Ratio, proportion and rates of change flashcards

Topic 3: Ratio, proportion and rates of change

How do you share an amount in a given ratio?

1. Add the parts of the ratio. 2. Divide the total by that number to get one part. 3. Multiply by each share of the ratio. 4. Check the shares add back to the total.

Two amounts are in the ratio 4 : 7 and differ by 18. Find them.

Difference = 3 parts, so one part = 18 ÷ 3 = 6. Amounts: 4 × 6 = 24 and 7 × 6 = 42.

Formulae for speed, density and pressure

Speed = distance ÷ time Density = mass ÷ volume Pressure = force ÷ area

Convert 72 km/h to m/s

× 1000 ÷ 3600, which is the same as ÷ 3.6: 72 000 m ÷ 3600 s = 20 m/s.

How do you find the original value after a percentage increase?

Divide by the multiplier. Example: £89.70 after a 15% rise: 89.70 ÷ 1.15 = £78.

Compound interest formula

Total accrued = P(1 + r/100)ⁿ, where P is the amount invested, r the percentage rate and n the number of years. It is not given in the exam. Example: £5000 at 4% for 3 years = 5000 × 1.04³ = £5624.32.

Simple interest compared with compound interest

Simple: the same amount of interest each year (£2400 at 3.5% for 4 years: 2400 × 0.035 × 4 = £336). Compound: multiply by the same factor each year, so interest is earned on interest.

Direct and inverse proportion: equations and graphs

y ∝ x: y = kx, a straight line through the origin; doubling x doubles y. y ∝ 1/x: y = k/x, a curve that never meets the axes; doubling x halves y.

y is inversely proportional to x, and y = 8 when x = 5. Find y when x = 16.Higher

y = k/x, so 8 = k/5 and k = 40. y = 40/16 = 2.5

Average rate of change compared with instantaneous rate of changeHigher

Average: the gradient of the chord joining two points on the curve. Instantaneous: the gradient of the tangent at a point. Draw it with a ruler and use two points far apart on it.