10 Vicky and Sarah can clean the house in 3 hours if they work together. The time that Vicky takes in
cleaning the house alone is 8 hours more than the time Sarah takes cleaning the same house. How long
does it take Vicky to clean the house alone?
A 6 hours
B. 8 hours
C. 10 hours
D 12 hours
Answer:
D. 12 Hours
Step-by-step explanation:
[tex] \\ \sf{} \frac{3}{x} + \frac{3}{x + 8} = 1[/tex]
LCD is x(x + 8). Multiply all the terms by the LCD.
[tex] \\ \sf{}x(x + 8) \bigg ( \frac{3}{x} + \frac{3}{x + 8} = 1 \bigg)[/tex]
[tex] \\ \sf{} \frac{3 \big(x(x + 8) \big)}{x} + \frac{3 \big(x (x + 8) \big)}{x + 8} = x(x + 8)[/tex]
[tex] \\ \sf{}3(x + 8) + 3x = x(x + 8)[/tex]
[tex] \\ \sf{}3x + 24 + 3x = {x}^{2} + 8x[/tex]
[tex] \\ \sf{}6x + 24 = {x}^{2} + 8x[/tex]
[tex] \\ \sf{}6x + 24- {x}^{2} – 8x = 0[/tex]
[tex] \\ \sf{} – 2x + 24 – {x}^{2} = 0[/tex]
[tex] \\ \sf{} {x}^{2} + 2x – 24= 0[/tex]
[tex] \\ \sf{}(x – 4)(x + 6) = 0[/tex]
[tex] \\ \sf{}x – 4 = 0 \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: x + 6 = 0[/tex]
[tex] \\ \sf{}x = 4 \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: x = – 6 \\ \\ [/tex]
X = 4, – 6
Since, time cannot be counted as negative, let’s eliminate – 6. Thus,
Sarah can clean the house in 4 Hours.
For Vicky:
= X + 8
= 4 + 8
= 12
Vicky can clean the house in 12 Hours.
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