If first, second and fourth terms of a proportion are `1/2, 1/3 and 1/6` respectively, find its third term.

Answer

To find the third term of the proportion when the first, second, and fourth terms are given as \( \frac{1}{2}, \frac{1}{3}, \) and \( \frac{1}{6} \) respectively, we can follow these steps:

### Step 1: Understand the Proportion
In a proportion, if we have four terms \( a, b, c, d \), they are said to be in proportion if:
\[ \frac{a}{b} = \frac{c}{d} \]
Here, we have:
- \( a = \frac{1}{2} \)
- \( b = \frac{1}{3} \)
- \( d = \frac{1}{6} \)
- \( c \) is the term we need to find.

### Step 2: Set Up the Equation
Using the property of proportions, we can set up the equation:
\[ a \cdot d = b \cdot c \]
Substituting the known values:
\[ \frac{1}{2} \cdot \frac{1}{6} = \frac{1}{3} \cdot c \]

### Step 3: Calculate \( a \cdot d \)
Now, we calculate the left side of the equation:
\[ \frac{1}{2} \cdot \frac{1}{6} = \frac{1 \cdot 1}{2 \cdot 6} = \frac{1}{12} \]
So, we have:
\[ \frac{1}{12} = \frac{1}{3} \cdot c \]

### Step 4: Solve for \( c \)
To find \( c \), we can rearrange the equation:
\[ c = \frac{1}{12} \div \frac{1}{3} \]
Dividing by a fraction is the same as multiplying by its reciprocal:
\[ c = \frac{1}{12} \cdot \frac{3}{1} = \frac{3}{12} = \frac{1}{4} \]

### Step 5: Conclusion
Thus, the third term \( c \) is:
\[ \frac{1}{4} \]

### Final Answer
The third term is \( \frac{1}{4} \).

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