numbers 2

Ratios 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 5/3 times more sweets as the girl, and the girl has 3/5 as many sweets as the boy. Ratios behave like fractions and can be simplified. Example: Simone made a scale model of a 'hot rod' car on a scale of 1 to 12.5 . The height of the model car is 10cm. (a) Work out the height of the real car. The ratio of the lengths is 1 : 12.5 . So for every 1 unit of length the small car is, the real car is 12.5 units. So if the small car is 10 units long, the real car is 125 units long. If the small car is 10cm high, the real car is 125cm high. (b) The length of the real car is 500cm. Work out the length of the model car. We know that model : real = 1 : 12.5 . However, the real car is 500cm, so 1 : 12.5 = x : 500 (the ratios have to remain the same). x is the length of the model car. To work out the answer, we convert the ratios into fractions:  1   =    x   12.5      500 multiply both sides by 500: 500/12.5 = x so x = 40cm Example: Alix and Chloe divide £40 in the ratio 3 : 5. How much do they each get? First, add up the two numbers in the ratio to get 8. Next divide the total amount by 8, i.e. divide £40 by 8 to get £5. £5 is the amount of each 'unit' in the ratio. To find out how much Alix gets, multiply £5 by 3 ('units') = £15. To find out how much Chloe gets, multiply £5 by 5 = £25. 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. You can manipulate these ratios if necessary.

Standard Form Standard form is a way of writing down very large or very small numbers easily. 10³ = 1000, so 4 × 10³ = 4000 . So 4000 can be written as 4 × 10³ . This idea can be used to write even larger numbers down easily in standard form. Small numbers can also be written in standard form. However, instead of the index being positive (in the above example, the index was 3), it will be negative. The rules when writing a number in standard form is that first you write down a number between 1 and 10, then you write × 10(to the power of a number). On a calculator, you usually enter a number in standard form as follows: Type in the first number (the one between 1 and 10). Press EXP . Type in the power to which the 10 is risen. Manipulation in Standard Form This is best explained with an example:

Basic surd manipulation Surds are numbers left in 'square root form' (or 'cube root form' etc). They are therefore irrational numbers. The reason we leave them as surds is because in decimal form they would go on forever and so this is a very clumsy way of writing them. Leaving them as surds is more mathematically precise. Addition and subtraction of surds:4Ö7 - 2Ö7 = 2Ö7. 5Ö2 + 8Ö2 = 13Ö2 Note: 5Ö2 + 3Ö3 cannot be manipulated because the surds are different (one is Ö2 and one is Ö3). Multiplication:Ö5 × Ö15 = Ö75 (= 15 × 5) = Ö25 × Ö3 = 5Ö3. (1 + Ö3) × (2 - Ö8)            [The brackets are expanded as usual] = 2 - Ö8 + 2Ö3 - Ö24 = 2 - 2Ö2 + 2Ö3 - 2Ö6 Rationalising the denominator: It is untidy to have a fraction which has a surd denominator. This can be 'tidied up' by multiplying the top and bottom of the fraction by a surd. This is known as rationalising the denominator, since surds are irrational numbers and so you are changing the denominator from an irrational to a rational number. Example: Rationalise the denominator of: a) 1    Ö2 . b) 1 + 2    1 - Ö2 a) Multiply the top and bottom of the fraction by Ö2. The top will become Ö2 and the bottom will become 2 (Ö2 times Ö2 = 2). b) In situations like this, look at the bottom of the fraction (the denominator) and change the sign (in this case change the plus into minus). Now multiply the top and bottom of the fraction by this. Therefore: 1 + 2   =  (1 + 2)(1 + Ö2)  =  1 + Ö2 + 2 + 2Ö2  =  3 + 3Ö2 1 - Ö2       (1 - Ö2)(1 + Ö2)     1 + Ö2 - Ö2 - 2             - 1 = -3(1 + Ö2)

Variation Bookmark this page Proportion If a is proportional to b, a µ b and a = kb, where k is a constant The value of k will be the same for all values of a and b. Example: If a µ b, and b = 10 when a = 5, find an equation connecting a and b. a = kb  (1) Substitute the values of 5 and 10 into the equation to find k: 5 = 10k so k = 1/2 substitute this into (1) a = ½b Similarly, if m is proportional to n², m = kn² Inverse Proportion If a and b are inversely proportionally to one another, a µ1/b\ a = k/b In these examples, k is known as the constant of variation. Example: If b is inversely proportional to the square of a, and when a = 3, b = 1, find the constant of variation. b = k/a² when a = 3, b = 1\ 1 = k/3²\ k = 9

ratios

standard form

surds

variaton