KS4. Direct & Inverse Proportion

Any questions which specifically mention direct proportion or inverse proportion require us to follow a number of steps as outlined below. With this type of question it is better to follow these steps precisely, rather than use alternative methods

Example 1 – y is directly proportional to x (question may say “y is directly proportion to x”, “y is proportional to x”, “y varies in proportion with x”, “y and x are in direct proportion”, or similar statements – the method will remain the same)

Question:

y is in direct proportion to x. When y is 34, x is 51.

(a) Find an equation for y in terms of x

(b) Find y when x is 7

Example 2 – y is inversely proportion to x

Question:

y is in inverse proportion to x. When y is 3, x is 5.

(a) Find an equation for y in terms of x

(b) Find y when x is 3

Example 3 – y is directly proportional to f(x)

y is in direct proportion to the cube root of x. When y is 12, x is 64.

(a) Find an equation for y in terms of x

(b) Find y when x is 125

Example 4 – y is inversely proportional to f(x)

y is in inverse proportion to the x⁴. When y is 5, x is 2.

(a) Find an equation for y in terms of x

(b) Find y when x is 10

Exercise

y is directly proportional to x. When y = 30, x = 6.
- Write an equation for y in terms of x;
- Find y when x = 7;
- Find x when y = 25.
y is proportional to x. When y is 64, x is 4.
- Write y in terms of x;
- Calculate y when x is 3;
- Calculate x when y is 80.
y is in direct proportion with x. When x = 13, then y = 52.
- Write an equation involving y and x;
- Calculate x when y is 24;
- Calculate y when x = 11.
A and B are directly proportional. When A is 35, then B is 5.
- Write A in terms of B;
- Calculate A when B = 9;
- Calculate B when A = 42.
z is directly proportional to w. z=21 when w=7.
- Write an equation showing z in terms of w;
- Calculate z when w is 16;
- Calculate w when z is 102.
y is inversely proportional to x. y = 2 when x = 3.
- Write y in terms of x;
- Calculate y when x = 6;
- Calculate x when y = 12.
y varies in inverse proportion with x. y = 1 when x = 7.
- Write an equation for y in terms of x;
- Find y when x is 35;
- Calculate x when y = 14.
C is inversely proportional to D. C = 4 when D = 2.
- Write C in terms of D;
- Find C when D is 8;
- Find D when C is 2.
y is in inverse proportion to x. y = 2 when x = 5.
- Write y in terms of x;
- Find x when y = 1/2;
- Calculate y when x = 100.
w and z are in inverse proportion. w = 3 when z = 3
- Write an equation for w in terms of z;
- Find w when z = 63;
- Find z when w = 5.
y is directly proportional to x². y = 16 when x =2
- Write y in terms of x;
- Calculate y when x = 10;
- Calculate x when y = 144.
y is proportional to the positive square root of x. y = 12 when x = 36.
- Write an equation involving y and x;
- Find y when x = 49;
- Find x when y = 10;
y varies in proportion to the cube of x-1. y is 40 when x is 3.
- Write an equation showing y in terms of x;
- Calculate x when y is 135;
- Calculate y when x is 1.
A is in direct proportion with the cube root of B. A = 30 when B = 125.
- Write A in terms of B;
- Calculate B when A is 1;
- Find A when B is 60.
y is directly proportional to . y is 12 when x is 2.
- Write y in terms of x;
- Find the value of x when y is 6;
- Calculate y when x is 11.
y is inversely proportional to the square of x. y is 5 when x is 2.
- Write y in terms of x;
- Find the value of y when x=10;
- Find the value of x when y = 80.
y varies inversely proportional to the square root of x. y = 4 when x = 9.
- Write an equation involving y and x;
- Find y when x = 36;
- Find x when y = 6.
y is in inverse proportion to x-1. When y is 8, x is 3.
- Write an equation for y in terms of x;
- Calculate y when x is 9;
- Calculate x when y = 16.
y is inversely proportional to the cube root of x. When y = 2, x = 8.
- Write y in terms of x;
- Calculate the value of y when x is 27;
- Find x when y is 4.
y varies in inverse proportion with x⁵. When y = 3, x = 2.
- Write an equation for y in terms of x;
- Calculate y when x = 1;
- Calculate x when y = 10 (round answer to 3 significant figures).

