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It takes 8 minutes for Byron to fill the kiddie pool in the backyard using only a handheld hose. When his younger sister is impatient, Byron also uses the lawn sprinkler to add water to the pool so it is filled more quickly. If the hose and sprinkler are used together, it takes 5 minutes to fill the pool. Which equation can be used to determine r, the rate in parts per minute, at which the lawn sprinkler would fill the pool if used alone? + 5r = 8 + 5r = 1 5() = r = 5r

1 answer

To find the rate \( r \) at which the lawn sprinkler would fill the pool if used alone, we can start by determining the filling rates for both the hose and the sprinkler.

  1. Rate of the hose:

    • The hose can fill the pool in 8 minutes, so its rate is \( \frac{1}{8} \) of the pool per minute.

  2. Rate of the sprinkler:

    • Let \( r \) be the rate at which the sprinkler fills the pool (in pools per minute).

  3. Combined Rate:

    • When both the hose and the sprinkler are used together, they can fill the pool in 5 minutes. Therefore, the combined rate is \( \frac{1}{5} \) of the pool per minute.

Now, we can set up the equation with the rates:

\[ \text{Rate of hose} + \text{Rate of sprinkler} = \text{Combined rate} \]

Substituting the values:

\[ \frac{1}{8} + r = \frac{1}{5} \]

This is the equation that can be used to determine \( r \).

To find \( r \), we can solve this equation:

  1. First, find a common denominator for the two fractions on the left side. The least common multiple of 8 and 5 is 40.

  2. Rewrite the equation: \[ \frac{5}{40} + r = \frac{8}{40} \]

  3. Subtract \( \frac{5}{40} \) from both sides: \[ r = \frac{8}{40} - \frac{5}{40} = \frac{3}{40} \]

So the lawn sprinkler's rate \( r \) is \( \frac{3}{40} \) pools per minute.

In summary, the equation you need is:

\[ \frac{1}{8} + r = \frac{1}{5} \]