Interpreting a Fundraising Stall's takings
A school charity stall sells two items: cupcakes at $3.50 each and jars of jam at $6 each. On Saturday the stall took $438 in total and sold 96 items altogether. The organisers also recorded that 40% of the cupcakes and 25% of the jam were sold in the first hour. The cost of ingredients was $0.90 per cupcake and $2.10 per jar, and the stall paid a $45 site fee. The group wants to know its profit and whether the first hour was its strongest trading period, given that in the last hour it sold 18 items in total.
1. Set up and solve simultaneous equations to find how many cupcakes and how many jars of jam were sold.
Read the worked answer
Let c = cupcakes, j = jars.
c + j = 96 (item count) 3.50c + 6j = 438 (takings)
From the first: c = 96 − j. Substitute: 3.50(96 − j) + 6j = 438 336 − 3.50j + 6j = 438 2.50j = 102 j = 40.8
This is not a whole number, so the recorded figures are inconsistent - you cannot sell 0.8 of a jar. A correct response must recognise this rather than reporting 40.8 jars. Checking: with j = 41, c = 55, takings = 3.50×55 + 6×41 = 192.50 + 246 = $438.50, which is $0.50 over. With j = 40, c = 56, takings = 196 + 240 = $436, which is $2 short. The judgement is that one recorded figure contains an error, and the takings or item count should be re-checked.
2. Assuming the corrected figures are 56 cupcakes and 40 jars, calculate the profit and the profit margin as a percentage of takings.
Read the worked answer
Takings = 3.50 × 56 + 6 × 40 = 196 + 240 = $436.
Costs: cupcakes = 0.90 × 56 = $50.40; jam = 2.10 × 40 = $84.00; site fee = $45. Total costs = 50.40 + 84 + 45 = $179.40.
Profit = 436 − 179.40 = $256.60.
Profit margin = 256.60 ÷ 436 × 100 = 58.9% (to 1 d.p.).
The reasoning point: margin is profit over takings, not over costs. Dividing by costs (256.60 ÷ 179.40 = 143%) would be a markup, a different measure, and would overstate the picture.
3. Using the corrected totals, determine how many items were sold in the first hour and compare with the last hour's 18 items to judge whether the first hour was the strongest.
Read the worked answer
First hour: cupcakes = 40% of 56 = 22.4; jam = 25% of 40 = 10. Total = 32.4 items.
Again a fractional count appears, so the percentages are approximate or the counts were rounded. Treating it as about 32 items, the first hour (≈32) clearly exceeds the last hour (18).
But item count is not the same as takings. First-hour takings ≈ 22.4 × 3.50 + 10 × 6 = 78.40 + 60 = $138.40. Last-hour takings are unknown because the split of the 18 items is not given; if all 18 were cupcakes, takings = $63; if all jam, $108. Either way the first hour's takings exceed the last hour's. The judgement is that comparing item counts and comparing takings can give different impressions, and here both favour the first hour, though the fractional counts signal the data is approximate.