Question

the table below shows the number of times the dishwasher was used in three different houses over three days
dishwasher use
house a house b house c
friday 1 2 2
saturday 2 3 4
sunday 2 2 3
on friday, 17.5 total gallons of water were used. on saturday, 31.5 total gallons were used. on sunday, 24.75 total gallons were used. what is the difference between the water use per load of the dishwasher that uses the most wa and the dishwasher that uses the least water?
0.5 gallon
0.75 gallon
3 gallons
4 gallons

Explanation:

Step1: Calculate total loads per day

Friday: \(1 + 2+2=5\) loads.
Saturday: \(2 + 3+4 = 9\) loads.
Sunday: \(2+2 + 3=7\) loads.

Step2: Calculate water - use per load per day

Friday: \(\frac{17.5}{5}=3.5\) gallons per load.
Saturday: \(\frac{31.5}{9}=3.5\) gallons per load.
Sunday: \(\frac{24.75}{7}=3.5357\cdots\) (approx \(3.54\)) gallons per load (House A: \(2\) loads, House B: \(2\) loads, House C: \(3\) loads).
For House A: Total loads \(=1 + 2+2=5\), total water \(17.5\times\frac{1}{5}+31.5\times\frac{2}{9}+24.75\times\frac{2}{7}\)
\(=3.5 + 7+7.0714\approx17.57\), per - load \(=\frac{17.57}{5}\approx3.51\)
For House B: Total loads \(=2 + 3+2=7\), total water \(17.5\times\frac{2}{5}+31.5\times\frac{3}{9}+24.75\times\frac{2}{7}\)
\(=7 + 10.5+7.0714\approx24.57\), per - load \(=\frac{24.57}{7}\approx3.51\)
For House C: Total loads \(=2 + 4+3=9\), total water \(17.5\times\frac{2}{5}+31.5\times\frac{4}{9}+24.75\times\frac{3}{7}\)
\(=7+14 + 10.6071\approx31.61\), per - load \(=\frac{31.61}{9}\approx3.51\) (Wait, wrong approach.
Alternative:
Let \(x\), \(y\), \(z\) be water per load for House A, B, C.
We have \(

$$\begin{cases}x + 2y+2z=17.5\\2x + 3y+4z=31.5\\2x+2y + 3z=24.75\end{cases}$$

\)
From the first equation \(x=17.5-(2y + 2z)\)
Substitute into the second: \(2(17.5-(2y + 2z))+3y + 4z=31.5\)
\(35-4y-4z+3y + 4z=31.5\), \(y = 3.5\)
Substitute \(y = 3.5\) into the first: \(x+2\times3.5+2z=17.5\), \(x+2z=10.5\)
Substitute \(y = 3.5\) into the third: \(2x+2\times3.5+3z=24.75\), \(2x+3z=17.75\)
Multiply \(x + 2z=10.5\) by \(2\): \(2x+4z=21\)
\((2x + 4z)-(2x+3z)=21 - 17.75\), \(z = 3.25\), then \(x = 10.5-2\times3.25 = 4\)

Step3: Find the difference

The most water per load is \(x = 4\) (House A), the least is \(z = 3.25\) (House C).
Difference \(=4 - 3.5=0.5\)

Answer:

0.5 gallon