Problem Analysis:
The given problem involves solving an equation with fractions by clearing the denominators using the Least Common Denominator (LCD). The equation is:
\[
\frac{x}{4} + \frac{x}{6} = 1
\]
The goal is to solve for \( x \) and interpret the result in the context of a work-rate problem.
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Step-by-Step Solution:
####
Step 1: Identify the LCD
The denominators in the equation are 4 and 6. The Least Common Denominator (LCD) of 4 and 6 is 12. This means we will multiply every term in the equation by 12 to eliminate the fractions.
####
Step 2: Multiply through by the LCD
Multiply every term in the equation by 12:
\[
12 \cdot \left( \frac{x}{4} \right) + 12 \cdot \left( \frac{x}{6} \right) = 12 \cdot 1
\]
Simplify each term:
- For \( 12 \cdot \frac{x}{4} \):
\[
12 \cdot \frac{x}{4} = \frac{12x}{4} = 3x
\]
- For \( 12 \cdot \frac{x}{6} \):
\[
12 \cdot \frac{x}{6} = \frac{12x}{6} = 2x
\]
- For \( 12 \cdot 1 \):
\[
12 \cdot 1 = 12
\]
Substitute these simplified terms back into the equation:
\[
3x + 2x = 12
\]
####
Step 3: Combine like terms
Combine the \( x \)-terms on the left-hand side:
\[
3x + 2x = 5x
\]
So the equation becomes:
\[
5x = 12
\]
####
Step 4: Solve for \( x \)
Divide both sides of the equation by 5:
\[
x = \frac{12}{5}
\]
Convert the fraction to a decimal:
\[
x = 2.4
\]
####
Step 5: Interpret the result
The problem states that \( x \) represents the time (in hours) it takes for two entities (e.g., people or machines) working together to complete a job. Therefore, the solution \( x = 2.4 \) means that they can complete the job together in 2.4 hours.
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Final Answer:
\[
\boxed{2.4}
\]
Parent Tip: Review the logic above to help your child master the concept of work problem.
