## 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

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?

## Answers ( )

Answer:There were

250cakes of selling price $0.30 and100cakes of selling price $0.40Step-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 xAll 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 ( [tex]\frac{1}{2}[/tex] × 0.40 = 0.20)

∴ The selling price of y cakes = [tex]\frac{1}{2}[/tex] y × 0.40 + [tex]\frac{1}{2}[/tex] 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 = 100Substitute 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 = 250There were 250 cakes of selling price $0.30 and 100 cakes of selling price $0.40