Simplify $\left[\frac{6}{7}+\frac{3}{8}-\frac{1}{2}\right] \frac{4}{3}$ and find its reciprocal.
Answer (2)
Simplify the expression inside the brackets by finding a common denominator: 7 6 + 8 3 − 2 1 = 56 41 .
Multiply the result by 3 4 : 56 41 × 3 4 = 168 164 .
Simplify the fraction: 168 164 = 42 41 .
Find the reciprocal of the simplified fraction: 41 42 .
Explanation
Understanding the Problem We are asked to simplify the expression [ 7 6 + 8 3 − 2 1 ] 3 4 and then find its reciprocal. Let's start by simplifying the expression inside the brackets.
Finding a Common Denominator To simplify the expression inside the brackets, we need to find a common denominator for the fractions 7 6 , 8 3 , and 2 1 . The least common multiple (LCM) of 7, 8, and 2 is 56. So, we will rewrite each fraction with a denominator of 56.
Rewriting Fractions Now, we rewrite the fractions with the common denominator: 7 6 = 7 × 8 6 × 8 = 56 48 8 3 = 8 × 7 3 × 7 = 56 21 2 1 = 2 × 28 1 × 28 = 56 28
Calculating the Sum Next, we calculate the sum inside the brackets: 56 48 + 56 21 − 56 28 = 56 48 + 21 − 28 = 56 41
Multiplying by 4/3 Now, we multiply the result by 3 4 :
56 41 × 3 4 = 56 × 3 41 × 4 = 168 164
Simplifying the Fraction We simplify the fraction 168 164 by dividing both the numerator and the denominator by their greatest common divisor (GCD). The GCD of 164 and 168 is 4. 168 ÷ 4 164 ÷ 4 = 42 41
Finding the Reciprocal Finally, we find the reciprocal of the simplified fraction 42 41 . The reciprocal is obtained by swapping the numerator and the denominator, which gives us 41 42 .
Final Answer Therefore, the simplified expression is 42 41 and its reciprocal is 41 42 .
Examples
Fractions and their reciprocals are fundamental in various real-life scenarios, such as scaling recipes, calculating proportions in construction, or determining gear ratios in mechanics. For instance, if you need to increase a recipe by a factor of 3 4 , you multiply each ingredient by this fraction. Conversely, if you want to reduce the recipe back to its original size, you multiply by the reciprocal, 4 3 . Understanding these concepts allows for accurate adjustments and ensures the desired outcome in practical applications.
To simplify [ 7 6 + 8 3 − 2 1 ] 3 4 , we find it equals 42 41 . The reciprocal of this fraction is 41 42 .
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