Let a,b,c are the A.M. between two numbers such that `a+b+c=15` and p,q,r be the H.M. between same numbers such that `1/p+1/q+1/r=5/3`, then the numbers are

Options

A. -1,-9
B. -3,-3
C. 3,3
D. 9,1

Answer

To solve the problem, we need to find two numbers \( x \) and \( y \) given the conditions related to their Arithmetic Mean (A.M.) and Harmonic Mean (H.M.).

### Step 1: Set Up the A.M. Equation
Given that \( a + b + c = 15 \) and \( a, b, c \) are the A.M. of the two numbers \( x \) and \( y \), we can express this as:
\[ a = \frac{x + y}{2}, \quad b = \frac{x + y}{2}, \quad c = \frac{x + y}{2} \]
Since there are three A.M.s, we have:
\[ 3 \cdot \frac{x + y}{2} = 15 \]
From this, we can simplify:
\[ \frac{3(x + y)}{2} = 15 \]
Multiplying both sides by 2:
\[ 3(x + y) = 30 \]
Dividing by 3:
\[ x + y = 10 \quad \text{(Equation 1)} \]

### Step 2: Set Up the H.M. Equation
Next, we know that \( p, q, r \) are the H.M. of the same numbers \( x \) and \( y \). The relationship for H.M. is given by:
\[ \frac{1}{p} + \frac{1}{q} + \frac{1}{r} = \frac{5}{3} \]
For two numbers, the H.M. can be expressed as:
\[ \frac{1}{p} + \frac{1}{q} + \frac{1}{r} = \frac{3}{\frac{2}{x + y}} = \frac{3(x + y)}{2xy} \]
Substituting \( x + y = 10 \) into the equation:
\[ \frac{3 \cdot 10}{2xy} = \frac{5}{3} \]
This simplifies to:
\[ \frac{30}{2xy} = \frac{5}{3} \]
Cross-multiplying gives:
\[ 30 \cdot 3 = 5 \cdot 2xy \]
\[ 90 = 10xy \]
Dividing both sides by 10:
\[ xy = 9 \quad \text{(Equation 2)} \]

### Step 3: Solve the System of Equations
Now, we have two equations:
1. \( x + y = 10 \)
2. \( xy = 9 \)

From Equation 1, we can express \( y \) in terms of \( x \):
\[ y = 10 - x \]
Substituting this into Equation 2:
\[ x(10 - x) = 9 \]
Expanding this gives:
\[ 10x - x^2 = 9 \]
Rearranging leads to:
\[ x^2 - 10x + 9 = 0 \]

### Step 4: Factor the Quadratic Equation
We can factor the quadratic:
\[ (x - 9)(x - 1) = 0 \]
Setting each factor to zero gives:
\[ x - 9 = 0 \quad \Rightarrow \quad x = 9 \]
\[ x - 1 = 0 \quad \Rightarrow \quad x = 1 \]

### Step 5: Find Corresponding \( y \) Values
Using \( x + y = 10 \):
- If \( x = 9 \), then \( y = 10 - 9 = 1 \).
- If \( x = 1 \), then \( y = 10 - 1 = 9 \).

Thus, the two numbers are \( 1 \) and \( 9 \).

### Final Answer
The numbers are \( 1 \) and \( 9 \).

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