Practice Solving Mixture Word Problems
buttons.Solve Mixture Word Problems
Now we'll solve some more general applications of the mixture model. Grocers and bartenders use the mixture model to set a fair price for a product made from mixing two or more ingredients. Financial planners use the mixture model when they invest money in a variety of accounts and want to find the overall interest rate. Landscape designers use the mixture model when they have an assortment of plants and a fixed budget, and event coordinators do the same when choosing appetizers and entrees for a banquet.
Our first mixture word problem will be making trail mix from raisins and nuts.
Example 3.32
Henning
is mixing raisins and nuts to make 10 pounds of trail mix. Raisins cost
$2 a pound and nuts cost $6 a pound. If Henning wants his cost for the
trail mix to be $5.20 a pound, how many pounds of raisins and how many
pounds of nuts should he use?
Solution
As before, we fill in a chart to organize our information.
The 10 pounds of trail mix will come from mixing raisins and nuts.
\(\text{Let x= number of pounds of raisins.} \)
\(\text{10-x= number of pounds of nuts} \)
We enter the price per pound for each item.
We multiply the number times the value to get the total value.
|
Type |
Number of pounds($) \(\cdot\) Price per pound ($) = Total Value |
||
|
Raisins |
\(X\) |
\(2\) |
\(2x\) |
|
Nuts |
\(10-x\) |
\(6\) |
\(6(10-x)\) |
|
Trail mix |
\(10\) |
\(5.20\) |
\(10(5.20)\) |
Notice that the last line in the table gives the information for the total amount of the mixture.
We know the value of the raisins plus the value of the nuts will be the value of the trail mix.
| Write the equation from the total values. | \(2 x+6(10-x)=10(5.20)\) |
| Solve the equation. | \(2 x+60-6 x=52\) |
| \(-4 x=-8\) | |
| \(x=2\) pounds of raisins | |
| Find the number of pounds of nuts. | \(10-x\) |
| \(10-2\) | |
| 8 pounds of nuts | |
| Check. \(\begin{aligned} 2(\$ 2)+8(\$ 6) & \stackrel{?}{=} 10(\$ 5.20) \\ \$ 4+\$ 48 & \stackrel{?}{=} \$ 52 \\ \$ 52 &=\$ 52 \text{✓} \end{aligned}\) |
|
| Henning mixed two pounds of raisins with eight pounds of nuts. | |
Try It 3.63
Orlando is mixing nuts and cereal squares to make a party mix. Nuts sell for $7 a pound and cereal squares sell for $4 a pound. Orlando wants to make 30 pounds of party mix at a cost of $6.50 a pound, how many pounds of nuts and how many pounds of cereal squares should he use?
Try It 3.64
Becca wants to mix fruit juice and soda to make a punch. She can buy fruit juice for $3 a gallon and soda for $4 a gallon. If she wants to make 28 gallons of punch at a cost of $3.25 a gallon, how many gallons of fruit juice and how many gallons of soda should she buy?
We can also use the mixture model to solve investment problems using simple interest. We have used the simple interest formula, \(I=Prt\), where \(t\) represented the number of years. When we just need to find the interest for one year, \(t=1\), so then \(I=Pr\).
Example 3.33
Stacey has $20,000 to invest in two different bank accounts. One account pays interest at 3% per year and the other account pays interest at 5% per year. How much should she invest in each account if she wants to earn 4.5% interest per year on the total amount?
Solution
We will fill in a chart to organize our information. We will use the simple interest formula to find the interest earned in the different accounts.
The interest on the mixed investment will come from adding the interest from the account earning 3% and the interest from the account earning 5% to get the total interest on the $20,000.
Let \(x = \text{amount invested at 3%}\).
\(20,000−x = \text{amount invested at 5%}\)
The amount invested is the principal for each account.
We enter the interest rate for each account.
We multiply the amount invested times the rate to get the interest.
| Type | \(\text{ Amount Invested · Rate = Interest}\) |
||
|---|---|---|---|
| 3% |
\(x\) | 0.03 | \(0.03x\) |
| 5% | \(20,000-x\) | 0.05 | \(0.05(20,000-x)\) |
| 4.5% | \(20,000\) | 0.045 | \(0.45(20,000)\) |
Notice that the total amount invested, 20,000, is the sum of the amount invested at 3% and the amount invested at 5%. And the total interest, 0.045(20,000), is the sum of the interest earned in the 3% account and the interest earned in the 5% account.
As with the other mixture applications, the last column in the table gives us the equation to solve.
| Write the equation from the interest earned. Solve the equation. |
\(0.03x+0.05(20,000−x) = 0.045(20,000)\) \(0.03x+1,000−0.05x = 900\) \(−0.02x+1,000= 900\) \(−0.02 x= −100\) \(x=5,000\) amount invested at 3% |
| Find the amount invested at 5% | |
| Check. \(0.03x+0.05(15,000+x) \stackrel{?}{=} 0.045(20,000)\) \(150+750 \stackrel{?}{=} 900\) \(900=900\) ✓ |
|
| Stacey should invest $5,000 in the account that earns 3% and $15,000 in the account that earns 5%. | |
Try It 3.65
Remy has $14,000 to invest in two mutual funds. One fund pays interest at 4% per year and the other fund pays interest at 7% per year. How much should she invest in each fund if she wants to earn 6.1% interest on the total amount?
Try It 3.66
Marco has $8,000 to save for his daughter's college education. He wants to divide it between one account that pays 3.2% interest per year and another account that pays 8% interest per year. How much should he invest in each account if he wants the interest on the total investment to be 6.5%?