Problem 62 - Introductory Algebra for College Students

Question

Use the given information to write an equation. Let x represent the number described in each exercise. Then solve the equation and find the number. The difference between 3 and \(\frac{2}{7}\) of a number is \(\frac{5}{4}\) of that number. Find the number.

Step-by-Step Solution

Answer

The number is \(\frac{84}{43}\).

Step 1: Decoding the problem

From the problem, we understand that the difference between 3 and \(\frac{2}{7}\) of a some number is \(\frac{5}{4}\) of that same number. We denote the unknown number as 'x'.

Step 2: Translating into mathematical equation

Express the problem into a mathematical equation. The problem gives '..the difference between 3 and \(\frac{2}{7}\) of a number..', so this part of the sentence becomes \(3 - \frac{2}{7}x\). The sentence continues with '...is \(\frac{5}{4}\) of that number...', so this would be = \(\frac{5}{4}x\), hence getting the full equation: \(3 - \frac{2}{7}x = \frac{5}{4}x\).

Step 3: Simplifying the equation to solve for x

Firstly, let's remove the fractions to simplify the problem by multiplying the whole equation by 28 (which is the lowest common multiple of 7 and 4). \(28(3 - \frac{2}{7}x) = 28(\frac{5}{4}x)\) we get \(84 - 8x = 35x\). Then add \(8x\) to both sides, which gives: \(84 = 43x\). Finally, divide both sides by 43 to get the solution: \(x = \frac{84}{43}\).

Key Concepts

Equation Solving

Solving equations involves finding the value of unknowns, like \( x \), that satisfy a given equation. In our example, we were tasked to find \( x \). The problem was expressed using the difference and relationship between fractions, leading to an equation. It is common in algebra to convert word problems into algebraic equations. This involves translating words, such as "difference" or "of," into mathematical symbols. Once you have a clear equation, you can use relationships, like addition or subtraction, to isolate \( x \).
Isolating the variable often involves getting all the terms involving \( x \) on one side of the equation and constants on the other. Perform operations in a sequence:

• Addition/Subtraction.
• Multiplication/Division.

This ordered approach makes solving for \( x \) systematic and reliable.

Fractions

Fractions are numbers expressed as the ratio of two integers. Understanding fractions is crucial in solving equations like ours. The given problem involves fractions, specifically \( \frac{2}{7} \) and \( \frac{5}{4} \). Fractions compare parts of a whole and are essential for representing division cases in algebra.
Operations with fractions often include finding a common denominator, especially when they appear in equations. In our example, by multiplying the whole equation by 28, we effectively removed the fractions: a technique known as clearing fractions.

• Multiplication and division can simplify the handling of fractions.
• Always convert fractions to a common base when adding or subtracting them.

Clearing fractions early in the equation process simplifies solving and helps make equations easier to understand.

Linear Equations

Linear equations are equations of the first degree, meaning they have the form \( ax + b = c \), where \( a \), \( b \), and \( c \) are constants. These equations graph to straight lines and involve variables with no exponents or roots. In our problem, we transformed a word problem into the linear equation \( 3 - \frac{2}{7}x = \frac{5}{4}x \).

Linear equations are straightforward to solve once simplified, usually through a few basic steps: moving terms and constant consolidation. After simplifying complex fractions, our example boiled down to the linear form \( 84 = 43x \). This form makes it easy to determine \( x \) by dividing both sides by the coefficient (43 in this case). Linear equations like this are foundational in algebra and are encountered in many contexts, making understanding them essential.