Problem 92 - Elementary Algebra

Question

For the following problems, find the solution. When the same number is added to both terms of the fraction \(\frac{13}{15}\), the result is \(\frac{8}{9}\). What is the number?

Step-by-Step Solution

Answer

Answer: 3

Step 1: Set up the equation

Let's denote the unknown number we're looking for as \(x\). We can set up the following equation to represent the problem: \(\frac{13 + x}{15 + x} = \frac{8}{9}.\)

Step 2: Cross multiply

To solve for \(x\), first cross multiply: \((13 + x) \cdot 9 = 8 \cdot (15 + x)\).

Step 3: Distribute and simplify the equation

Now distribute the numbers on both sides of the equation: \(117 + 9x = 120 + 8x\).

Step 4: Solve for x

Subtract \(8x\) from both sides of the equation to isolate the term with \(x\) on the left side: \(117 + x = 120.\) Then, subtract 117 from both sides to solve for \(x\): \(x = 3\). The number we're looking for is 3, so when we add 3 to both terms of the fraction \(\frac{13}{15}\), we get the result \(\frac{8}{9}\).

Key Concepts

Cross Multiplication

Cross multiplication is a vital technique used for solving equations involving fractions. When two fractions are set equal to each other, cross multiplication involves multiplying the numerator of the first fraction by the denominator of the second and vice versa, then setting the products equal to each other. This method effectively removes the fractions, simplifying the process of finding the variable.

For example, consider the equation \( \frac{a}{b} = \frac{c}{d} \). By cross multiplying, you get \( ad = bc \), which is an equation without fractions and is easier to solve. In our textbook problem, cross multiplication was used to turn the equation \( \frac{13 + x}{15 + x} = \frac{8}{9} \) into \( (13 + x) \cdot 9 = 8 \cdot (15 + x) \), a simple linear equation.

Equation Solving

Equation solving is the process of finding the value of the variable that makes the equation true. The main goal is to isolate the variable on one side of the equation, ultimately determining its value. To do so, we perform operations that simplify and reorganize the equation, maintaining balance by doing the same to both sides.

Common steps include distributing multiplication over addition, combining like terms, and using addition or subtraction to get the variable with its coefficient by itself. In our original exercise, after cross-multiplying, we distributed and combined like terms to get \( 117 + 9x = 120 + 8x \), then we isolated the variable \( x \) by subtracting \( 8x \) and \( 117 \) from both sides, leading to \( x = 3 \) as the final solution.

Algebraic Fractions

Algebraic fractions are fractions that include variables in their numerators, denominators, or both. They are manipulated using the same principles applied to numerical fractions, but special care must be taken when dealing with variables, particularly to ensure that we do not divide by zero.

When solving equations with algebraic fractions, one common approach is to find a common denominator in order to combine terms or to use cross multiplication, as we did in our example. It's important to remember that with algebraic fractions, finding the lowest common denominator or cross multiplying simplifies the process by eliminating the fractions and turning the equation into a more manageable format.

Improving Understanding

Students should practice recognizing equivalent algebraic fractions and simplifying complex fractions to improve their overall command of the concept.