Rates, ratios & proportions

Skills and methods in this topic, easy to hard

1. 1RatesA rate is one quantity ÷ another, with its units

   Method36 ÷ 4 = 9 ($/kg), 9 × 6 = 54

   Another way: The multiplier6 ÷ 4 = 1.5, 36 × 1.5 = 54

2. 2Density, pressure and exchange ratesThe unit says what to do: g/cm³ is g ÷ cm³

   Method540 ÷ 200 = 2.7 (g/cm³)

   Another way: The formula triangleMass = density × volume = 2.7 × 200 = 540

3. 3Ratios and proportionsMultiply or divide both terms alike, and the ratio stays

   Methodx : 6 = 4 : 3, 3x = 6 × 4 = 24, x = 8

   Another way: Spot the multiplier6 ÷ 3 = 2, 4 × 2 = 8

4. 4Three-term ratios and sharingMake the shared term the same, then join the ratios

   Methoda : b = 8 : 12, b : c = 12 : 15, a : b : c = 8 : 12 : 15

   Another way: Give the shared term a valueb = 12: a = 12 × 23 = 8, c = 12 × 54 = 15

5. 5Direct proportionDirect proportion: y ÷ x stays the same, y = kx

   Method60 ÷ 5 = 12, 12 × 8 = 96

   Another way: A proportionx : 8 = 60 : 5, 5x = 480, x = 96

6. 6Inverse proportionInverse proportion: x × y stays the same

   Method6 × 10 = 60, 60 ÷ 15 = 4

   Another way: The multiplier15 ÷ 6 = 2.5, 10 ÷ 2.5 = 4

7. 7Average speed and mixturesTotal over total, not the average of the rates

   Method(40 × 2 + 60 × 3) ÷ 5 = 52

   Another way: Weight by the shares40 × 25 + 60 × 35 = 16 + 36 = 52

8. 8Scales on maps and plansLengths times the scale; areas times the scale squared

   Method6 × 50000 = 300000 cm = 3 km

   Another way: 1 cm first1 cm stands for 500 m: 1 cm² for 500 × 500 = 250000 m²

Worked examples

From different skills in this topic. Have a go first, then open each one for the answer and the working.

1. Example 1Rates →

   7 kg of rice costs $70: unit price = $kg

   ▶Show the answer and working

   Answer: 10

   Unit price = price ÷ weight = 70 ÷ 7 = 10, that is $10 a kg.

2. Example 2Direct proportion →

   In which set of values is y in direct proportion to x?

   • A. (x, y) = (1, 24), (2, 12), (4, 6)
   • B. (x, y) = (1, 10), (2, 11), (4, 13)
   • C. (x, y) = (1, 6), (2, 12), (4, 24)
   • D. (x, y) = (1, 15), (2, 21), (4, 33)

   Choose the right answer

   ▶Show the answer and working

   Answer: C. (x, y) = (1, 6), (2, 12), (4, 24)

   Direct proportion needs the same y ÷ x for every pair: 6 ÷ 1 = 12 ÷ 2 = 24 ÷ 4 = 6.

3. Example 3Scales on maps and plans →

   A map's scale is 1 : 25000: 7 cm on it stand for km

   ▶Show the answer and working

   Answer: 1.75

   7 × 25000 = 175000 cm; 1 km = 100000 cm, so it is 1.75 km.