Proportion flashcards

Key points

• When one is multiplied by a number, the other is multiplied by the same number.
• So y ÷ x is the same for every pair of values: that constant is k in y = kx.
• Zero of one goes with zero of the other, like the cost of 0 kg of apples.

x: 2, 5, 8 and y: 6, 15, 24

Key points

• When one is multiplied by a number, the other is divided by the same number: double one, the other halves.
• So x × y is the same for every pair of values.
• Speed and journey time: twice as fast covers the same distance in half the time.

x: 2, 4, 8 and y: 12, 6, 3

Key points

• Go down to one unit first: divide by the number you have.
• Then go up to the number you need: multiply.
• Work in one unit throughout, such as pence or grams.

Example

Key points

• More workers means less time, so the logic flips.
• Find how long one worker would take: multiply the workers by the time.
• Then divide by the number of workers you need.

3 painters take 8 hours

Key points

• Work out the price of one unit for each pack, such as pence per gram.
• The lower price per unit is the better value.
• Or find how much you get per penny: then the bigger amount wins.

400 g for £1.20 or 650 g for £1.82

Key points

• A straight line through the origin, (0, 0).
• The gradient is the constant k: the price per kilogram, or the exchange rate.
• Read one pair of values off the line, then scale it.

An exchange rate: £50 = €58

Key points

• A curve that gets closer and closer to both axes but never touches them.
• It is the shape of y = $\frac{k}{x}$: as x grows, y shrinks, and neither ever reaches 0.
• Every point on the curve gives the same value of x × y.

Points on y = $\frac{12}{x}$

Key points

• Proportional means "equals a constant times": y ∝ x² becomes y = kx².
• Write the equation first, then put in the pair of values you are given to find k.
• Write the equation again with k in, then use it to answer the question.

y = 48 when x = 4. Find y when x = 5

Key points

• Inverse proportion puts x on the bottom: y = $\frac{k}{x}$, or y = $\frac{k}{x^2}$ for the square.
• Find k by multiplying: k = y × x, or k = y × x².
• Then substitute the new value of x.

y ∝ $\frac{1}{x^2}$, and y = 4 when x = 3

Key points

• y ∝ x²: multiply x by 2 and y is multiplied by 2² = 4.
• y ∝ x³: multiply x by 2 and y is multiplied by 2³ = 8.
• y ∝ $\sqrt{x}$: multiply x by 4 and y is multiplied by $\sqrt{4}$ = 2.
• Inverse: y ∝ $\frac{1}{x^2}$ means multiplying x by 2 divides y by 4.

y ∝ x³ and x is multiplied by 3