Mr Kidman's Classroom
Revise the key ideas
Ratios and fractions
- A ratio compares quantities in a specified order. Red : blue = 2 : 3 means two red parts for every three blue parts; reversing the order changes the ratio.
- Simplify by dividing all parts by their HCF: 18 : 24 = 3 : 4. For three parts, 12 : 18 : 30 = 2 : 3 : 5.
- Convert quantities to the same units before comparing: 50 cm : 2 m = 50 : 200 = 1 : 4.
- A part-to-part ratio 2 : 3 gives part-to-whole fractions and . The first part is of the second part, but of the total.
Ratio bar for sharing eighty-four pounds - To express one quantity as a fraction of another, use with matching units. The fraction can exceed 1.
Sharing and finding totals
- Worked example: Share £84 in ratio 2 : 5. There are 7 parts; each is £12, so the shares are £24 and £60. Check both their total and their simplified ratio.
- Worked example: A : B = 3 : 7 and B is 28. One part is 28 ÷ 7 = 4, so A = 12 and the total is 40. Do not divide 28 by all 10 parts.
- Worked example: Two shares are in ratio 3 : 5 and differ by £18. Two parts represent £18, so one part is £9; the shares are £27 and £45.
- For mixing, preserve the ratio when scaling quantities. A drink mixed concentrate : water = 1 : 4 needs 200 ml concentrate and 800 ml water for 1 litre of drink.
- If a ratio describes concentrations or mixtures, be clear whether it is a mass ratio or volume ratio. Do not combine incompatible units silently.
Linking ratios and relationships
- Equivalent ratios have equal quotients: a : b = c : d means = where denominators are non-zero. This links ratios to proportional relationships.
- If x : y = 2 : 5, then y = x. On a y-against-x graph this is a straight line through the origin.
- To combine A : B = 2 : 3 and B : C = 4 : 5, make B match: 8 : 12 and 12 : 15, so A : B : C = 8 : 12 : 15.
Multiply both parts of 2 : 3 by 4 to get 8 : 12. Multiply both parts of 4 : 5 by 3 to get 12 : 15. Now both ratios use 12 parts for B, so they can be joined.
- Adding the same amount to both quantities generally changes their ratio. A 2 : 3 mixture does not stay in that ratio if you add 1 litre to each component.
- After a change, work with actual quantities or parts before simplifying the new ratio. Multiplying both quantities by the same factor preserves the ratio; adding does not.
