**12.1- Introduction**

If the ratio of one length to another is 1 : 2, this means that the second length is twice as large as the first. If a boy has 5 sweets and a girl has 3, the ratio of the boy’s sweets to the girl’s sweets is 5 : 3 . The boy has\(\frac{5}{3}\) times more sweets as the girl, and the girl has \(\frac{3}{5}\) as many sweets as the boy. Ratios behave like fractions and can be simplified.

Map Scales

If a map has a scale of 1 : 50 000, this means that 1 unit on the map is actually 50 000 units across the land. So 1cm on the map is 50 000cm along the ground (= 0.5km).

So 1cm on the map is equivalent to half a kilometre in real life.

For 1 : 25 000, 1 unit on the map is the same as 25 000 units on the land. So 1 inch on the map is 25 000 inches across the land, or 1cm on the map is 25 000 cm in real life.

**12.2- Proportion**

If a “is proportional” to b (which is the same as ‘a is in direct proportion with b’) then as b increases, a increases. In fact, there is a constant number k with a = kb. We write a ∝ b if a is proportional to b.

The value of k will be the same for all values of a and b and so it can be found by substituting in values for a and b.

Inverse Proportion (HIGHER TIER)

If a and b are inversely proportionally to one another,

a ∝ \(\frac{1}{b}\)

therefore a = \(\frac{k}{b}\)

**12.3- Ratio and Scales**

A scale factor is a number which is used as a multiplier when scaling.

Scale factors can be used to scale objects in 1, 2 or 3 dimensions. They can be written as either ratios, decimals, fractions or percentages.

Linear Scale Factor

Using linear scale factor a shape can be transformed into another similar shape by changing the size of all its dimensions (either enlargement or reduction) by using a scale factor.

To enlarge or reduce any shape you must begin by working out the scale factor, this is calculated by using the following formula:

For an enlargement = large number ÷ small number

For a reduction = small number ÷ large number

**12.4- Percentage change**

Repetitive rate of change

With repetitive rates of change the percentage change is applied more than once. You therefore have to calculate one step at a time.

Question 1 : The cost of a ticket increases by 10% to £19.25 .Calculate the original cost ?

**12.5- Summary and Review**

Simple and Compound Interest

Simple Interest and Compound Interest are different forms of interest, usually either paid by a bank to someone saving money or paid by the borrower of a loan such as a mortgage.

Question 1 : Calculate an investment of 4500 $ for 5 years at a compound interest of 4 % p.a ?

**12.6- Assessment 12**

Question 1 : Bertrand wants to buy these tickets for a show. 4 adult tickets at £15 each and 2 child tickets at £10 each. A 10% booking fee is added to the ticket price. 3% is then added for paying by credit card. Calculate the total charge for these tickets when paying by credit card ?

Question 2 : The value of y is 20% more than the value of x. Calculate the ratio x : y?

Question 3 : £360 is shared between Abby, Ben, Chloe and Dan. The ratio of the amount Abby gets to the amount Ben gets is 2 : 7 Chloe and Dan each get 1.5 times the amount Abby gets. Calculate the amount of money that Ben gets ?

**12.7- Revision exercise 2**

Question 1 : The height of Zak is 1.86 metres. The height of Fred is 1.6 metres. Calculate the height of Zak as a fraction of the height of Fred?

Question 2 : Calculate the value of 2^{14}÷(2^{9})^{2}

Question 3 : Calculate the value of \(25^{\frac{3}{2}}\) ?

Question 4 : b is two thirds of c. 5a = 4c. Calculate the ratio a : b : c ?