Problem 13 - Elementary and Intermediate Algebra

Question

Solve each of these number problems. See Example \(1 .\) If the same number is added to both the numerator and the denominator of \(\frac{2}{5},\) the result is \(\frac{2}{3} .\) Find the number.

Step-by-Step Solution

Answer

The number is 4.

Step 1: Introduce Variables

Let the original numerator be 2 and the original denominator be 5. Let the number to be added to both be \( x \). Thus, the new fraction will be \( \frac{2+x}{5+x} \).

Step 2: Set Up the Equation

According to the problem, the new fraction \( \frac{2+x}{5+x} \) should equal \( \frac{2}{3} \). Therefore, set up the equation as follows: \( \frac{2+x}{5+x} = \frac{2}{3} \).

Step 3: Cross-Multiply

To eliminate the fractions, cross-multiply: \( 3(2 + x) = 2(5 + x) \). This results in \( 6 + 3x = 10 + 2x \).

Step 4: Solve for x

Isolate \( x \) on one side of the equation: \( 3x - 2x = 10 - 6 \). Simplifying this gives \( x = 4 \).

Step 5: Verify the Solution

Substitute \( x = 4 \) back into the original equation to verify: The new fraction becomes \( \frac{2+4}{5+4} = \frac{6}{9} = \frac{2}{3} \), confirming that the solution is correct.

Key Concepts

Understanding Fractions

Fractions are a way to represent parts of a whole. They consist of two numbers separated by a slash. - The top number is called the numerator. It represents how many parts we have.- The bottom number is the denominator. It tells us how many equal parts the whole is divided into.In the fraction \(\frac{2}{5}\), 2 is the numerator and 5 is the denominator. This means that we have 2 out of 5 equal parts. Using and manipulating fractions is a fundamental skill in algebra, which helps in solving various problems involving parts of a whole. When you add the same number to both the numerator and the denominator, it changes the fraction but keeps a relationship that can often be analyzed through equations. This manipulation can help solve problems, such as adjusting recipes or comparing ratios.

Equations in Algebra

An equation is like a balance scale, where both sides must be equal. Equations use algebraic symbols to represent relationships between different quantities. An equation can have variables, which stand for numbers you don't know yet. The goal is to find the values of these variables that make the equation true.In the given problem, we have an equation set up when we state that \(\frac{2+x}{5+x}\) is equal to \(\frac{2}{3}\). Here, \(x\) is the unknown variable we need to find. With equations, we often solve for the variable by performing operations that keep the equation balanced, like addition, subtraction, multiplication, or division.Understanding equations allows us to solve number problems systematically and find unknown values in both simple and complex mathematical situations.

The Magic of Cross-Multiplication

Cross-multiplication is a powerful technique used to solve equations involving fractions. It helps us get rid of the fractions to simplify the problem. When you have an equation where two fractions are set equal, you can cross-multiply to eliminate the fractions.Here's how it works: Suppose you have two fractions, \(\frac{a}{b} = \frac{c}{d}\). You can multiply across the equals sign: \(a \times d = b \times c\). This turns the problem into an easier equation without fractions.In our example, we started with \(\frac{2+x}{5+x} = \frac{2}{3}\). By cross-multiplying, we turned it into a simpler equation, \(3(2 + x) = 2(5 + x)\). This removal of fractions streamlines the solving process and makes complex algebraic problems more manageable for students learning the ropes of algebra.