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Which of these shows this fraction simplified?
$\frac{\frac{2 x}{12}}{\frac{2 x}{6}}$
Answer (1)
Simplify the numerator: 12 2 x = 6 x .
Simplify the denominator: 6 2 x = 3 x .
Divide the simplified numerator by the simplified denominator: 6 x ÷ 3 x = 6 x ⋅ x 3 .
Simplify the resulting expression: 6 x 3 x = 2 1 .
The simplified fraction is 2 1 .
Explanation
Understanding the Problem We are given a complex fraction 6 2 x 12 2 x and a simplified fraction x 2 18 . Our goal is to simplify the complex fraction and see if it matches the given simplified fraction.
Simplifying Numerator and Denominator First, let's simplify the numerator and the denominator of the complex fraction separately. The numerator is 12 2 x , which simplifies to 6 x . The denominator is 6 2 x , which simplifies to 3 x .
Rewriting as Division and Multiplying by Reciprocal Now, we can rewrite the complex fraction as a division problem: 6 x ÷ 3 x . To divide fractions, we multiply by the reciprocal of the second fraction: 6 x ⋅ x 3 .
Simplifying the Expression Next, we simplify the expression by canceling common factors. We have 6 x ⋅ x 3 = 6 x 3 x . We can cancel the x terms, which gives us 6 3 . Simplifying this fraction, we get 2 1 .
Comparing and Solving for x^2 Now, we compare the simplified fraction 2 1 with the given simplified fraction x 2 18 . We want to determine if 2 1 = x 2 18 . To solve for x 2 , we can cross-multiply: x 2 = 2 ⋅ 18 = 36 . Thus, x 2 = 36 .
Final Answer The simplified form of the given complex fraction is 2 1 . The other expression given is x 2 18 . We found that x 2 18 = 2 1 only when x 2 = 36 . Therefore, the complex fraction 6 2 x 12 2 x simplifies to 2 1 , which is not the same as x 2 18 in general.
Determining the Simplified Fraction However, the question asks which of the expressions shows the simplified fraction. The simplified form of \frac{\frac{2x}{12}}{\frac{\frac{2x}{6}}} is 2 1 . The expression x 2 18 is only equal to 2 1 when x 2 = 36 . Therefore, the simplified fraction is 2 1 .
Examples
Fractions appear in various real-life situations, such as when calculating proportions in recipes, determining discounts while shopping, or understanding probabilities in games. Simplifying complex fractions makes these calculations easier and more intuitive. For example, if you are halving a recipe that already calls for fractional amounts, simplifying the fractions beforehand can prevent errors and ensure accurate measurements. This skill is also useful in financial calculations, such as determining the actual interest rate on a loan or investment.
