Back to Math Problems (Ages 16 to 17)
Mia owns a candy shop and is preparing special Christmas packs of licorice allsorts. She creates three different mixes, each containing a combination of black licorice, coconut rolls, and fruit pastilles. All mixes are sold at $12 per 500g pack.
If she combines black licorice, coconut rolls, and fruit pastilles in the ratio of 2:2:1 by weight, she makes a profit of 25%.
If she combines black licorice, coconut rolls, and fruit pastilles in the ratio of 1:3:1 by weight, she makes a profit of 20%.
If she combines black licorice, coconut rolls, and fruit pastilles in the ratio of 3:1:2 by weight, she makes a profit of 15%.
(a) What is the cost price (in dollars per kg) for Mia for each of the black licorice, coconut rolls, and fruit pastilles?
(b) What percentage profit would Mia make if she creates a mix with black licorice, coconut rolls, and fruit pastilles in the ratio of 1:2:2 by weight?
Let’s denote:
– b = cost price per gram of black licorice in cents per gram
– c = cost price per gram of coconut rolls in cents per gram
– f = cost price per gram of fruit pastilles in cents per gram
First Mix (Ratio 2:2:1, Profit 25%):
1. Amounts in 500g Pack:
– Black licorice: 200g
– Coconut rolls: 200g
– Fruit pastilles: 100g
2. Cost Equation:
– Selling Price (SP): $12 = 1200 cents
– Cost Price (CP): SP / 1.25 = 960 cents
– Equation: 200b + 200c + 100f = 960
Second Mix (Ratio 1:3:1, Profit 20%):
1. Amounts in 500g Pack:
– Black licorice: 100g
– Coconut rolls: 300g
– Fruit pastilles: 100g
2. Cost Equation:
– CP: SP / 1.20 = 1000 cents
– Equation: 100b + 300c + 100f = 1000
Third Mix (Ratio 3:1:2, Profit 15%):
1. Amounts in 500g Pack:
– Black licorice: 250g
– Coconut rolls: 83.33g
– Fruit pastilles: 166.67g
2. Cost Equation:
– CP: SP / 1.15 = 1043.48 cents
– Multiply the equation by 3 to eliminate fractions:
750b + 250c + 500f = 3130.43
Solving the Equations:
Subtracting the first equation from the second:
-100b + 100c = 40
Simplifies to:
-b + c = 0.4 (Equation 1)
Using the third equation and simplifying:
b – c + f = 2.92 (Equation 2)
Substitute c = b + 0.4 (from Equation 1) into Equation 2:
b – (b + 0.4) + f = 2.92
-0.4 + f = 2.92
f = 3.32
Substitute f back into one of the equations to find b:
b + 3(b + 0.4) + 3.32 = 10
4b + 4.52 = 10
b = 1.37
Find c:
c = b + 0.4 = 1.37 + 0.4 = 1.77
Cost Prices per Kilogram:
– Black licorice: 1.37 cents/g × 1000 = $13.70/kg
– Coconut rolls: 1.77 cents/g × 1000 = $17.70/kg
– Fruit pastilles: 3.32 cents/g × 1000 = $33.22/kg
(b) Calculating the Profit Percentage for the New Mix
New Mix Ratio (1:2:2):
1. Amounts in 500g Pack:
– Black licorice: 100g
– Coconut rolls: 200g
– Fruit pastilles: 200g
2. Total Cost:
– Cost of black licorice: 100 × 1.37 = 137 cents
– Cost of coconut rolls: 200 × 1.77 = 354 cents
– Cost of fruit pastilles: 200 × 3.32 = 664 cents
– Total CP: 137 + 354 + 664 = 1155 cents ($11.55)
3. Profit Calculation:
– SP: $12
– Profit: $12 – $11.55 = $0.45
– Profit Percentage: (0.45 / 11.55) × 100% = 3.90%
Answer:
(a)
– Black licorice cost: $13.70 per kg
– Coconut rolls cost: $17.70 per kg
– Fruit pastilles cost: $33.22 per kg
(b) Mia would make approximately a 3.90% profit on the new mix with a 1 : 2 : 2 ratio.
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