Problem 17 - Elementary and Intermediate Algebra

Question

Solve each of these number problems. See Example \(1 .\) If a number is added to the numerator of \(\frac{3}{4},\) and twice as much is added to the denominator, the result is \(\frac{4}{7} .\) Find the number.

Step-by-Step Solution

Answer

The number is 5.

Step 1: Set up the equation using the problem description

Let's denote the unknown number as \( x \). According to the problem, adding \( x \) to the numerator of the fraction \( \frac{3}{4} \), and twice as much, which is \( 2x \), to the denominator results in the fraction \( \frac{4}{7} \). Thus, we can set up the equation: \[ \frac{3+x}{4+2x} = \frac{4}{7} \]

Step 2: Cross-multiply to clear the fraction

To eliminate the fractions, cross-multiply. This gives us the equation: \[ 7(3 + x) = 4(4 + 2x) \]

Step 3: Distribute and simplify the equation

Start by distributing each side of the equation. The left-hand side becomes \( 21 + 7x \), and the right-hand side becomes \( 16 + 8x \). So the equation is: \[ 21 + 7x = 16 + 8x \]

Step 4: Solve for \( x \)

To find the value of \( x \), first subtract \( 7x \) from both sides of the equation: \[ 21 = 16 + x \]. Next, subtract 16 from both sides: \[ 5 = x \]. This shows that the unknown number is \( x = 5 \).

Step 5: Verify the solution

To ensure the solution is correct, plug \( x = 5 \) back into the original setup. Adding 5 to the numerator and 10 to the denominator (twice the number) gives us \( \frac{3+5}{4+10} = \frac{8}{14} \), which simplifies to \( \frac{4}{7} \), confirming our solution.

Key Concepts

Understanding Fractions

Fractions represent a part of a whole and consist of two parts: the numerator and the denominator. The numerator is the top number and it signifies how many parts are considered. The denominator is the bottom number and it indicates how many equal parts the whole is divided into. For example, in the fraction \( \frac{3}{4} \), "3" is the numerator and "4" is the denominator, meaning you have 3 parts out of a total of 4 equal parts.
Fractions are a fundamental concept in algebra because they help us express quantities that are not whole numbers. When dealing with problems in algebra involving fractions, it's crucial to understand how fractions behave during operations like addition, subtraction, multiplication, and division. Always remember that you can only directly add or subtract fractions when they have the same denominator. When performing other operations or solving equations involving fractions, you often need to manipulate these numbers to make the math easier or possible.

Cross-Multiplication Technique

Cross-multiplication is a powerful technique used to solve equations involving fractions. This method helps you to clear the fractions by multiplying terms across the equals sign. Let's look at an example.
For an equation like \( \frac{a}{b} = \frac{c}{d} \), you can cross-multiply to obtain \( a \times d = b \times c \). What this does is transform the equation into a simpler form without fractions, making it easier to solve.

• Clear the equation of fractions by multiplying each numerator by the opposite side's denominator.
• This technique is particularly useful in simplifying complex fraction problems.
• After cross-multiplication, the resulting equation often becomes a simple linear equation that can be easily solved.

In our solved problem, we used cross-multiplication to rid the equation of fractions: \( 7(3 + x) = 4(4 + 2x) \). This step was crucial for further simplifying and solving the equation. Cross-multiplication makes handling fractions straightforward by converting the equation into a familiar and easier to work with format.

Solving Algebraic Equations

Solving algebraic equations involves finding an unknown value that makes the equation true. After cross-multiplying in a fraction equation like the one in our example, the next step is to simplify and solve the resulting equation.
Here’s a simplified step-by-step method to solve equations:

• Begin by distributing any numbers that multiply terms inside parentheses.
• Combine like terms on either side of the equation where possible.
• Isolate the variable by performing operations like addition or subtraction on both sides.
• Lastly, solve for the variable by performing division or multiplication if needed.

In our example, distribution and simplification lead to: \( 21 + 7x = 16 + 8x \). We then isolated \( x \) by subtracting \( 7x \) from both sides and solving the simplified equation \( 21 = 16 + x \), leading to \( x = 5 \).
Solving equations can require more advanced steps for more complex ones, but the key ideas remain: simplify, isolate, and solve for the variable. Practicing these skills builds a strong foundation for understanding and tackling various types of algebraic problems.