Simplify the complex fraction.
$\frac{\frac{1}{2}+\frac{1}{4}}{\frac{1}{8}-\frac{3}{7}}$
Answer (1)
Simplify the numerator: 2 1 + 4 1 = 4 3 .
Simplify the denominator: 8 1 − 7 3 = 56 − 17 .
Divide the simplified numerator by the simplified denominator: 56 − 17 4 3 = 4 3 × − 17 56 .
Simplify the resulting fraction: − 17 42 .
− 17 42
Explanation
Analyze the problem We are asked to simplify the complex fraction 8 1 − 7 3 2 1 + 4 1 To do this, we will first simplify the numerator and the denominator separately.
Simplify the numerator The numerator is 2 1 + 4 1 . To add these fractions, we need a common denominator, which is 4. So we rewrite 2 1 as 4 2 . Then we have 2 1 + 4 1 = 4 2 + 4 1 = 4 2 + 1 = 4 3 Thus, the numerator simplifies to 4 3 .
Simplify the denominator The denominator is 8 1 − 7 3 . To subtract these fractions, we need a common denominator, which is 8 × 7 = 56 . So we rewrite 8 1 as 56 7 and 7 3 as 56 24 . Then we have 8 1 − 7 3 = 56 7 − 56 24 = 56 7 − 24 = 56 − 17 Thus, the denominator simplifies to 56 − 17 .
Divide the numerator by the denominator Now we have the complex fraction 56 − 17 4 3 To simplify this, we divide the numerator by the denominator, which is the same as multiplying by the reciprocal of the denominator: 56 − 17 4 3 = 4 3 ÷ 56 − 17 = 4 3 × − 17 56 = 4 × − 17 3 × 56 = − 17 3 × 14 = − 17 42 = − 17 42 Thus, the simplified complex fraction is − 17 42 .
State the final answer The simplified form of the complex fraction is − 17 42 .
Examples
Complex fractions might seem abstract, but they appear in various real-world scenarios. For instance, when calculating the equivalent resistance of parallel circuits in electronics, you often encounter complex fractions. Also, in physics, when dealing with Doppler effect calculations involving relative velocities, complex fractions can arise. Simplifying these fractions makes the calculations more manageable and helps in understanding the underlying relationships between different quantities.