Answers

(a) y = 5x, (b) y = 35, (c) x = 5;
(a) y = 16x, (b) y = 48, (c) x = 5;
(a) y = 4x, (b) x = 6, (c) y = 44;
(a) A = 7B, (b) B = 63, (c) A = 6;
(a) z = 3w, (b) z = 48, (c) w = 34;
(a) y = 6/x, (b) y = 1, (c) x = 1/2;
(a) y = 7/x, (b) y = 1/5, (c) x = 1/2;
(a) C = 8/D, (b) C = 1, (c) D = 4;
(a) y = 10/x, (b) x = 20, (c) y = 1/10;
(a) w = 9/z, (b) w = 1/7, (c) $z = 1 \frac{4}{5}$ ;
(a) y = 4x², (b) y = 400, (c) x = 6 or -6;
(a) $y = 2 \sqrt{x}$ , (b) y = 17, (c) x = 25;
(a) y=5(x-1)³, (b) x = 4, (c) y = 0;
(a) $b = 6 \sqrt[3]{A}$ , (b) B = 6, (c) A = 1000;
(a) $y = 4 \frac{x+1}{x-1}$ , (b) x = 5, (c) y = 4.8;
(a) $y = \frac{20}{x^2}$ , (b) y=1/5, (c) x = 1/2 or -1/2;
(a) $y = \frac{12}{\sqrt{x}}$ , (b) y = 2, (c) x = 4;
(a) $y = \frac{16}{x-1}$ , (b) y = 2, (c) x = 2;
(a) $y = \frac{4}{\sqrt[3]{x}}$ , (b) $y = 1 \frac{1}{3}$ , (c) x = 1;
(a) $y = \frac{96}{x^5}$ , (b) y = 96, (c) $y = \sqrt[5]{9.6} \approx 1.57$ .

Solution	Comment
$y \propto x$	1. Rewrite the first sentence using mathematical notation.
y = kx	2. Rewrite this statement as an equation, including the constant of proportionality, k.
34 = k x 51 So k = 34/51 or 2/3	3. Use the numbers given to calculate k
$y = \frac{2}{3} x$	4. Write the equation (i.e. in comment 2 above) again, but this time replace k with the value you have calculated for it. This answers part (a)
(b) $y = \frac{2}{3} \times 7$ $y = \frac{14}{3} = 4 \frac{2}{3}$ (c) $10 = \frac{2}{3} \times x$ 30 = 2x x=15	5. For each of part (b) and (c) substitute the value you are given into this new equation to find the value that you want.

Solution	Comment
$y \propto \frac{1}{x}$	1. Rewrite the first sentence using mathematical notation.
$y = \frac{k}{x}$	2. Rewrite this statement as an equation, including the constant of proportionality, k.
$3 = \frac{k}{5}$ So k = 15	3. Use the numbers given to calculate k
$y = \frac{15}{x}$	4. Write the equation (i.e. in comment 2 above) again, but this time replace k with the value you have calculated for it. This answers part (a)
(b) $y = \frac{15}{3}$ So y = 5 (c) $\frac{1}{2} = \frac{15}{x}$ So x = 30	5. For each of part (b) and (c) substitute the value you are given into this new equation to find the value that you want.

Solution	Comment
$y \propto \sqrt[3]{x}$	1. Rewrite the first sentence using mathematical notation.
$y = k \sqrt[3]{x}$	2. Rewrite this statement as an equation, including the constant of proportionality, k.
$12 = k \sqrt[3]{64}$ $12 = k \times 4$ So k = 3	3. Use the numbers given to calculate k
$y = 3 \sqrt[3]{x}$	4. Write the equation (i.e. in comment 2 above) again, but this time replace k with the value you have calculated for it. This answers part (a)
(b) $y = 3 \sqrt[3]{125}$ y = 3 x 5 y = 15 (c) $18 = 3 \sqrt[3]{x}$ $6 = \sqrt[3]{x}$ 6³ = x x = 216	5. For each of part (b) and (c) substitute the value you are given into this new equation to find the value that you want.

Solution	Comment
$y \propto \frac{1}{x^4}$	1. Rewrite the first sentence using mathematical notation.
$y = \frac{k}{x^4}$	2. Rewrite this statement as an equation, including the constant of proportionality, k.
$5 = \frac{k}{2^4}$ $5 = \frac{k}{16}$ So k = 80	3. Use the numbers given to calculate k
$y = \frac {80}{x^4}$	4. Write the equation (i.e. in comment 2 above) again, but this time replace k with the value you have calculated for it. This answers part (a)
(b) $y = \frac{80}{10^4}$ $y = \frac{80}{10000}$ So y = 0.008 (c) $10 = \frac{80}{x^4}$ $x^4 = \frac{80}{10}$ x⁴ = 8 $x \approx 1.68$	5. For each of part (b) and (c) substitute the value you are given into this new equation to find the value that you want.

KS4. Direct & Inverse Proportion

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