The rates at which three pipes, A, B, and C, can fill a pool are in the ratio 3:4:6, respe… — ACT

The rates at which three pipes, $A$ , $B$ , and $C$ , can fill a pool are in the ratio $3:4:6$ , respectively. To fill an empty pool, pipes $A$ and $B$ are turned on. After $2\text{ hours}$ , pipe $A$ is turned off and pipe $C$ is turned on. It takes another $3\text{ hours}$ for pipes $B$ and $C$ to finish filling the pool. What fraction of the pool's total volume was filled by pipe $B$ ?

A
$\frac{4}{13}$
B
$\frac{3}{11}$
$\frac{5}{11}$ Cevap
D
$\frac{8}{17}$
E
$\frac{20}{41}$

Cevap

The fraction of the pool's total volume filled by Pipe B is

\frac{5}{11}

To find the fraction of the pool's volume filled by Pipe B, we determine the total volume of the pool and the volume contributed by Pipe B. If the rates of pipes

A

B

, and

C

are

3r

4r

, and

6r

per hour, the first

2\text{ hours}

fill

2(3r+4r) = 14r

units, and the next

3\text{ hours}

fill

3(4r+6r) = 30r

units, for a total pool volume of

44r

units. Since Pipe B runs for the entire

5\text{ hours}

, it fills

5 \times 4r = 20r

units. The fraction is

\frac{20r}{44r} = \frac{5}{11}

Adım Adım Çözüm

Define the rates of the three pipes using a constant multiplier

r

Let the filling rates of pipes

A

B

, and

C

3r

4r

, and

6r

units of volume per hour, respectively.

This translates the given ratio

3:4:6

into algebraic expressions for their individual rates.

Calculate the volume of the pool filled during the first

2\text{ hours}

when pipes

A

and

B

are active.

Volume 1 =

2\text{ hours} \times (3r + 4r) = 2 \times 7r = 14r

units.

The rate of pipes

A

and

B

combined is the sum of their individual rates, and volume is rate multiplied by time.

Calculate the volume of the pool filled during the next

3\text{ hours}

when pipes

B

and

C

are active.

Volume 2 =

3\text{ hours} \times (4r + 6r) = 3 \times 10r = 30r

units.

The rate of pipes

B

and

C

combined is the sum of their individual rates, and volume is rate multiplied by time.

Find the total volume of the pool.

Total Volume =

14r + 30r = 44r

units.

The total volume is the sum of the volumes filled in both time intervals.

Calculate the total volume filled specifically by Pipe

B

across both periods.

Pipe

B

was active for the entire duration of

2 + 3 = 5\text{ hours}

. Volume filled by Pipe

B

5\text{ hours} \times 4r = 20r

units.

To find Pipe B's contribution, multiply its rate by the total time it operated.

Divide the volume filled by Pipe

B

by the total volume of the pool to find the fraction.

Fraction =

\frac{20r}{44r} = \frac{20}{44} = \frac{5}{11}

The fraction is the ratio of the part filled by Pipe B to the whole volume of the pool.

Anahtar Kavram

Using rates and ratios to find the fraction of work completed by a specific component in a multi-stage work problem.

Alternatif Yöntem

Instead of using a variable

r

, you can assume a concrete rate for the pipes that simplifies the arithmetic. For example, let the rates of pipes

A

B

, and

C

3

4

, and

6

gallons per hour, respectively. In the first

2\text{ hours}

, pipes

A

and

B

fill

2 \times (3 + 4) = 14\text{ gallons}

. In the next

3\text{ hours}

, pipes

B

and

C

fill

3 \times (4 + 6) = 30\text{ gallons}

. The total pool capacity is

14 + 30 = 44\text{ gallons}

. Pipe

B

works for the entire

5\text{ hours}

, filling

5 \times 4 = 20\text{ gallons}

. The fraction of the pool filled by Pipe

B

\frac{20}{44} = \frac{5}{11}

Tahmini Süre:3m 0s

Soru

Cevap

Adım Adım Çözüm

Anahtar Kavram

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Alternatif Yöntem

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