What is the ratio of the sum of the even positive integers between 1 and 100 (inclusive) and the sum of the odd positive integers between 100 and 150?

To find the ratio of the sum of the even positive integers between 1 and 100 (inclusive) to the sum of the odd positive integers between 100 and 150, we need to calculate each sum separately.

First, let's find the sum of even positive integers between 1 and 100:

The first even positive integer in this range is 2, and the last one is 100. These numbers form an arithmetic sequence with a common difference of 2. We can use the formula for the sum of an arithmetic series:

Sum = (n/2) * [2a + (n - 1)d]

Where:

• n is the number of terms,
• a is the first term, and
• d is the common difference.

In this case, n = (100 - 2) / 2 + 1 = 50 (since we start from 2 and end at 100), a = 2, and d = 2.

Sum of even positive integers = (50/2) * [2(2) + (50 - 1)(2)] = 25 * [4 + 98] = 25 * 102 = 2550

Now, let's find the sum of odd positive integers between 100 and 150:

The first odd positive integer in this range is 101, and the last one is 149. Again, these numbers form an arithmetic sequence with a common difference of 2.

n = (149 - 101) / 2 + 1 = 25 (since we start from 101 and end at 149) a = 101 d = 2

Sum of odd positive integers = (25/2) * [2(101) + (25 - 1)(2)] = 12.5 * [202 + 48] = 12.5 * 250 = 3125

Now, let's find the ratio:

Ratio = (Sum of even positive integers) / (Sum of odd positive integers) = 2550 / 3125 = 102/125

So, the correct answer is 102/125.