1.  A supermarket estimates that it sells sliced bread and packets of bread rolls in the ratio 8:2. If 500 loaves of sliced bread are sold in a day, how many packets of bread rolls can the supermarket expect to sell?

The ratio 8:2 in its simplest terms is 4:1 (problems are much easier to solve if you first reduce the ratio to its simplest form - in this case 2 can be divided into 8 four times and 2 once)

Add the 4 and 1 together which give you 5. Now you can look at each part of the ratio as a fraction (which is much easier): where sliced bread is 4/5 of all bread sold and bread rolls are 1/5 of all bread sold.

If 500 loaves of sliced bread are sold this = 4/5 of the total. We need to find 1/5 so we simply divide 500 by 4 which = 125 packets.

ratio is 63:45:27

You need to "spot" that each number can be divided by 9 (it helps to know your times table for these questions):

So all you do now is divide each number by 9:

7:5:3

Don't get caught out with the terminology - proportion and ratio are taken to mean the same thing.

3.  The formula for making concrete is:
1 part cement
2 parts sand
4 parts aggregate

How much sand is needed to make 35m³ of concrete?

The ratio is 1:2:4

Converting to fractions 1/7 is cement, 2/7 is sand and 4/7 is aggregate.

If the total = 35m³, then the proportion of sand required is 2/7 multiplied by 35 = 70/7 = 10 parts.

4.  A GCSE maths course consists of a written examination paper which accounts for 80% of the marks and coursework which accounts for the remaining 20%.

Write the ratio of written examination to coursework in its lowest terms.

Jane gets an overall mark of 56%. If the ratio of her examination marks to coursework marks was 9:3, what mark did she get for her coursework?

80:20 in its lowest terms is 4:1

Find the fraction for the coursework marks = 3/12

Now multiply by the total mark = 3/12 x 56% = 14%

5. A triangle ABC has sides of lengths:
AB = 4cm, BC = 6cm and CA = 12cm.

A similar triangle has side AB of length 7cm. What are the lengths of the sides BC and CA?

If triangles are similar then the ratio of their sides is in proportion (this comes from geometry):

we have 4:6:12 is similar to 7:a:b

Simplify 4:6:12 giving 2:3:6 to make things a bit easier

looking at the proportions to get 3 we multiply 2 by 1.5 and to get 6 we multiply 2 by 3.

So if we have 7 in the place of 2, what do we have in the place of 3 (shown as "a" in the first ratio - see above)?

It's 1.2 x 7 = 10.5 cm

Likewise we have 3 x 7 = 21cm.

BC = 10.5cm and CA = 21cm

6. If 5 miles is approximately equal to 8 km, what is:
a) 60 miles per hour in km per hour ?
b) 70 km per hour in miles per hour?

We have a ratio of 5:8

With questions like this it's often easiest to work out how much 1 is.

If 5 miles = 8km then 1 mile = 8/5km

So, 60 miles = 60 x 8/5km = 96km

So, 60miles per hour = 96km per hour

Similarly 1km = 5/8 miles

giving 70km = 70 x 5/8 = 43.75 miles

So, 70 km per hour = 43.75 miles per hour

7.  A computer programmer works out that every week, on average she spends 4 hours in meetings, 8 hours writing documentation, and 28 hours writing software. write this ratio in its lowest terms.

What fraction of the week is spent in meetings?
What percentage of the week is spent writing software?

4:8:28

simplified is 1:2:7

Fraction of the week spent in meetings is 1/(1 + 2 + 7)

= 1/10

Fraction of the week spent writing software = 7/10

7/10 as a percentage = 70%

8. Paul and Danny buy a lottery ticket together. Paul can only put 15p towards the ticket so Danny agrees to pay the other 85p under the condition that if they win that the prize money will be shared out in the same proportions as the amount of money each of them invested in the ticket.

If they win £1,500, how much does Danny get?

If they win £1,360,000 how much does Paul get?

15:85

simplified is 3:17

Danny gets 17/20 x £1,500 = £1,275

Paul gets 3/20 x £1,360,000 = £204,000