Question

If 5 / y = 7 / x = 24 and 12 / y + 2 / x = 24, find the ratio of x to y.

A. 1 / 6
B. 2 / 7
C. 5 / 12
D. 5 / 7

Answer 1

• D) 5/7

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Given equations

• 5/y + 7/x = 24
• 12/y + 2/x = 24

Clear fractions

• 5/y + 7/x = 24 ⇒ (5x + 7y)/(xy) = 24 ⇒ 5x + 7y = 24xy
• 12/y + 2/x = 24 ⇒ (12x + 2y)/(xy) = 24 ⇒ 12x + 2y = 24xy

Compare the two equations, find the x:y ratio

• 5x + 7y = 12x + 2y
• 12x - 5x = 7y - 2y
• 7x = 5y
• x/y = 5/7

Correct choice is D

Answer 2

`\textsf{D)} \quad (5)/(7)`

Explanation:

Given equations:

`\begin{cases}(5)/(y)+(7)/(x)=24\\\\(12)/(y)+(2)/(x)=24\end{cases}`

Rearrange the first equation to eliminate the fractions:

`(5)/(y)+(7)/(x)=24`

`\implies (5x)/(xy)+(7y)/(xy)=24`

`\implies (5x+7y)/(xy)=24`

`\implies 5x+7y=24xy`

Rearrange the second equation to eliminate the fractions:

`(12)/(y)+(2)/(x)=24`

`\implies (12x)/(xy)+(2y)/(xy)=24`

`\implies (12x+2y)/(xy)=24`

`\implies 12x+2y=24xy`

Substitute the first equation into the second equation:

`\implies 12x+2y=5x+7y`

Rearrange so that x is on the left side and y is on the right side:

`\implies 12x-5x=7y-2y`

`\implies 7x=5y`

Therefore, the ratio of x to y is:

`\implies (x)/(y)=(5)/(7)`