the value of k.

And again, as with Direct Proportion questions, it may be x² or x³ that y is inversely proportional to.

c varies inversely as the square of d and when d = 5, c = 9. What's the value of c when d = 3?

We're told that c α 1/d² which we can write as: c = k/d² where kis a constant. Putting in the figures we've been give, 9 = k/25 so k = 225 So, when d = 3, c = 225/3² = 25