A network router transmits data from two sources, Server A and Server B, at a constant rat… — SAT

A network router transmits data from two sources, Server $A$ and Server $B$ , at a constant ratio of $5$ megabytes ( $\text{MB}$ ) from Server $A$ for every $3\text{ MB}$ from Server $B$ . In a single session, the router transmitted a combined total of $160\text{ MB}$ of data from both servers. During the next session, the router transmits data at the same ratio, but the amount of data transmitted from Server $A$ is $20\%$ greater than the amount of data transmitted from Server $A$ in the first session. How many megabytes of data are transmitted from Server $B$ in the second session?

Answer: 72 MB

Answer

The amount of data transmitted from Server B in the second session is 72 megabytes.

To find the data transmitted from Server B in the second session, we first determine the individual data amounts from the first session. Given the ratio of data from Server A to Server B is

5:3

and the total is

160\text{ MB}

, we can express their data amounts as

5x

and

3x

. Solving

5x + 3x = 160

yields

x = 20

. Thus, Server A transmitted

5 \times 20 = 100\text{ MB}

and Server B transmitted

3 \times 20 = 60\text{ MB}

in the first session. In the second session, the data from Server A increased by

20\%

, yielding

100 \times 1.20 = 120\text{ MB}

. Since the transmission ratio between Server A and Server B remains constant at

5:3

, we can set up the proportion

\frac{120}{B} = \frac{5}{3}

. Cross-multiplying gives

5B = 360

, which solves to

B = 72

megabytes.

Step-by-Step Solution

Set up an equation for the total data in the first session.

5x + 3x = 160 \implies 8x = 160 \implies x = 20

The ratio of data from Server A to Server B is

5:3

, meaning the amounts can be represented as

5x

and

3x

for some multiplier

x

Calculate the data transmitted from Server A in the first session.

A_1 = 5(20) = 100\text{ MB}

Multiply the ratio term for Server A by the scale factor

x

to find its initial data contribution.

Calculate the increased data from Server A in the second session.

A_2 = 100 \times (1 + 0.20) = 120\text{ MB}

Increase the first session's data from Server A by

20\%

Use the constant ratio to find the data from Server B in the second session.

B_2 = 72\text{ MB}

Set up the proportion

\frac{120}{B_2} = \frac{5}{3}

and solve for

B_2

by cross-multiplying.

Key Concept

Solving part-to-part and part-to-whole ratio problems under proportional changes

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