The sum of two numbers is 2490. If `6.5%` of one number is equal to `8.5%` of the other, then the numbers are :

Options

A. A)`989, 1501`
B. B)`1011,1479`
C. C)`1401, 1089`
D. D)`1411,1079`

Answer

To solve the problem step by step, we will follow the given information systematically.

### Step 1: Define the Variables
Let the two numbers be \( A \) and \( B \).

### Step 2: Set Up the Equations
From the problem, we know that:
1. The sum of the two numbers is \( A + B = 2490 \).
2. \( 6.5\% \) of \( A \) is equal to \( 8.5\% \) of \( B \). This can be expressed as:
\[ 0.065A = 0.085B \]

### Step 3: Simplify the Second Equation
To eliminate the decimals, we can multiply both sides of the equation \( 0.065A = 0.085B \) by \( 1000 \) to get:
\[ 65A = 85B \]
Now, we can rearrange this to find the ratio of \( A \) to \( B \):
\[ \frac{A}{B} = \frac{85}{65} = \frac{17}{13} \]

### Step 4: Express \( A \) and \( B \) in Terms of a Common Variable
Let \( A = 17x \) and \( B = 13x \) for some variable \( x \).

### Step 5: Substitute into the Sum Equation
Substituting \( A \) and \( B \) into the sum equation:
\[ 17x + 13x = 2490 \]
This simplifies to:
\[ 30x = 2490 \]

### Step 6: Solve for \( x \)
Now, divide both sides by \( 30 \):
\[ x = \frac{2490}{30} = 83 \]

### Step 7: Find the Values of \( A \) and \( B \)
Now we can find \( A \) and \( B \):
\[ A = 17x = 17 \times 83 = 1411 \]
\[ B = 13x = 13 \times 83 = 1079 \]

### Step 8: Write the Final Answer
The two numbers are \( A = 1411 \) and \( B = 1079 \).

### Summary of the Solution
The two numbers are \( 1411 \) and \( 1079 \).

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