Directions: Two quantities A and B are given in the following questions. You have to find the value to both A and B by using your knowledge of mathematics and choose the most suitable relation between the magnitude of A and B from the given options.

Let m be a perfect square less than 50, and n be a perfect cube greater than 50 but less than 100. The difference between m² and n² is 1695. Another number, o, is a perfect square smaller than m and more than 20, but difference between m and o is more than difference between m and n. and p is a two-digit perfect square less than o.

Quantity I: find the value of m + n + 2p=?

Quantity II: Find the value of m + 2n + o=?

Given:

m is a perfect square less than 50

n is a perfect cube greater than 50 but less than 100

The difference between m² and n² is 1695

Another number, o, is a perfect square smaller than m and more than 20, but the difference between m and o is more than the difference between m and n

p is a two-digit perfect square less than o

Formula used:

Perfect square: m = x²

Perfect cube: n = y³

Difference between squares: a² - b² = (a - b)(a + b)

Calculations:

Let m be a perfect square less than 50, so m can be 1, 4, 9, 16, 25, 36, or 49.

Let n be a perfect cube greater than 50 but less than 100. The perfect cubes between 50 and 100 are 64 and 125. So, n = 64.

We are given that the difference between m² and n² is 1695:

m² - n² = 1695

Using the difference of squares formula:

(m - n)(m + n) = 1695

Substitute n = 64:

(m - 64)(m + 64) = 1695

Let’s try different values for m (perfect squares less than 50) and see which one satisfies this equation.

For m = 49:

(49 - 64)(49 + 64) = (-15)(113) = -1695

This satisfies the equation. So, m = 49 and n = 64.

Next, we need to find the value of o, which is a perfect square smaller than m and more than 20. o must be a perfect square between 20 and 49. The possible values for o are 25 and 36. Since the difference between m and o is more than the difference between m and n, we choose o = 25 because the difference between m and 25 is 24, while the difference between m and n is 15.

Finally, p is a two-digit perfect square less than o. The perfect squares less than 25 are 1, 4, 9, 16. The largest two-digit perfect square less than 25 is 16, so p = 16.

Now, let’s compute the quantities:

Quantity I: m + n + 2p = 49 + 64 + 2 × 16 = 49 + 64 + 32 = 145

Quantity II: m + 2n + o = 49 + 2 × 64 + 25 = 49 + 128 + 25 = 202

Conclusion:

Quantity I = 145

Quantity II = 202

The relation between the magnitudes of Quantity I and Quantity II is that Quantity II is greater than Quantity I.