Linear Equation Word Problem Miles per Gallon

Question

Here's a specific question I could not answer.

"On a trip, Jennifer noticed that her car averaged 21 mi/gal of gas except for the days she used the air conditioning, and then it averaged only 17 mi/gal. If she used 91 gal of gas to drive 1751 mi, on how many of those miles did she use the air conditioning?"

I tried many things from using some methods from age problem and work problem but to no avail. I tried to make a table of it to understand but it couldn't just make that kind of click in my mind. Still I couldn't make an equation out of it. How should I solve it? Thanks.

Answer 1

Hint:

She has used a total of $91$ gallons of gas. Take $x$ as the gallons of gas used with air conditioning and it follows that she used $91-x$ gallons without it. We know that: $$\text{Gallons used }\times \text{ Speed} = \text{Total Distance} $$ $$\implies 17x + 21(91-x) = 1751$$

Answer 2

We can use the assumption method. Assume she used air conditioning the whole time. Then she would have done $91\times17=1547$mi. Since she did $1751$mi, she spent some time without air conditioning, and that time is given by $\frac{1751-1547}{21-17}=51$. She spent $51$gal without air conditioning, meaning she spend $40$gal with air conditioning, which is $680$mi.

Side note: How is it that using an air conditioner use less gallons?