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Chapter 4: Problem 81

A librarian has $$\$ 45,000$$ to spend on print journals and on-line access to journals. If one-fourth as much money will be spent on print journals as on on-line access to journals, find the amount that will be spent on print journals and the amount that will be spent on on-line access to journals.

Short Answer

Expert verified
Amount spent on on-line access: \( 36{,}000 \) dollars, Amount spent on print journals: \( 9{,}000 \) dollars.

Step by step solution

01

Identify Variables

Let the amount spent on on-line access to journals be \( x \). Therefore, the amount spent on print journals will be \( \frac{x}{4} \).
02

Set Up The Equation

The total amount to be spent is \( \$45{,}000 \). The equation can be set up as: \( x + \frac{x}{4} = 45{,}000 \).
03

Combine Like Terms

To eliminate the fraction, multiply the entire equation by \( 4 \). \[ 4 \times \bigg( x + \frac{x}{4} \bigg) = 4 \times 45{,}000 \] This simplifies to: \[ 4x + x = 180{,}000 \]
04

Solve For x

Combine like terms: \[ 5x = 180{,}000 \] Now, solve for \( x \): \[ x = \frac{180{,}000}{5} = 36{,}000 \]
05

Find Amount Spent on Print Journals

The amount spent on print journals is \( \frac{x}{4} \). Substitute \( x = 36{,}000 \): \[ \frac{36{,}000}{4} = 9{,}000 \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Variable Identification
Variables are symbols that represent unknown values. In this problem, we need to identify the amount of money spent on on-line access to journals and print journals. By assigning a variable to the unknown, we can solve the problem systematically.
First, let's designate a variable to the on-line access amount. Let鈥檚 call it \( x \). Now, the problem states that the amount spent on print journals is one-fourth of the amount spent on online access.
Therefore, the amount spent on print journals can be represented as \( \frac{x}{4} \). This step is crucial because it sets the foundation for our equation.
Setting Up Equations
Setting up the equation is where we translate the word problem into mathematical form. From the problem, we know the total amount to be spent is \( \$45{,}000 \).
Since the total budget is the sum of the amounts spent on on-line access and print journals, we can write:
\[ x + \frac{x}{4} = 45{,}000 \]
This equation shows that the total expenditure is split into the two categories, perfectly matching the budget constraints given.
Solving Linear Equations
Solving the equation involves isolating the variable. To get rid of the fraction, we multiply the entire equation by 4:
\[ 4 \times \left( x + \frac{x}{4} \right) = 4 \times 45{,}000 \]
Simplifying, we get:
\[ 4x + x = 180{,}000 \]
Now the fraction is eliminated, and we sum the variables:
\[ 5x = 180{,}000 \]
Dividing both sides by 5 will isolate \( x \):
\[ x = \frac{180{,}000}{5} = 36{,}000 \]
So, the amount spent on on-line access is \( \$36{,}000 \).
Combining Like Terms
Combining like terms means adding or subtracting terms with the same variable to simplify the equation. In our case, after eliminating the fraction, we had:
\[ 4x + x = 180{,}000 \]
Here, both terms are 'like' because they contain the variable \( x \). Adding them gives:
\[ 5x = 180{,}000 \]
This makes it easier to solve the equation by isolating \( x \).
Finally, to find the amount spent on print journals, we substitute \( x = 36{,}000 \) back into \( \frac{x}{4} \):
\[ \frac{36{,}000}{4} = 9{,}000 \]
Thus, \( \$9{,}000 \) will be spent on print journals.

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