Question

it takes \\( \frac{2}{5} \\) hour to fill a gallon container with liquid soap from a dispenser in a factory. yesterday, the dispenser ran for \\( 3 \frac{1}{3} \\) hours. which multiplication equation can be used to find the number of gallons of soap \\( w \\) that were dispensed?\
\\( \boldsymbol{\text{a) } 3 \frac{1}{3} w = \frac{2}{5}} \\)\
\\( \boldsymbol{\text{b) } \frac{2}{5} w = 3 \frac{1}{3}} \\)\
\\( \boldsymbol{\text{c) } w = \frac{2}{5}\left(3 \frac{1}{3}\
ight)} \\)\
\\( \boldsymbol{\text{d) } \frac{2}{5} w = 3 \frac{1}{3} w} \\)

Explanation:

Step1: Understand the rate

The time to fill 1 gallon is $\frac{2}{5}$ hour. So the rate (gallons per hour) is $1\div\frac{2}{5}=\frac{5}{2}$ gallons per hour. Wait, no, actually, if $\frac{2}{5}$ hour fills 1 gallon, then in $t$ hours, the number of gallons $w$ is related by the rate. Wait, let's re - express: Let the time taken be $t$ hours, and gallons be $w$. The time per gallon is $\frac{2}{5}$ hour/gallon. So if the dispenser runs for $t = 3\frac{1}{3}$ hours, then the number of gallons $w$ is given by the formula: time per gallon $\times$ number of gallons= total time. So $\frac{2}{5}w=t$, where $t = 3\frac{1}{3}$. So $\frac{2}{5}w = 3\frac{1}{3}$, which is option B.

Step2: Analyze each option

• Option A: $3\frac{1}{3}w=\frac{2}{5}$: This would imply that the rate is $3\frac{1}{3}$ gallons per hour, which is incorrect as we know the time per gallon is $\frac{2}{5}$ hour.
• Option B: $\frac{2}{5}w = 3\frac{1}{3}$: As we derived, time per gallon ($\frac{2}{5}$ hour/gallon) times number of gallons ($w$) equals total time ($3\frac{1}{3}$ hours), so this is correct.
• Option C: $w=\frac{2}{5}(3\frac{1}{3})$: This would be the case if the rate was $\frac{2}{5}$ gallons per hour, but we know that $\frac{2}{5}$ hour gives 1 gallon, so the rate is $\frac{5}{2}$ gallons per hour, not $\frac{2}{5}$. So this is incorrect.
• Option D: $\frac{2}{5}w = 3\frac{1}{3}w$: This equation would imply $\frac{2}{5}=3\frac{1}{3}$ which is false.

Answer:

B) $\frac{2}{5}w = 3\frac{1}{3}$