# Thread: Systems of Equations word problem

1. ## Systems of Equations word problem

Hello,

Will anyone help me set these two problems up? tHanks so muchhhh!!!

1. John can paint a house in 30 hours and Jerald can paint a house in 5 hours. How long would it take the two of them to paint a house together?

2. In mixing some fuel, a scientist combines a 50% ethanol sotluion with a 90% ethanol solution to get 40 liters 80% ethanol solution. How much of the 50% solution did the scientist use in the mixture?

2. Originally Posted by jenjen
Hello,

Will anyone help me set these two problems up? tHanks so muchhhh!!!

1. John can paint a house in 30 hours and Jerald can paint a house in 5 hours. How long would it take the two of them to paint a house together?

2. In mixing some fuel, a scientist combines a 50% ethanol sotluion with a 90% ethanol solution to get 40 liters 80% ethanol solution. How much of the 50% solution did the scientist use in the mixture?
1.
In 1 hour, John and Jerald can paint $\displaystyle \frac{1}{30}$ and $\displaystyle \frac{1}{5}$ of the house, respectively.
If they do it together for 1 hour, they can paint $\displaystyle \frac{1}{30} + \frac{1}{5}$ of the house, or $\displaystyle \frac{7}{30}$
Let H be the number of hours for them working together:
$\displaystyle \frac{7}{30}$ x H = 1
where 1 represent 100% of the work (painting the house)
Therefore, H = $\displaystyle \frac{30}{7}$ hours or $\displaystyle 4\frac{2}{7}$ hours

3. Originally Posted by jenjen
2. In mixing some fuel, a scientist combines a 50% ethanol sotluion with a 90% ethanol solution to get 40 liters 80% ethanol solution. How much of the 50% solution did the scientist use in the mixture?
Let x and y be volumes (L) of 50% and 90% solutions used, respectively.
x + y = 40 ...(1)
0.5x + 0.9y = 0.8(x + y)
5x + 9y = 8x + 8y
y = 3x ...(2)

(2) into (1) you have 4x = 40, hence x = 10 (litres)
and y = 30 litres

4. To learn how to set these up for yourself (that is, to learn the reasoning by which they obtained the hand-in solutions they provided), try this lesson on "work" problems and this lesson on "mixture" problems.