Problem 86 - 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 30 is subtracted from seven-eighths of a number, the result is equal to one-half of the number. What is the number?

Step-by-Step Solution

Answer

\(x = 80\). The number is 80.

Step 1: Translate the Statement into Equation

Translate the given statement into an algebraic equation. Let \(x\) be the unknown number. The statement 'When 30 is subtracted from seven-eighths of a number, the result is equal to one-half of the number.' can be translated into '\(\frac{7}{8}x - 30 = \frac{1}{2}x\)'.

Step 2: Simplify the Equation

Start by trying to get rid of the fractions by multiplying the entire equation by 8, the least common denominator, this gives '7x - 240 = 4x'. Then, subtract 4x from both sides of the equation to form a new equation '3x - 240 = 0'.

Step 3: Solve for x

Add 240 to both sides of the equation which gives '3x = 240'. Then, divide both sides by 3 to solve for \(x\). Therefore, \(x = 240 / 3 = 80\).