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?

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

Step-by-step 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 ( × 0.40 = 0.20)

∴ The selling price of y cakes = y × 0.40 + 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