Simplify [
6
7
+
3
8
−
1
2
]
4
3
and find its reciprocal.

Let's break this problem down and simplify it step by step:

### Given expression:
\[
\left[ \frac{6}{7} + \frac{3}{8} - \frac{1}{2} \right] \div \frac{4}{3}
\]

### Step 1: Simplify the expression inside the brackets

First, find a common denominator for the fractions inside the brackets. The denominators are 7, 8, and 2. The least common denominator (LCD) is 56.

Convert each fraction to have the denominator 56:
\[
\frac{6}{7} = \frac{48}{56}, \quad \frac{3}{8} = \frac{21}{56}, \quad \frac{1}{2} = \frac{28}{56}
\]

Now combine the fractions:
\[
\frac{48}{56} + \frac{21}{56} - \frac{28}{56} = \frac{48 + 21 - 28}{56} = \frac{41}{56}
\]

### Step 2: Divide by \(\frac{4}{3}\)

Dividing by a fraction is the same as multiplying by its reciprocal, so:
\[
\frac{41}{56} \div \frac{4}{3} = \frac{41}{56} \times \frac{3}{4} = \frac{41 \times 3}{56 \times 4} = \frac{123}{224}
\]

### Step 3: Find the reciprocal

The reciprocal of \(\frac{123}{224}\) is \(\frac{224}{123}\).

### Final Answer:
The simplified value is \(\frac{123}{224}\), and its reciprocal is \(\frac{224}{123}\).

Would you like more details or have any questions?

Here are 5 related questions to deepen your understanding:
1. How do you find the least common denominator for three fractions?
2. What is the process of converting a division by a fraction to multiplication?
3. Why is finding the reciprocal important in division problems?
4. How would this process change if one of the terms was a mixed number?
5. Can you verify the calculation by using decimals instead of fractions?

**Tip:** Always simplify fractions and expressions inside brackets before performing operations outside the brackets.

Simplify [6/7 + 3/8 - 1/2] ÷ 4/3 and find its reciprocal.

Learn how to simplify the expression [6/7 + 3/8 - 1/2] ÷ 4/3 step by step and find its reciprocal. This problem covers fractions, common denominators, and the reciprocal theorem, perfect for students in Grades 6-8.

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