Here are some different ways to think about the solution to the problem:
How many pounds of coffee worth $7 per pound must be mixed with 12 pounds of coffee worth $4 per pound to make a mixture worth $6 per pound?
1. Creating an Equation:
Let \(x\) represent the number of pounds of the $7 per pound coffee. There are 12 pounds of the $4 per pound coffee and \(x\) pounds of $7 per pound mixture means that the total is \(x+12\)pounds.
Knowing the cost for each type of coffee gives us this equation to solve:
\[\begin{align}4(12)+7(x)&=6(x+12)\\\\48+7x&=6x+72\\\\x&=24\end{align}\]
, \(24\) pounds of the \($7\) per pound coffee is needed.
2. Creating a Table:
| Number of Pounds | Cost Per Pound | Total Cost | |
|---|---|---|---|
| Cheaper Coffee | \(12\) | \($4\) | \($48\) |
| Expensive Coffee | \(x\) | \($7\) | \($7x\) |
| Combined Coffee | \(12+x\) | \($6\) | \($6(12+x)\) |
Use the Total Cost Column to create this equation to solve:
\[\begin{align}48+7(x)&=69x+12)\\\\48+7x&=6x+72\\\\x&=24\end{align}\]
Notice the solution is \(24\).
3. Using a Balance :
http://www.gardenista.com/products/teeter-totter
\(12\) \(12+x\) \(x\) weight
________________________
\($4\) \($6\) \($7\) cost per lb
Set up a proportion using the weight in pounds and difference between price per pound.
\[($6–$4 =2) \text{ and } ($7–$6 = 1)\]
\[\dfrac{12}{1}=\dfrac{x}{2}\]
Solve the proportion:
\(x=24\)
Return to Explore to complete the problem set.