To address your request, we'll first simplify a rational expression involving multiple operations, and then we will create examples of expressions that represent real-world situations using rational numbers.
Part 1: Simplifying a Rational Expression
Let's simplify the following rational expression:
\[ \frac{2x^2 - 8}{4x} \]
Step 1: Factor the numerator The numerator \(2x^2 - 8\) can be factored. We can take out a common factor of 2:
\[ 2(x^2 - 4) = 2(x - 2)(x + 2) \]
Step 2: Rewrite the expression Now, we can rewrite the rational expression:
\[ \frac{2(x - 2)(x + 2)}{4x} \]
Step 3: Simplify Next, we simplify the expression by canceling out common factors:
\[ \frac{2}{4} \cdot \frac{(x - 2)(x + 2)}{x} = \frac{1}{2} \cdot \frac{(x - 2)(x + 2)}{x} \]
So, the simplified expression would be:
\[ \frac{(x - 2)(x + 2)}{2x} \]
Part 2: Real-World Situations Using Rational Numbers
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Example 1: Cooking Recipe A recipe requires \( \frac{3}{4} \) cup of sugar per batch of cookies, and you want to make \(2\) batches. The total sugar needed can be represented by the expression:
\[ \text{Total Sugar} = 2 \cdot \frac{3}{4} = \frac{6}{4} = \frac{3}{2} \text{ cups} \]
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Example 2: Distance and Time A train travels \( \frac{150}{2} \) miles in \(2\) hours. To find the speed of the train, you can express it as:
\[ \text{Speed} = \frac{\text{Distance}}{\text{Time}} = \frac{75 \text{ miles}}{1 \text{ hour}} = 75 \text{ miles per hour} \]
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Example 3: Budgeting You earn \( \frac{1200}{3} \) dollars per month from a part-time job. Over \(4\) months, your total earnings can be represented as:
\[ \text{Total Earnings} = 4 \cdot \frac{1200}{3} = \frac{4800}{3} = 1600 \text{ dollars} \]
These examples illustrate how rational numbers can be used in real-world situations, making calculations relevant to everyday life.