Solve Mixture Applications with Systems of Equations: example 5.46 · Physics Playground

Example 5.46 from Elementary Algebra, 5.5 Solve Mixture Applications with Systems of Equations

Solve Mixture Applications with Systems of Equations: example 5.46

Translate to a system of equations and solve:

Priam has a collection of nickels and quarters, with a total value of $7.30. The number of nickels is six less than three times the number of quarters. How many nickels and how many quarters does he have?

Work it out on paper first. Then open the solution one step at a time, and stop as soon as you can finish on your own.

Your turn

The book's Try It right after this example: same idea, new numbers. Only the answer is given.

Translate to a system of equations and solve:

Matilda has a handful of quarters and dimes, with a total value of $8.55. The number of quarters is 3 more than twice the number of dimes. How many dimes and how many quarters does she have?

How did it go?

Step 1. Read the problem.	We will create a table to organize the information.
Step 2. Identify what we are looking for.	We are looking for the number of nickels and the number of quarters.
Step 3. Name what we are looking for.	Let $n =$ the number of nickels. $q =$ the number of quarters
A table will help us organize the data. We have two types of coins, nickels and quarters.	Write n and q for the number of each type of coin.
Fill in the Value column with the value of each type of coin.	The value of each nickel is $0.05. The value of each quarter is $0.25.
The number times the value gives the total value, so, the total value of the nickels is n (0.05) = 0.05n and the total value of quarters is q(0.25) = 0.25q. Altogether the total value of the coins is $7.30.
Step 4. Translate into a system of equations.
The Total value column gives one equation.
We also know the number of nickels is six less than three times the number of quarters. Translate to get the second equation.
Now we have the system to solve.
Step 5. Solve the system of equations We will use the substitution method. Substitute n = 3q − 6 into the first equation. Simplify and solve for q.





To find the number of nickels, substitute q = 19 into the second equation.


Step 6. Check the answer in the problem. $\begin{matrix} 19 quarters at $ 0.25 & = & $ 4.75 \\ 51 nickels at $ 0.05 & = & $ 2.55 \\ Total & = & $ 7.30 ✓ \\ 3 \cdot 19 - 16 & = & 51 ✓ \end{matrix}$
Step 7. Answer the question.	Priam has 19 quarters and 51 nickels.

Solution

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