Problem 59 - Understanding Elementary Algebra with Geometry

Question

Solve each of the following problems algebraically. Be sure to label what the variable represents. Marina does one third of her homework problems when she comes home, 8 problems just before supper, and the remaining two fifths of the problems after supper. How many homework problems did she have?

Step-by-Step Solution

Answer

Marina had 30 homework problems.

Step 1 - Define the variable

Let the total number of homework problems be represented by the variable \( x \).

Step 2 - Set up the equation

According to the problem, Marina completes her homework in three parts:1. One third when she comes home: \( \frac{1}{3} x \)2. Eight problems before supper3. Two fifths of the total after supper: \( \frac{2}{5} x \)Set up the equation to add these parts together: \[ \frac{1}{3} x + 8 + \frac{2}{5} x = x \]

Step 3 - Combine like terms

To simplify the equation, combine the terms involving \( x \): \[ \frac{1}{3} x + \frac{2}{5} x = x - 8 \]

Step 4 - Find a common denominator

To combine the fractions, find a common denominator (which is 15): \[ \frac{5}{15} x + \frac{6}{15} x = \frac{11}{15} x \] Thus, the equation now reads: \[ \frac{11}{15} x + 8 = x \]

Step 5 - Solve for x

Subtract \( \frac{11}{15} x \) from both sides to isolate the term involving \( x \): \[ 8 = x - \frac{11}{15} x \] \[ 8 = \frac{15}{15} x - \frac{11}{15} x \] \[ 8 = \frac{4}{15} x \] To solve for \( x \), multiply both sides by \( \frac{15}{4} \): \[ x = 8 \cdot \frac{15}{4} \] \[ x = 30 \]

Key Concepts

Algebraic Equations

Algebraic equations are mathematical statements that show the equality between two expressions. These equations are used to find unknown values, represented by variables like x. In algebra word problems, you convert the situation described in words into an algebraic equation. For example, in a homework problem, if Marina completes her homework in parts, we set up an algebraic equation to represent this. The main idea is to translate the text into mathematical expressions and solve for the unknown variable step by step.

Solving for Variables

When solving for variables, you aim to find the value of the unknowns. Let's say we have the equation: \[\frac{1}{3} x + 8 + \frac{2}{5} x = x\]. The variable here is x, which represents the total number of homework problems Marina needs to complete.

• Step 1: Combine like terms: \[\frac{1}{3} x + \frac{2}{5} x = x - 8\]
• Step 2: Find a common denominator to simplify the fractions: \[\frac{5}{15} x + \frac{6}{15} x = \frac{11}{15} x\]
• Step 3: Subtract to isolate the term with x: \[8 = \frac{4}{15} x\]
• Step 4: Solve for x by multiplying both sides by \(\frac{15}{4}\): \[x = 8 \cdot \frac{15}{4} = 30\]

Breaking down each step helps in simplifying the equation, ensuring you systematically solve for the variable.

Fractions in Algebra

Dealing with fractions can seem tricky, but it's manageable with practice. In our problem, Marina's homework completion is expressed in fractions: like one third (\(\frac{1}{3}\)) and two fifths (\(\frac{2}{5}\)). To work with these fractions in algebraic equations:

• Step 1: Express each part of the word problem as a fraction of the total.
• Step 2: Find a common denominator to combine fractions.
• Step 3: Simplify fractions when adding or subtracting them.

In our example, combining \(\frac{1}{3}x\) and \(\frac{2}{5}x\) translates to using a common denominator of 15: \(\frac{5}{15}x + \frac{6}{15}x\). The sum \(\frac{11}{15}x\) is easier to manage and helps you solve the equation systematically. Remember, the objective is to simplify while preserving the equation's balance.