What is the cost per unit of gas.

Question

During one month, a homeowner used 400 units of electricity and 100 units of gas for a total cost of $314. The next month, 300 units of electricity and 250 units of gas were used for a total cost of $302. Find the cost per unit of gas.

Answer 1

Let's start by breaking the problem down and identifying our different variables in equations. In the problem we are given the same information for both months: the units used of each utility, and the total monthly cost. Because this information is the same, we can use a system of equations to find our cost per unit of gas, which will be constant across both months.

Let's call our cost per unit of gas X and our cost per unit of electricity Y.

If we write out the bill for each month as an equation we get:

Month 1: (100 * X) + (400 * Y) = $314

Month 2: (250 * X) + (300 * Y) = $302

When solving for a variable in a 2 equation system, pick one equation, and re-arrange it to isolate the other variable, like I do below with Month 2.

M2: 300Y = 302 - 250X [Subtract (250*X) from both sides]

M2: Y = 1.0067 - 0.8333X [Divide both sides by 300]

Now, because the cost per unit for electricity and gas are constant between months, we can take this new value for Y and insert it into the Y value in Month 1 like so to then solve for the remaining variable X.

M1: 100X + 400(1.0067 - 0.8333X) = 314 [Insert our rearranged M2 into the Y position of M1]

M1: 100X + 402.667 - 333.333X = 314 [Distribute and multiply the 400 inside the parentheses]

M1: 100X - 333.333X = -88.667 [Subtract 402.667 from both sides]

M1: 233.333X = -88.667 [Subtract 333.333X from 100X]

M1: X = 0.38 [Divide -88.667 by -233.333]

Now having solved for X we can see that our Cost Per Unit of Gas is $0.38, or 38 cents. If we want to solve for the Cost Per Unit of Electricity, we can plug this value into our modified M2 equation:

M2: Y = 1.0067 - 0.8333 * (0.38) [Insert 0.38 in our X position]

M2: Y = 1.0067 - 0.3167 [Multiply 0.8333 by 0.38]

M2: Y = 0.69 [Subtract 0.3167 from 1.0067]

We can see that our Y value is $0.69, or 69 cents. With both X and Y found we can lastly check our work by plugging both equations in to either original M1 or M2 equation. If the solved value is equal to the monthly value given, then the solutions are correct.

M1: (100 * 0.38) + (400 * 0.69) = $314

M2: (250 * 0.38) + (300 * 0.69) = $302

M1: 38 + 276 = $314

M2: 95 + 207 = $302

M1: 314 = $314

M2: 302 = $302

Answer 2

This question is about systems of equations. One way of being able to see this is that the problem gives you two different versions of the same thing. They give you two different versions of the total cost using some amount of electricity and some amount of gas. You can set up an equation for each version but you'll need variables for the cost of one unit of electricity and the cost of one unit of gas. Let's call these c and g.

400 units of electricity will cost 400 * c and 100 units of gas will cost 100 * g. Together this will add to 314 dollars.

The first equation is: 400c + 100g = 314

The second equation is: 300c + 250g = 302

Now you can find the cost per unit gas, g, that solves this system of equations. There are two ways to do this. The first is by elimination. Here, I will use the second way which is substitution.

The first step is to get c in terms of g. You can use either of the two equations to do this. Let's say I use the first equation. Solve for c by:

400c + 100g = 314

-100g -100g

400c = 314 - 100g

\400 \400

c = (314 - 100g) / 400

Next, plug c into the second equation.

300c + 250g = 302

300 ((314 - 100g) / 400) + 250g = 302

Now you have an equation where g is the only variable in it. Solve for g by:

(300/400) * (314 - 100g) + 250g = 302

(3/4) * (314 - 100g) + 250g = 302

((3/4) * 314) - ((3/4) * 100g) + 250g = 302

942/4 - (300/4) * g + 250g = 302

942/4 - 75g + 250g = 302

942/4 + 175g = 302

-942/4 -942/4

175g = 302 - 942/4

175g = 302 - 235.5 942/4 = 235.5

175g = 66.5

/175 /175

g = 0.38 dollars