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# help

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The taylor family and the simmons family each used sprinklers last summer. The water output rate for the taylor family's sprinkler was 40L per hour. The water output rate for the simmons family's sprinkler was 35L per hour. The families used their sprinklers for a combined total of 35 hours, resulting in a water output of 1300L. How long was each sprinkler used?

Jan 20, 2020

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The taylor family and the simmons family each used sprinklers last summer.
The water output rate for the taylor family's sprinkler was $$40~l$$ per hour.
The water output rate for the simmons family's sprinkler was $$35~l$$ per hour.
The families used their sprinklers for a combined total of $$35~ hours$$, resulting in a water output of $$1300~l$$.
How long was each sprinkler used?

I assume:

$$\begin{array}{|rcll|} \hline \dfrac{40~l}{h}*t_1 + \dfrac{35~l}{h}*t_2 &=& 1300~ l \\ && t_1+t_2 = 35~ h \\ && \text{or} \quad \boxed{t_2 = 35~h-t_1} \\\\ \dfrac{40~l}{h}*t_1 + \dfrac{35~l}{h}*(35~h-t_1) &=& 1300~ l \\\\ \dfrac{40~l}{h}*t_1 + 35^2~l-\dfrac{35~l}{h}*t_1 &=& 1300~ l \\\\ \dfrac{40~l}{h}*t_1-\dfrac{35~l}{h}*t_1 &=& 1300~l-35^2~l \\\\ \dfrac{5~l}{h}*t_1 &=& 75~l \\\\ t_1 &=& \dfrac{75~l}{5~l}~h \\\\ \mathbf{t_1} &=& \mathbf{15~h} \\ \hline t_2 &=& 35~h-t_1 \\ t_2 &=& 35~h-15~h \\ \mathbf{t_2} &=& \mathbf{20~h} \\ \hline \end{array}$$

taylor family's sprinkler was $$\mathbf{15~h}$$ used.
simmons family's sprinkler was $$\mathbf{20~h}$$ used.

Jan 20, 2020