Question

Two different types of cake are on sale at prices of $0.30 and $0.40 each. The cakes that are being sold for $0.30 cost 0.20 to make and the ones being sold for $0.40 cost $0.25 to make. No cakes can be kept to be sold the next day, so all of the cakes are being reduced to half price 2 hours before the sale finish. All of the $0.30 cakes sold out before any of the prices were reduced, and all of the $0.40 were eventually sold out, even though only half had been sold when the price was reduced. The overall profit at the end of the day was $30.00, but it could have been $40.00 if all of the cakes had been sold before the prices were reduced. How many of each type of cake were there in the sale?

Answer 1

There were 250 cakes of selling price $0.30 and 100 cakes of selling price $0.40

Explanation:

Assume that the number of cakes that price $0.30 is x and the number of cakes that price $0.40 is y

∵ The cakes that are being sold for $0.30 cost 0.20 to make

- The profit = selling price - cost price

∴ The profit of each cake is = 0.30 - 0.20 = 0.10

∵ The number of that cakes is x

∴ The profit of x cakes = 0.10 x

All of the cakes are being reduced to half price 2 hours before the sale finish

∵ Half of y cakes had been sold when the price was reduced

∵ The cost of each one is $0.40

- That means the selling price of half y is 0.40 and the selling

price of other half y is (
`(1)/(2)` × 0.40 = 0.20)

∴ The selling price of y cakes =
`(1)/(2)` y × 0.40 +
`(1)/(2)` y × 0.20

∴ The selling price of y cakes = 0.20 y + 0.10 y

∴ The selling price of y cakes = 0.30 y

∵ The ones of y costs $0.25 to make

∴ The total cost of y cakes = 0.25 y

∴ The profit of y cakes = 0.30 y - 0.25 y

∴ The profit of y cakes = 0.05 y

∵ The overall profit at the end of the day was $30.00

- Add the profits of x and y, then equate the sum by 30

∴ 0.10 x + 0.05 y = 30 ⇒ (1)

It could have been $40.00 if all of the cakes had been sold before the prices were reduced

∵ The profit of all x = 0.10 x

∵ The profit of all y = (0.40 - 0.25) × y

∴ The profit of all y = 0.15 y

- Add 0.10 x and 0.15 y, then equate the sum by 40

∴ 0.10 x + 0.15 y = 40 ⇒ (2)

Now we have a system of equation to solve it

Subtract equation (1) from equation (2) to eliminate x

∵ (0.10 x - 0.10 x) + (0.15 y - 0.05 y) = (40 - 30)

∴ 0.10 y = 10

- Divide both sides by 0.10

∴ y = 100

Substitute the value of y in equation (1) to find x

∵ 0.10 x + 0.05(100) = 30

∴ 0.10 x + 5 = 30

- Subtract 5 from both sides

∴ 0.10 x = 25

- Divide both sides by 0.10

∴ x = 250

There were 250 cakes of selling price $0.30 and 100 cakes of selling price $0.40