Q) A mixture of 125 galllons of wine and water contains 20% of water.How much water must be added to the mixture in order to increase the percentage of the water to 25% of the new mixture?
Thanks
Hello, a69356!
When I first learned "Mixture Problems", I made a chart . . .A mixture of 125 galllons of wine and water contains 20% water.
How much water must be added to increase the percentage of the water to 25%?
The chart has the formula across the top:
. . $\displaystyle \text{[Gallons of mixture]} \times \text{[Percent water]} \:=\:\text{[Gallons of water]}$
. . $\displaystyle \begin{array}{c||ccccc}
\qquad & \text{Gallons} & \times & \text{Percent} & = & \text{Water} \\ \hline\hline
& &|& &|& \\ \hline
& &|& &|& \\ \hline \hline
& &|& &|& \end{array}$
We start with 125 gallons of mixture which is 20% water.
. . We have: .$\displaystyle 20\% \times 125 \:=\:25$ gallons of water.
Write this in the first row:
. . $\displaystyle \begin{array}{c||ccccc}
& \text{Gallons} & \times & \text{Percent} & = & \text{Water} \\ \hline\hline
\text{Start} & 125 &|& 20\% &|& 25 \\ \hline
& &|& &|& \\ \hline \hline
& &|& &|& \end{array}$
We add $\displaystyle x$ gallons of water (which is 100% water).
. . We add: .$\displaystyle 100\% \times x \:=\:x$ gallons of water.
Write this in the second row.
. . $\displaystyle \begin{array}{c||ccccc}
& \text{Gallons} & \times & \text{Percent} & = & \text{Water} \\ \hline\hline
\text{Start} & 125 &|& 20\% &|& 25 \\ \hline
\text{Add} & x &|& 100\% &|& x \\ \hline \hline
& &|& &|& \end{array}$
The final mixture has $\displaystyle x+125$ gallons which is 25% water.
. . It contains: .$\displaystyle 25\% \times (x+125) \:=\:0.25(x+125)$ gallons of water.
Write this in the third row.
. . $\displaystyle \begin{array}{c||ccccc}
& \text{Gallons} & \times & \text{Percent} & = & \text{Water} \\ \hline\hline
\text{Start} & 125 &|& 20\% &|& 25 \\ \hline
\text{Add} & x &|& 100\% &|& x \\ \hline \hline
\text{Mixture} & x+125 &|& 25\% &|& 0.25(x+125) \end{array}$
Read down the last column.
We start with 25 gallons of water, then add $\displaystyle x$ gallons of water.
. . Hence, the mixture contains $\displaystyle x+25$ gallons of water. [1]
Read across the last row.
We have $\displaystyle x+125$ gallons which is 25% water.
. . Hence, the mixture contains $\displaystyle 0.25(x+125)$ gallons of water. [2]
We just described the final amount of water in two ways.
There is our equation! $\displaystyle \hdots\quad x + 25 \:=\:0.25(x+125)$
Got it?