Solving Equations Involving Fractions and Operations with Numbers

When dealing with problems that involve fractions and operations with numbers, such as adding or subtracting fractions to or from a part of a number, the approach often involves setting up and solving algebraic equations. This guide walks you through the process of solving such equations and provides solutions for three different scenarios.

Problem 1: Solving for the Number When Given Operations with Fractions

Problem Statement

The problem states: '14 is added to 2/3 of a number. The result is 1 whole number 1/4 times the original number. What is the number?'

Solution Steps

Let the unknown number be x. The given equation is: (frac{2}{3}x 14 frac{5}{4}x). Convert (frac{5}{4}) to an improper fraction, which is already done. Eliminate fractions by multiplying the entire equation by the least common multiple (LCM) of 3 and 4, which is 12: (12 times left( frac{2}{3}x 14 right) 12 times frac{5}{4}x) This simplifies to: (8x 168 15x) Solve for (x): (168 15x - 8x) (168 7x) (x frac{168}{7} 24) The number is (boxed{24}).

Problem 2: Solving for the Number with a Replaced Fraction

In another situation, consider the problem: '2/3 of a number increased by 14 is equal to the whole number 1/4 of the number.' The problem can be represented as:

Let the unknown number be (u). The equation becomes: (frac{2}{3}u 14 frac{1}{4}u). Subtract (frac{1}{4}u) from both sides: (14 frac{1}{4}u - frac{2}{3}u) Rearrange and combine like terms: (14 frac{3}{12}u - frac{8}{12}u) (14 -frac{5}{12}u) Multiply both sides by (-frac{12}{5}) to solve for (u): (u 14 times -frac{12}{5} -frac{168}{5} -33frac{3}{5}) The number is (boxed{-33.6}).

Problem 3: Solving for the Unknown with Another Interpretation

Another approach involves interpreting the problem with a different equation: '14 added to 2/3 of a number equals 1/4 of the number.' This can be written as:

Let the unknown number be (u). The equation is: (frac{2}{3}u 14 frac{1}{4}u). Subtract (frac{1}{4}u) from both sides: (14 frac{1}{4}u - frac{2}{3}u) Convert and combine like terms: (14 frac{3}{12}u - frac{8}{12}u) (14 -frac{5}{12}u) Multiply both sides by (-frac{12}{5}) to solve for (u): (u 14 times -frac{12}{5} -frac{168}{5} -33frac{3}{5}) The number is (boxed{-33.6}).

These solutions demonstrate the step-by-step process of solving equations involving fractions and operations with numbers. Understanding and practicing these techniques can help in solving more complex algebraic problems efficiently.