The ratio between 2 numbers is 4:5 and the sum of their squares is 1025 , then the numbers are

Options

A. 20 and 25
B. 24 and 30
C. 8 and 10
D. 40 and 50

Answer

To solve the problem step by step, we will follow the given conditions: the ratio of two numbers is 4:5, and the sum of their squares is 1025.

### Step 1: Define the Variables
Let the two numbers be \( x \) and \( y \).
According to the ratio provided:
\[ \frac{x}{y} = \frac{4}{5} \]

### Step 2: Express One Variable in Terms of the Other
From the ratio, we can express \( x \) in terms of \( y \):
\[ x = \frac{4}{5}y \]

### Step 3: Set Up the Equation for the Sum of Squares
We are given that the sum of their squares is 1025:
\[ x^2 + y^2 = 1025 \]

### Step 4: Substitute the Expression for \( x \)
Substituting \( x = \frac{4}{5}y \) into the equation:
\[ \left(\frac{4}{5}y\right)^2 + y^2 = 1025 \]

### Step 5: Simplify the Equation
Calculating \( \left(\frac{4}{5}y\right)^2 \):
\[ \frac{16}{25}y^2 + y^2 = 1025 \]
Now, convert \( y^2 \) into a fraction with a common denominator:
\[ \frac{16}{25}y^2 + \frac{25}{25}y^2 = 1025 \]
Combine the fractions:
\[ \frac{41}{25}y^2 = 1025 \]

### Step 6: Solve for \( y^2 \)
Multiply both sides by 25 to eliminate the fraction:
\[ 41y^2 = 1025 \times 25 \]
Calculating \( 1025 \times 25 \):
\[ 41y^2 = 25625 \]
Now, divide both sides by 41:
\[ y^2 = \frac{25625}{41} \]
Calculating this gives:
\[ y^2 = 625 \]

### Step 7: Find \( y \)
Taking the square root of both sides:
\[ y = \sqrt{625} = 25 \]

### Step 8: Find \( x \)
Now substitute \( y \) back into the equation for \( x \):
\[ x = \frac{4}{5} \times 25 = 20 \]

### Conclusion
Thus, the two numbers are:
\[ \text{The numbers are } 20 \text{ and } 25. \]