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Graphs and proportion - Higher

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# Inverse proportion

Inverse proportion is when one value increases as the other value decreases.

A simple example of inversely proportional quantities is the lengths and widths of rectangles with the same area. As the length of one side doubles, the width has to be halved for the area to stay the same.

## Example

y is inversely proportional to x. When y = 3, x = 12 .

Find the constant of proportionality, and the value of x when y = 8.

• y ∝ 1/x
• y = k/x
• So xy = k
• Substitute the values x = 12 and y = 3 into xy = k
• 3 × 12 = 36
• So k = 36
• To find the value of x when y = 8, substitute k = 36 and y = 8 into xy = k
• 8x = 36
• So x = 4.5

Again, you can have questions involving squares, cubes or other powers of the variables.

Question

v is inversely proportional to r3. When r = 2, v = 25. Find r when v = 60.

v = ∝ so v =

Re-arrange the above to get k on its own.

• k = vr3
• k = 25 × 23
• So k = 200
• When v = 60
• 60r3 = 200
• r3 = 200/60
• r3 = 3.333

So r equals the cube root of 3.333

So r = 1.494

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