For how many integer values of n does the expression 4n + 7 take a value that is greater than 1 and less than 200?

Introduction / Context:
This question tests your understanding of linear expressions and inequalities. You are given an expression in terms of an integer variable n, namely 4n + 7, and asked to count how many integer values of n make this expression lie strictly between 1 and 200. Problems like this are common in aptitude tests to check comfort with inequality manipulation and counting integers in a range.

Given Data / Assumptions:

- The expression is 4n + 7. - n must be an integer (it can be negative, zero or positive). - The value of 4n + 7 must be greater than 1 and less than 200. - We need to count all integer n that satisfy these conditions.

Concept / Approach:
The main idea is to convert the condition on 4n + 7 into a double inequality involving n. Once we isolate n, we obtain a range of real numbers. We then identify all integers lying within this open interval. Finally, we count how many such integers exist using a simple counting formula. Throughout, we treat the inequality carefully to preserve the strict greater than and less than signs.

Step-by-Step Solution:
Step 1: Start from the condition 1 < 4n + 7 < 200. Step 2: Subtract 7 from all three parts of the inequality to isolate the term with n: 1 - 7 < 4n < 200 - 7. Step 3: Simplify the bounds: -6 < 4n < 193. Step 4: Divide all parts by 4 (a positive number, so inequality directions do not change): -6 / 4 < n < 193 / 4. Step 5: Simplify the fractions: -1.5 < n < 48.25. Step 6: Since n must be an integer, n can take any integer value strictly greater than -1.5 and strictly less than 48.25. Step 7: The smallest integer greater than -1.5 is -1. Step 8: The largest integer less than 48.25 is 48. Step 9: Count the integers from -1 to 48 inclusive: total = 48 - (-1) + 1 = 50.

Verification / Alternative check:
To verify, check the boundary values. For n = -1, 4n + 7 = 4(-1) + 7 = 3, which is between 1 and 200, so -1 is included. For n = 48, 4n + 7 = 192 + 7 = 199, which is also between 1 and 200, so 48 is included. For n = -2, 4n + 7 = -8 + 7 = -1, which is not greater than 1, so -2 is excluded. For n = 49, 4n + 7 = 203, which is larger than 200, so 49 is excluded. This confirms that -1 through 48 are exactly the valid integers, giving 50 values.

Why Other Options Are Wrong:
Options 48, 49 and 51 result from miscounting the integer range or from mistakenly including or excluding one of the boundary values. For example, stopping at 47 instead of 48 gives 49 values, while incorrectly starting at 0 gives 49 as well. Only 50 matches the careful inequality analysis.

Common Pitfalls:
Typical mistakes include using greater than or equal to instead of strict greater than, which changes whether boundary values are allowed. Another common error is to forget that n can be negative, leading to counting only positive integers. Writing the inequality properly and explicitly listing a few values near the boundaries helps prevent such errors.

Final Answer:
The number of integer values of n for which 4n + 7 lies between 1 and 200 is 50, corresponding to option C.