Question

The ratio between three positive numbers is \[\dfrac{1}{4}:\dfrac{1}{3}:\dfrac{1}{2}\]. When the square of the middle number is subtracted from the sum of the squares of the other numbers, the result is 725. Find the numbers.

Answer 1

Three positive number: \[\frac{1}{4} : \frac{1}{3} : \frac{1}{2}\] 

Multiply each term by the LCM of 4, 3, and 2, which is 12:

\[\frac{1}{4} \times 12 : \frac{1}{3} \times 12 : \frac{1}{2} \times 12 = 3 : 4 : 6\]

Let the three numbers be = 3x, 4x, 6x where x > 0

Here, 4x is the middle number.

According to the problem:

\[(3x)^{2} + (6x)^{2} - (4x)^{2} = 725\]

\[9x^{2} + 36x^{2} - 16x^{2} = 725\]

\[29x^{2} = 725\]

\[x^{2} = 25\]

\[x = \pm 5\]

Since the numbers are positive, x = 5

And, numbers are = 3x, 4x, 6x

\[\text{First number} = 3x = 3 \times 5 = 15\]

\[\text{Second number} = 4x = 4 \times 5 = 20\]

\[\text{Third number} = 6x = 6 \times 5 = 30\]

So the numbers are 15, 20 and 30.