Find the value of * in the following `1 (2)/(3) // (2)/(7) xx ("*")/(7) = 1 (1)/(4) xx (2)/(3) // (1)/(6)`

Options

A. `(1)/(6)`
B. 0.6
C. 0.006
D. 6

Answer

To find the value of * in the equation

\[ 1 \frac{2}{3} \div \frac{2}{7} \times \frac{*}{7} = 1 \frac{1}{4} \times \frac{2}{3} \div \frac{1}{6} \]

we will follow these steps:

### Step 1: Convert Mixed Numbers to Improper Fractions
First, we convert the mixed numbers into improper fractions.

- For \(1 \frac{2}{3}\):
\[ 1 \frac{2}{3} = \frac{3 \times 1 + 2}{3} = \frac{5}{3} \]

- For \(1 \frac{1}{4}\):
\[ 1 \frac{1}{4} = \frac{4 \times 1 + 1}{4} = \frac{5}{4} \]

Now, the equation looks like this:
\[ \frac{5}{3} \div \frac{2}{7} \times \frac{*}{7} = \frac{5}{4} \times \frac{2}{3} \div \frac{1}{6} \]

### Step 2: Simplify the Right Side of the Equation
Now, we simplify the right side of the equation:
\[ \frac{5}{4} \times \frac{2}{3} \div \frac{1}{6} \]

Dividing by a fraction is the same as multiplying by its reciprocal:
\[ \frac{5}{4} \times \frac{2}{3} \times 6 \]

Calculating \(6\) as \(\frac{6}{1}\):
\[ = \frac{5}{4} \times \frac{2}{3} \times \frac{6}{1} \]

### Step 3: Multiply the Fractions
Now we multiply the fractions:
\[ = \frac{5 \times 2 \times 6}{4 \times 3 \times 1} = \frac{60}{12} = 5 \]

### Step 4: Simplify the Left Side of the Equation
Now we simplify the left side:
\[ \frac{5}{3} \div \frac{2}{7} \times \frac{*}{7} \]

Dividing by a fraction:
\[ = \frac{5}{3} \times \frac{7}{2} \times \frac{*}{7} \]

The \(7\) in the numerator and denominator cancels out:
\[ = \frac{5 \times *}{3 \times 2} = \frac{5*}{6} \]

### Step 5: Set the Two Sides Equal
Now we set both sides equal to each other:
\[ \frac{5*}{6} = 5 \]

### Step 6: Solve for *
To isolate *, we multiply both sides by 6:
\[ 5* = 30 \]

Now divide by 5:
\[ * = 6 \]

### Final Answer
The value of * is \(6\).

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