Unit 9
9 marks · NormalA supermarket sells two products in bulk and uses a mathematical model for pricing.
Toffee A: 1.35 kg for $9.99. Toffee B: 2.1 kg for $15.82.
Work out the cost per kg for each and decide which is better value for money.
The cost per kg, , for a pack weighing kg with total price dollars is given by .
If a special almond pack has weight and price dollars, show, by algebraic reasoning, that the cost per kg increases as decreases.
A third product, caramel fudge, is priced so that for a pack of kg, the price is dollars.
The company claims it is never better value per kg than Toffee B for packs up to 4 kg. Is their claim correct? Justify your answer with calculations and reasoning.
Work this out on paper, then compare with the answer and worked solution below. With a free account you can submit written answers for marking.
Show answer
(a) Toffee A: $7.40/kg; Toffee B: $7.53/kg; Toffee A is better value. dollars per kg · must show cost per kg for both, identification of better value
(b) C = (2w + 3)/w = 2 + 3/w; as w decreases, 3/w increases, so C increases. · must show correct expression, explanation of effect as w decreases
(c) Cost per kg for fudge: (7.5y-2)/y = 7.5 - 2/y. For y = 4, cost = 7.0 < 7.53, so claim is incorrect for y = 4, fudge is cheaper. · must show calculation for y = 4, logical check, counter-example
Common mistakes: Using incorrect formula for cost per kg; Not checking all valid values for part (c)
Show mark scheme and worked solution
Marking points
M1 (a) · 1 mark · Converts weights and calculates unit price for both products correctly
A1 (a) · 1 mark · Accurate comparison and conclusion about value (follow-through)
A2 (a) · 1 mark · All calculation steps shown (follow-through)
M2 (b) · 1 mark · Derives correct expression C = 2 + 3/w
R1 (b) · 2 marks · Explanation shows C increases as w decreases, with logical reasoning
M3 (c) · 1 mark · Correct expression for cost per kg for fudge
A3 (c) · 1 mark · Evaluates at boundary (y=4), valid comparison to Toffee B (follow-through)
R2 (c) · 1 mark · Provides valid justification or counter-example; correct conclusion (follow-through)
Worked solution
(a) Toffee A: $9.99 ÷ 1.35 = $7.40/kg (2 d.p.); Toffee B: $15.82 ÷ 2.1 = $7.53/kg. Toffee A is better value.
(b) For the almond pack: cost per kg = (2w+3)/w = 2 + 3/w. As w decreases, 3/w increases, so the cost per kg increases.
(c) Cost per kg for fudge: (7.5y-2)/y = 7.5 - 2/y. At y=4: 7.5-2/4 = 7.0, which is less than $7.53 (Toffee B). Thus, for y=4, fudge is cheaper, so the company claim is not always correct.