How many of each $45 per ton and $50 per ton should be added?

Question

A 24-ton mixture of crushed rock is needed in a construction job. It will cost $1200. If the mixture is composed of rocks costing $45 per ton and $50 per ton, how many of each should be added? Solve using 2 equations with two variables.

Answer 1

This is a system of equations problem. First, we want to think about our unknowns to define our variables. We want to know how many tons of each kind of rock should be added. So let's

1. Let x= rocks costing $45 per ton and let y= rocks costing $50 per ton .

Now, we want to set up a system of equations to represent the information in the word problem using the variables we have defined (x & y).

We know that a 24-ton mixture is needed, so together, the rocks will add to 24 tons. Therefore,

1. 24 = x + y

We also know that the cost totals $1,200 and that x costs $45 per ton and y costs $50 per ton. Therefore,

1. $45x + $50y = $1200.

When you have two equations with two variables, you should solve the system using either elimination, equal values or the substitution method. Let's use the substitution method here. Since neither x or y have coefficients in the first equation, it's easy to isolate them. Let's get y by itself by subtracting x on both sides:

24 - x = y

Now y is really (24 - x), so I can substitute its value into the other equation:

$45x + $50 (24 - x) = $1200

Now I only have one variable, x, and I can solve by distributing and then isolating x:

$45x + $50 * 24 - $50x = $1200 Next, combine like terms

-$5x +$1200 = $1200 Now isolate the x term

-$5x = 0 so x = 0

Now that we have solved for x, we want to plug this value back into either of our original equations to solve for y.

24 = (0) + y so y = 24

Looking back at how we defined our variables, we can now answer the question in a complete sentence.

24 tons of $50/ton rocks and no $45/ton rocks should be added in order to spend $1200 on a 24 ton mixture of rocks.