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:
\[4(12)+7(x)=6(x+12)\]\[48+7x=6x+72\]\[x=24\]
, \(24\) 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:
\[48+7(x)=69x+12)\]\[48+7x=6x+72\]\[x=24\]
Notice the solution is \(24\).
3) Using a Balance :
http://www.gardenista.com/products/teeter-totter
\(12\) \(12+x\) \(x\) weight
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\($4\) \($6\) \($7\) cost per lb
Set up a proportion using the weight in pounds and difference between price per pound.
\[($6 – $4 =2) \ and \ ($7 – $6 = 1)\]
\[\frac{12}{1}=\frac{x}{2}\]
Solve the proportion:
\(x=24\)