If 243 is the third proportional of a and b, where b:a = 9:1, then find the value of a and b.

Options

A. 3, 27
B. 2, 18
C. 4, 36
D. 1, 9

Explanation

To find the values of \( a \) and \( b \) given that 243 is the third proportional of \( a \) and \( b \), and that the ratio \( b:a = 9:1 \), we can follow these steps:

### Step 1: Understand the relationship of the third proportional
The third proportional \( c \) to two numbers \( a \) and \( b \) means that:
\[ \frac{a}{b} = \frac{b}{c} \]
In our case, \( c = 243 \). Therefore, we can write:
\[ \frac{a}{b} = \frac{b}{243} \]

### Step 2: Express \( b \) in terms of \( a \)
From the ratio \( b:a = 9:1 \), we can express \( b \) in terms of \( a \):
\[ \frac{b}{a} = 9 \implies b = 9a \]

### Step 3: Substitute \( b \) in the proportional relationship
Now, substitute \( b = 9a \) into the proportional relationship:
\[ \frac{a}{9a} = \frac{9a}{243} \]

### Step 4: Simplify the equation
This simplifies to:
\[ \frac{1}{9} = \frac{9a}{243} \]

### Step 5: Cross-multiply to solve for \( a \)
Cross-multiplying gives:
\[ 243 = 9 \times 9a \]
\[ 243 = 81a \]

### Step 6: Solve for \( a \)
Now, divide both sides by 81:
\[ a = \frac{243}{81} = 3 \]

### Step 7: Find \( b \)
Now that we have \( a \), we can find \( b \):
\[ b = 9a = 9 \times 3 = 27 \]

### Final Answer
Thus, the values of \( a \) and \( b \) are:
\[ a = 3, \quad b = 27 \]