Ratio Sharing, Algebraic Direct Proportion & Inverse Proportion Equations

1

Sharing in a Ratio (Total, One Share, or Difference)

Always find the value of 1 part first: 1. If given Total T for ratio a : b, 1 part = T / (a + b). 2. If given that A's share is V, 1 part = V / a. 3. If given that B gets D more than A (where b > a), 1 part = D / (b - a). Once you know 1 part, multiply by the required number of parts.

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2

Combining Two Ratios (A:B and B:C) & 1:n Form

To write a ratio a : b in the form 1 : n, divide both sides by a -> 1 : (b/a). To write in the form n : 1, divide both sides by b -> (a/b) : 1. To combine A : B and B : C into A : B : C, scale both ratios so that the middle shared term B equals the Lowest Common Multiple (LCM) of its two values.

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3

Algebraic Changing Ratios (Higher Tier)

When an initial ratio of A : B is a : b, represent the initial quantities as ax and bx. After adding/subtracting amounts (e.g., A loses 4 and B gains 10 to give a new ratio c : d), equate the fraction to the new ratio: (ax - 4) / (bx + 10) = c / d. Cross-multiply to form a linear equation: d(ax - 4) = c(bx + 10) and solve for x.

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4

Direct Proportion Equations (y ∝ xⁿ)

If y is directly proportional to x (y ∝ x), then y = kx, and the graph of y against x is a straight line passing through the origin (0,0). On the Higher Tier, pay close attention to powers/roots: 'y is directly proportional to the square of x' means y = kx²; 'y is directly proportional to the cube of x' means y = kx³; 'y is directly proportional to the square root of x' means y = k√x.

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5

Inverse Proportion Equations (y ∝ 1 / xⁿ)

If y is inversely proportional to x (y ∝ 1/x), then y = k / x (or xy = k), and the graph is a reciprocal hyperbola in the first quadrant. Higher Tier variations: 'y is inversely proportional to the square of x' means y = k / x²; 'y is inversely proportional to the cube root of x' means y = k / ³√x.

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6

Chaining three proportional variables (higher tier)

If y is inversely proportional to x² (y = a / x²) and x is directly proportional to √z (x = b√z), substitute the expression for x directly into the equation for y (remembering to square the whole bracket): y = a / (b√z)² = a / (b²z) = k / z, showing y is inversely proportional to z.

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1. Q1

   In a bag of counters, the ratio of red counters to blue counters is 3 : 7. When 12 more red counters are added to the bag, the ratio of red counters to blue counters becomes 3 : 5. Work out the number of blue counters in the bag. (Higher Tier) [4 marks]

   Check Mark Scheme

   Click “Check Mark Scheme” to compare your answer, 1. Let the original number of red counters be 3x and the original number of blue counters be 7x [1 mark]. 2. After adding 12 red counters, the number of red counters is (3x + 12) and the number of blue counters is still 7x, so we form the equation: (3x + 12) / (7x) = 3 / 5 [1 mark]. 3. Cross-multiply to eliminate fractions: 5(3x + 12) = 3(7x) -> 15x + 60 = 21x [1 mark]. 4. Solve for x: 60 = 6x -> x = 10; therefore, the number of blue counters is 7x = 7 x 10 = 70 [1 mark].

2. Q2

   y is inversely proportional to the square of x. When x = 4, y = 5. (a) Find an equation for y in terms of x. (b) Work out the positive value of x when y = 20. (Higher Tier) [4 marks]

   Check Mark Scheme

   Click “Check Mark Scheme” to compare your answer, 1. (a) Write the inverse square proportion equation: y = k / x² [1 mark]. 2. Substitute x = 4 and y = 5 to find k: 5 = k / 4² -> 5 = k / 16 -> k = 5 x 16 = 80; therefore, the equation is y = 80 / x² [1 mark]. 3. (b) Substitute y = 20 into the formula: 20 = 80 / x² [1 mark]. 4. Rearrange and solve for positive x: x² = 80 / 20 = 4 -> x = √4 = 2 [1 mark].

3. Q3

   Three friends, Priya, Qasim, and Ryan, share a cash prize. The ratio of Priya's share to Qasim's share is 2 : 3. The ratio of Qasim's share to Ryan's share is 5 : 4. Ryan receives £180. Work out how much Priya receives. [4 marks]

   Check Mark Scheme

   Click “Check Mark Scheme” to compare your answer, 1. Write down the two separate ratios: P : Q = 2 : 3 and Q : R = 5 : 4 [1 mark]. 2. Make the shared middle term Q equal to the LCM of 3 and 5 (which is 15): multiply P : Q by 5 to get 10 : 15, and multiply Q : R by 3 to get 15 : 12, giving the combined ratio P : Q : R = 10 : 15 : 12 [1 mark]. 3. Ryan's 12 parts equal £180, so 1 part = £180 ÷ 12 = £15 [1 mark]. 4. Priya receives 10 parts: 10 x £15 = £150 [1 mark].

4. Q4

   Given that P is directly proportional to the cube of Q, and when Q = 2, P = 40: (a) find a formula for P in terms of Q, and (b) calculate the percentage increase in P when Q is increased by 50%. (Higher Tier) [4 marks]

   Check Mark Scheme

   Click “Check Mark Scheme” to compare your answer, 1. (a) Write P = kQ³ and substitute Q = 2, P = 40: 40 = k(2³) = 8k -> k = 5, so P = 5Q³ [1 mark]. 2. (b) Increasing Q by 50% multiplies Q by the decimal multiplier 1.5, so Q_new = 1.5Q [1 mark]. 3. Substitute Q_new = 1.5Q into the formula: P_new = 5(1.5Q)³ = 3.375 x (5Q³) = 3.375P [1 mark]. 4. A multiplier of 3.375 represents a percentage increase of (3.375 - 1) x 100 = 237.5% [1 mark].

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