Noah M. answered • 12/08/21
Licensed Mathematics Teacher
This is a two-part question. So let's focus on the first part: "What is the price per liter of each kind of gas?" There are two types of gas in this problem: unleaded gas and supreme gas. Since we don't currently know the price for the two gases, I'm going to assign a variable to each one: U = price of unleaded, S = price of supreme. Now let's see if we can rewrite the information we're given into equations:
- "Suppose that 2 liters of unleaded gas and 3 liters of supreme gas cost a total of $2.52."
- 2•U + 3•S = 2.52
- "Five liters of unleaded gas and 4 liters of supreme gas cost a total of $4.48"
- 5•U + 4•S = 4.48
Now we have two equations and two unknown variables. There are many methods you can use to solve this. I'm going to use substitution:
- 2•U + 3•S = 2.52 → 2•U = 2.52 - 3•S → U = 1.26 - 3/2•S
- 5•U + 4•S = 4.48 → 5•(1.26 - 3/2•S) + 4•S = 4.48 → (6.30 - 15/2•S) + 4•S = 4.48 → 6.30 - 15/2•S + 8/2•S = 4.48 → 6.30 - 7/2•S = 4.48 → -7/2•S = -1.82 → S = 0.52
- U = 1.26 - 3/2•S → U = 1.26 - 3/2•(0.52) → U = 1.26 - (0.78) → U = 0.48
Now we have found the value for our (previously) unknown variables. Remember those variables represented the price of the two gases. So the answer to the first part is Unleaded gas = $0.48 per liter, Supreme gas = $0.52 per liter.
On to the second part: "If Ted filled his pickup with 30 liters of unleaded gas and also filled his car with 25 liters of supreme gas, how much would he have to pay for the gas? Assume prices have not changed." Once again, I'm going to make an equation out of this sentence using the same variables as before:
- 30•U + 25•S = ?
Since the prices have not changed, we can use our answer from the first part U = 0.48, S = 0.52. If we plug those numbers into the equation it should tell us how much Ted has to pay for the gas:
- 30•U + 25•S = 30•(0.48) + 25•(0.52) = (14.40) + (13.00) = 27.40
And there we go, problem solved. Ted has to pay $27.40 for the gas.
