Problem 10 - Elementary Algebra

Question

When the same number is added to both the numerator and denominator of the fraction \(\frac{7}{9},\) the result is \(\frac{2}{3}\). What is the number?

Step-by-Step Solution

Answer

Answer: -3

Step 1: Define the variables

Let \(x\) be the number we need to find. When \(x\) is added to both the numerator and denominator of \(\frac{7}{9}\), we get \(\frac{2}{3}\). Our equation becomes: $$\frac{7+x}{9+x} = \frac{2}{3}$$

Step 2: Cross-multiply

To solve the equation, cross-multiply the fractions: $$(7+x)\cdot 3 = (9+x)\cdot 2$$

Step 3: Expand and simplify the equation

Distribute on both sides of the equation: $$21+3x = 18+2x$$ Now, simplify by moving the terms with variable \(x\) to one side of the equation and keeping the constants on the other side. $$(3x-2x) = (18-21)$$

Step 4: Solve for x

Continue to solve the equation: $$x=-3$$ Therefore, the number we are looking for is \(-3\). When \(-3\) is added to both the numerator and denominator of \(\frac{7}{9}\), the result is \(\frac{2}{3}\).

Key Concepts

Cross Multiplication

Cross-multiplication is a valuable technique when solving equations that involve two fractions. This method effectively eliminates the fractions by multiplying across the equal sign.
Here's how it works: Take two fractions set equal to each other, for example, \( \frac{a}{b} = \frac{c}{d} \). Cross-multiplication involves multiplying the numerator of the first fraction by the denominator of the second fraction and vice versa.
This gives us the equation:

• \( a \times d = b \times c \)

In our exercise, we used cross-multiplication on \( \frac{7+x}{9+x} = \frac{2}{3} \). By cross-multiplying, we transformed this equation into \( (7+x) \times 3 = (9+x) \times 2 \).
This step is crucial because it allows us to eliminate the fractions quickly, making it simpler to solve the rest of the equation.

Numerator and Denominator

Understanding numerators and denominators is key when working with fractions. The

• numerator

is the top part of a fraction, and it indicates the number of parts we have. For example, in the fraction \( \frac{7}{9} \), 7 is the numerator, representing the number of parts we consider.

• denominator

is the bottom part of a fraction, indicating the total number of equal parts the whole is divided into. In this case, 9 is the denominator, implying the whole is divided into nine equal parts.
Within the context of the problem, we adjusted both the numerator and the denominator: we added the unknown number \( x \) to each. This required understanding this relationship, which ensures that the resulting fraction correctly reflects the problem conditions.

Variable Definition

Setting up a variable is usually the first and one of the most important steps in solving algebraic problems. A variable is a symbol, often a letter like \( x \), that represents an unknown value we need to find.
In our exercise, we defined \( x \) as the number that, when added to both the numerator and denominator of the fraction \( \frac{7}{9} \), results in \( \frac{2}{3} \).
By introducing this variable, we create a bridge from the word problem to a solvable equation. Defining the right variable at the start allows us to model the problem mathematically, which is a critical step in finding the solution. This approach not only organizes the information in a structured way but also helps streamline the entire process of solving the problem.