Lesson 6 rational expressions word problems | Math, Precalculus ... - Free Printable

The problem involves solving the equation:

\[
\frac{1}{4} + \frac{1}{3} = \frac{1}{x}
\]

This equation is typically used in work-rate problems, where \( x \) represents the time it takes for two entities (e.g., people or machines) working together to complete a task. Let's solve it step by step.

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Step 1: Understand the equation

The equation states that the combined rate of two entities working together is equal to the sum of their individual rates. Here:
- The first entity works at a rate of \( \frac{1}{4} \) (task per unit time).
- The second entity works at a rate of \( \frac{1}{3} \) (task per unit time).
- Together, they work at a rate of \( \frac{1}{x} \) (task per unit time).

We need to find \( x \), the time it takes for them to complete the task together.

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Step 2: Eliminate the denominators

To simplify the equation, we multiply through by the least common denominator (LCD) of the fractions. The denominators are 4, 3, and \( x \). The LCD is \( 12x \).

Multiply every term by \( 12x \):

\[
12x \cdot \frac{1}{4} + 12x \cdot \frac{1}{3} = 12x \cdot \frac{1}{x}
\]

Simplify each term:
- \( 12x \cdot \frac{1}{4} = \frac{12x}{4} = 3x \)
- \( 12x \cdot \frac{1}{3} = \frac{12x}{3} = 4x \)
- \( 12x \cdot \frac{1}{x} = 12 \)

So the equation becomes:

\[
3x + 4x = 12
\]

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Step 3: Combine like terms

Combine the \( x \)-terms on the left-hand side:

\[
7x = 12
\]

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Step 4: Solve for \( x \)

Divide both sides by 7:

\[
x = \frac{12}{7}
\]

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Step 5: Convert to decimal (if needed)

The fraction \( \frac{12}{7} \) can be converted to a decimal:

\[
x \approx 1.7142
\]

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Step 6: Interpret the result

The value \( x = \frac{12}{7} \) represents the time it takes for the two entities to complete the task together. Since the problem mentions "paint the room together," we interpret \( x \) as the time in hours.

Convert \( \frac{12}{7} \) hours to hours and minutes:
- \( \frac{12}{7} \approx 1.7142 \) hours.
- The integer part is 1 hour.
- The fractional part is \( 0.7142 \) hours. Convert this to minutes by multiplying by 60:

\[
0.7142 \times 60 \approx 42.852 \text{ minutes}
\]

Rounding to the nearest minute gives approximately 43 minutes.

Thus, the total time is 1 hour and 43 minutes.

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Final Answer:

\[
\boxed{1.7142}
\]

The boxed answer \( x = 1.7142 \) corresponds to the time in hours, and the interpretation in the image confirms it as 1 hour and 43 minutes.