Inverse proportion for AQA A-level Maths
This page covers the following topics:
1. Inverse proportion
2. Compound units
Two quantities are inversely proportional to each other when one is a multiple of the reciprocal of the other. This means that when one quantity increases, the other decreases. Two quantities being inversely proportional is equivalent to a ∝ 1/b. Inverse proportion can be displayed on a graph as a hyperbola that has a shape similar to letter L or C.
Compound units are ones made up of more than one unit. By adjusting each of the component units of a compound one, one can freely move between compound units. The units in the denominator have to be converted using direct proportionality rules. For changes of units in the denominator, if the units get bigger, the number gets greater and vice versa.
Rachel wants to calculate the density of water in g/cm³. Given that the density of water is 997 kg/m³, derive the value Rachel is looking for.
density = (997 × 1000)/(1 × 100³) = 0.997 g/cm³
Ibrahim is trying to find the density of a block of aluminium in kg/m³. Given that the density is 2.7 g/cm³, calculate the value in the units Ibrahim wants to use.
density = (2.7 ÷ 1000)/(1 ÷ 100³) = 2700 kg/m³
Elliot is jogging at 18 km/h. Calculate their speed in m/s.
speed = (18 × 1000)/(1 × 3600) = 5 m/s
The number of meat-eaters a restaurant hosts in a day is inversely proportional to the number of vegetarian dishes they make on that day. Given that when they host 15 meat-eaters, they make 4 vegetarian dishes, calculate the number of vegetarian dishes they will make when they get 20 meat-eaters.
Let x = the number of meat-eaters they host in a day and y = the number of vegetarians they host in a day.
Since x and y are inversely proportional, x = k/y, where k is a constant of proportionality.
Substituting x = 15, y = 4 into this gives 15 = k/4, thus k = 60.
Therefore, the equation is given by x = 60/y.
When x = 20, y = 60/20 = 3 vegetarian dishes.
Given that x and y are inversely proportional such that when x = 7, y = 3, calculate the gradient of the graph of x against 1/y.
Since x is inversely proportional to y, x = k/y, where k is a constant of proportionality.
Substituting y = 3, x = 7 into this gives 7 = k/3, therefore k = 21.
Thus, the equation is given by x = 21/y.
Plotting x against 1/y is a straight line with gradient 21.
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