The ratio between two numbers is 2 : 3 and the sum of their squares is 2197. The numbers are:a)26 and 39b)15 and 20c)26 and 42d)36 and 48Correct answer is option 'A'. Can you explain this answer?

Question

Answer

The given problem states that the ratio between two numbers is 2:3 and the sum of their squares is 2197. We need to find the two numbers that satisfy these conditions. Let's solve the problem step by step.

Step 1: Set up the problem
Let's assume the two numbers as x and y, where x is the smaller number and y is the larger number. According to the problem, the ratio between these numbers is 2:3. Therefore, we can write the equation as:
x/y = 2/3

Step 2: Express one variable in terms of the other
To solve the problem, we need to eliminate one variable. Let's express x in terms of y using the given ratio:
x = (2/3)y

Step 3: Substitute the expression in the equation
Now, substitute the expression of x in terms of y into the equation of the sum of their squares. The sum of their squares is given as 2197, so we have:
(x^2) + (y^2) = 2197

Substituting the expression of x, we get:
((2/3)y)^2 + (y^2) = 2197

Simplifying the equation, we have:
(4/9)y^2 + y^2 = 2197

Step 4: Solve the equation
To solve the equation, we can multiply through by 9 to eliminate the fraction:
4y^2 + 9y^2 = 19773

Combining like terms, we have:
13y^2 = 19773

Dividing both sides by 13, we get:
y^2 = 19773/13

Taking the square root of both sides, we have:
y = √(19773/13)
y ≈ 39

Step 5: Find the other variable
Now that we have found the value of y, we can substitute it back into the equation to find x:
x = (2/3)y
x = (2/3)(39)
x ≈ 26

Therefore, the two numbers that satisfy the given conditions are approximately 26 and 39. Hence, option A is the correct answer.

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