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A small business in Melbourne sells handmade candles. The cost to produce each candle is $4.50, and they sell each candle for $9.00.
The business also has a fixed monthly expense of $360 for rent and utilities. How many candles must they sell in a month to make a profit of exactly $450?
Use a linear model to find the exact number of candles.
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Worked solution
- The business sells each candle for $9.00 and incurs a production cost of $4.50 per candle. The profit per candle is calculated as selling price minus cost: $9.00 - $4.50 = $4.50 per candle.
- The business has a fixed monthly expense of $360 for rent and utilities. To make a profit of $450, the total revenue from candle sales must cover both the fixed costs and the desired profit. Total required revenue = fixed costs + desired profit = $360 + $450 = $810.
- Let x be the number of candles sold. The total profit from selling x candles is given by the linear model: profit = (profit per candle) × x - fixed costs. Substituting known values: $4.50x - $360.
- Set the profit equation equal to the desired profit: $4.50x - $360 = $450.
- Add $360 to both sides to isolate the term with x: $4.50x = $450 + $360 = $810.
- Divide both sides by $4.50 to solve for x: x = $810 / $4.50.
- Perform the division: $810 ÷ $4.50. First, express both numbers as fractions: 810 / 4.50 = 81000 / 450 (multiply numerator and denominator by 100 to eliminate decimal).
- Simplify 81000 / 450 by dividing both numerator and denominator by 10: 8100 / 45.
- Divide 8100 by 45: 45 × 180 = 8100. Therefore, x = 180.
- Verify the solution: 180 candles × $4.50 profit per candle = $810. Subtract fixed cost: $810 - $360 = $450. This matches the desired profit.
- The business must sell exactly 180 candles to make a profit of $450.
- The solution is valid because the profit per candle is constant, and the model accounts for both variable cost and fixed cost. All units are consistent (dollars and number of candles).
Reference answer: 180
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