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

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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?

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Ayla 3 months 2022-02-19T07:38:02+00:00 1 Answer 0 views 0

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    2022-02-19T07:39:10+00:00

    Answer:

    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 ( [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 = 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

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