William says to Gracie, “If you give me \$7.50,we will have and equal amount of money. “Grace responds, “That may be true, but if you give me

Question

William says to Gracie, “If you give me \$7.50,we will have and equal amount of money. “Grace responds, “That may be true, but if you give me \$ 7.50 instead, I will have twice as much money as you. “How much money did William and Grace each have in the beginning? Please show work

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2 weeks 2022-01-11T15:21:27+00:00 1 Answer 0 views 0

Step-by-step explanation:

William says to Gracie:

• “If you give me \$7.50, we will have an equal amount of money”

Grace responds:

• “That may be true, but if you give me \$ 7.50 instead, I will have twice as much money as you”

Assume that William has \$x and Gracie has \$y at the beginning

∵ William has \$x

∵ Gracie has \$y

∵ When Gracie gives William \$7.5, they will have the same

amount of money

– That means subtract 7.5 from Gracie and add 7.5 to William

∴ x + 7.5 = y – 7.5

– Subtract 7.5 from both sides

x = y – 15 ⇒ (1)

∵ When William gives Gracie \$7.5, Gracie will have twice as

much money as William

– That means add 7.5 to Gracie and equate the sum

by 2 times the difference of William and 7.5

∴ y + 7.5 = 2(x – 7.5)

∴ y + 7.5 = 2x – 15

– Subtract 7.5 from both sides

y = 2x – 22.5 ⇒ (2)

Now we have system of equations to solve it

Substitute y in equation (1) by equation (2)

∵ x = 2x – 22.5 – 15

– Add the like terms in the right hand side

∴ x = 2x – 37.5

– Subtract 2x from both sides

∴ – x = -37.5

– Divide both sides by -1

x = 37.5

– Substitute the value of x in equation (2) to find y

∵ y = 2(37.5) – 22.5

y = 52.5