RATIO & PROPORTION

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Numbers can be compared to each other by ratios, scales and proportions. The basic rules of fractions apply as these methods are other ways of expressing fractions.

Ratio and Scale: Ratio: if two numbers A and B are in the ratio 12 it means that A is half the size of B or B is twice the size of A. The rules for equivalent fractions and lowest terms also apply.

Example: A ratio 400650. Using the rules for fractions (equivalent, or lowest terms) equals: 813. The first number is 8/13 of the second number, and the second number is 13/8 of the first number.

Scale: a scale is a measurement that represents another value of measurement.

Example: A scaled model is 110 the original size. Therefore the model is 10 times smaller or the original is 10 times greater. Every measurement on the model is scaled to 1/10 of the original size. ie. a section measuring 20cm long on the model would be 2m long on the full size.

Example: £1000 is shared by a family in the ratio of 532. How much does each person receive.

Answer: Firstly total the shares: A has 5 shares, B has 3 shares and C has 2 shares. Total of 10 shares. Finding the value of one share: £100010 shares = £100 for each share. The £100 can now be multiplied by each individual's number of shares. This gives: £500 £300 £200.

Example: Scale map shows a road 2km long, measuring it was found to be 4cm, find the scale of the map.

Answer: 2km : 4cm. Convert 2km to cm = 200000cm. Ratio of: 4200000. Finding 1 by equivalent fractions gives 150000. Or a scale of km to 1cm.

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Proportion: Two quantities may increase at the same rate, they are proportional, if they increase at opposite rates they are inversely proportional.

Example: proportional: 10 cds cost £150 how much would 20 cds cost ? Answer: 10150 what is 20 increase by a factor of 2, giving: 20300. Therefore 20 cds cost £300.

Example: Inversely proportional: A driver travels to work at 30mph and it takes 30 minutes. If he cycled at 10mph how long would it take ?

Answer: 3030 what is 10 (inversely proportional) if one is decreased by a factor of 3 then the other is increased by a factor 3, giving 1090. Therefore time is 90 minutes.

Example: x is proportional to they. y = 16 when x = 2. Find y in terms of x.

Answer: Squaring both sides: xis proportional to y. (2= 16) The x or 2 has to be multiplied by 4 to balance the equation. (4 x 2= 16) Therefore y = 4x.

Example: x is proportional to they. y = 16 when x = 2. Find y when x = 4.

Answer: Using the result from the previous question giving an expression for y, substitute x = 2 for x = 4. (4 x 4= 64).Therefore y = 64 and still equivalent to y = 4x.

Example: x is proportional to the y. y = 16 when x = 2. Find x when y = 4.

Answer: Using the result from the previous question giving an expression for y, substitute y = 16 for y = 4. (1 x 4= 16) or reduce by factor of 4. Therefore x = 1 and still equivalent to y = 4x.

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