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Zorluk: ZorRatios, Rates, and Proportions

Three water pumps, PP, QQ, and RR, operate at constant individual rates. The ratio of the rate of pump PP to the rate of pump QQ is 2:32 : 3. When all three pumps operate simultaneously, their combined rate is 33 times the rate of pump PP alone. If pump RR working alone can drain a full reservoir in 2424 hours, how many hours would pump QQ working alone take to drain the same full reservoir?

  1. 88 hoursCevap
  2. B
    1212 hours
  3. C
    1616 hours
  4. D
    3232 hours
  5. E
    5656 hours

Cevap

8 hours
The correct answer is 8 hours. By setting the rate of pump Q as 32\frac{3}{2} times the rate of pump P, the combined rate of all three pumps is rP+32rP+rR=52rP+rRr_P + \frac{3}{2} r_P + r_R = \frac{5}{2} r_P + r_R. Setting this equal to 3rP3 r_P shows that pump R's rate is 12rP\frac{1}{2} r_P. Since pump R takes 24 hours (rR=124r_R = \frac{1}{24}), pump P's rate is 112\frac{1}{12} (taking 12 hours), and pump Q's rate is 32×112=18\frac{3}{2} \times \frac{1}{12} = \frac{1}{8} (taking 8 hours).

Adım Adım Çözüm

1
Express the rates of pumps P and Q in terms of a common variable.
Let rPr_P, rQr_Q, and rRr_R be the rates of pumps PP, QQ, and RR in reservoirs per hour. Given rP:rQ=2:3r_P : r_Q = 2 : 3, we have rQ=32rPr_Q = \frac{3}{2} r_P.
Relating pump rates using the given ratio simplifies the system of equations to one variable.
2
Set up the combined rate equation and solve for rRr_R in terms of rPr_P.
rP+rQ+rR=3rP  ⟹  rP+32rP+rR=3rP  ⟹  52rP+rR=3rP  ⟹  rR=12rPr_P + r_Q + r_R = 3 r_P \implies r_P + \frac{3}{2} r_P + r_R = 3 r_P \implies \frac{5}{2} r_P + r_R = 3 r_P \implies r_R = \frac{1}{2} r_P.
The total rate is the sum of individual rates, allowing us to express pump R's rate in terms of pump P's rate.
3
Calculate rPr_P and rQr_Q using the given rate for pump R.
Since pump RR takes 2424 hours alone, rR=124r_R = \frac{1}{24}. Therefore, 12rP=124  ⟹  rP=112\frac{1}{2} r_P = \frac{1}{24} \implies r_P = \frac{1}{12}. Then rQ=32×112=18r_Q = \frac{3}{2} \times \frac{1}{12} = \frac{1}{8}.
Knowing pump R's explicit numerical rate allows finding the numerical rates for pumps P and Q.
4
Determine the time required for pump Q alone to drain the reservoir.
\text{Time for } Q = \frac{1}{r_Q} = \frac{1}{1/8} = 8 \text{ hours}.
The time required to complete one full job is the reciprocal of the rate.

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