Problem 11 - Algebra A Combined Function

Question

Solve each equation. Check each solution. $$ \frac{1}{3}+f=\frac{3}{4} $$

Step-by-Step Solution

Answer

The solution is \(f = \frac{5}{12}\).

Step 1: Isolate the Variable

To solve for \(f\), we need to isolate it on one side of the equation. This means we will subtract \(\frac{1}{3}\) from both sides of the equation:\[f = \frac{3}{4} - \frac{1}{3}\]

Step 2: Find a Common Denominator

The fractions \(\frac{3}{4}\) and \(\frac{1}{3}\) have different denominators, so we need to find a common denominator to subtract them. The least common multiple of 4 and 3 is 12. Convert each fraction:\(\frac{3}{4} = \frac{9}{12}\) and \( \frac{1}{3} = \frac{4}{12}\)

Step 3: Subtract the Fractions

Now that the fractions have a common denominator, subtract them:\[f = \frac{9}{12} - \frac{4}{12} = \frac{5}{12}\] Thus, \(f = \frac{5}{12}\).

Step 4: Check the Solution

To check if \(f = \frac{5}{12}\) is correct, substitute it back into the original equation:\[\frac{1}{3} + \frac{5}{12} \]First, convert \(\frac{1}{3}\) into a fraction with a denominator of 12:\(\frac{1}{3} = \frac{4}{12}\)Then add \(\frac{4}{12} + \frac{5}{12} = \frac{9}{12}\), which simplifies to \(\frac{3}{4}\). The solution is correct as it satisfies the original equation.

Key Concepts

Fractions

Fractions are a way to represent parts of a whole. They consist of two parts: a numerator, which is the top number, and a denominator, which is the bottom number. The numerator indicates how many parts we have, while the denominator tells us into how many parts the whole is divided. For instance, in the fraction \( \frac{1}{3} \), "1" is the numerator, and "3" is the denominator. Understanding fractions is essential when solving equations that involve them.
It's important to remember that fractions can only be added or subtracted directly if they have the same denominator. This ensures that we're working with equal-sized parts. If the denominators aren't the same, we must first find a common denominator before performing any arithmetic.

Common Denominator

A common denominator is a shared multiple of the denominators of two or more fractions. It allows us to add or subtract fractions more easily. When dealing with fractions like \( \frac{3}{4} \) and \( \frac{1}{3} \), it's crucial to identify a common denominator to proceed with operations like subtraction or addition.
In this example, the denominators '4' and '3' have a least common multiple of 12. So, we convert both fractions to have a denominator of 12:

• \( \frac{3}{4} \) becomes \( \frac{9}{12} \) because \( 3 \times 3 = 9 \).
• \( \frac{1}{3} \) becomes \( \frac{4}{12} \) because \( 1 \times 4 = 4 \).

This conversion is key to simplifying or solving equations with fractions.

Isolate Variable

Isolating the variable is a fundamental step in solving equations, especially when dealing with unknowns. It means getting the variable by itself on one side of the equation. By isolating the variable, we simplify the equation, making it easier to identify the solution.
In the equation \( \frac{1}{3}+f=\frac{3}{4} \), our goal is to solve for \( f \). We can do this by subtracting \( \frac{1}{3} \) from both sides of the equation. This means the equation becomes:

• \( f = \frac{3}{4} - \frac{1}{3} \)

Now, with \( f \) isolated, we have a clearer path to find its value.

Subtracting Fractions

Once fractions have a common denominator, you can proceed with subtraction by working with the numerators alone. This process ensures that we maintain the integrity of the parts we're dealing with.
For example, subtract the fractions in the equation \( f = \frac{9}{12} - \frac{4}{12} \). Since they both share the denominator of 12, focus solely on the numerators:

• Subtract the numerators: \( 9 - 4 = 5 \).

Thus, the result is \( \frac{5}{12} \). The denominator remains unchanged, and we have accurately performed the subtraction, leading to the solution of the original problem.