Problem 84 - 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. When two-fifths of a number is added to one-fourth of the number, the sum is 13. What is the number?

Step-by-Step Solution

Answer

The number is 20.

Step 1: Translate the problem into an equation

The number is represented by \(x\). The problem tells us 'two-fifths of a number (\(2/5\cdot x\)) is added to one-fourth of the number (\(1/4\cdot x\))', and the result is 13. Writing these information leads to this equation: \(2/5\cdot x + 1/4\cdot x = 13\).

Step 2: Solve the equation

To solve the equation, first add the fractions on the left-hand side by taking Least Common Multiple (LCM) of 4 and 5, which is 20. So, \(8/20\cdot x + 5/20\cdot x = 13\), simplifying farther to \(13/20\cdot x = 13\). Next, to isolate \(x\), divide both sides of the equation by \(13/20\). Thus, \(x = 13 / (13/20) = 20\).

Step 3: Verify the solution

Substitute \(x = 20\) back into the original equation to verify it. \(2/5*20 + 1/4*20 = 8 + 5 = 13\), which confirms that the solution is correct.