Twenty one times a positive number is less than the square of that number by 100. What is the value of this positive number?

Introduction / Context:
This question is a straightforward quadratic equation problem expressed in words. It checks whether the learner can convert a verbal relationship into an algebraic equation and then solve the resulting quadratic. Such questions are very common in aptitude tests, engineering entrance exams, and school level mathematics, and they provide good practice for handling word problems involving squares and linear terms together.

Given Data / Assumptions:

• Let the positive number be n.
• Twenty one times this number is less than its square by 100.
• This means n^2 is larger than 21n by 100.
• We are told n is positive and we need its value.

Concept / Approach:
The key is to translate the sentence into a precise equation. The phrase less than its square by 100 means n^2 - 21n = 100. This leads to a quadratic equation in standard form. We solve it either by factorization or by using the quadratic formula. Then we choose the positive root, because the question explicitly mentions a positive number.

Step-by-Step Solution:
Let the number be n.Twenty one times the number is 21n.The statement says that 21n is less than n^2 by 100, which is written as n^2 - 21n = 100.Rearrange to standard quadratic form: n^2 - 21n - 100 = 0.Factor the quadratic if possible. We look for two numbers whose product is -100 and sum is -21.These numbers are -25 and 4 because -25 * 4 = -100 and -25 + 4 = -21.So n^2 - 21n - 100 = (n - 25)(n + 4) = 0.Thus, n = 25 or n = -4.Since the problem specifies a positive number, we take n = 25.

Verification / Alternative check:
Check with n = 25.Compute n^2 = 25^2 = 625.Compute 21n = 21 * 25 = 525.Difference n^2 - 21n = 625 - 525 = 100, which matches the statement twenty one times the number is less than its square by 100.Thus n = 25 is fully consistent with the given condition.

Why Other Options Are Wrong:
26, 42, and 41 do not satisfy the equation n^2 - 21n = 100 when substituted.For example, if n = 26, then n^2 - 21n = 676 - 546 = 130, which is not 100.Only 25 satisfies the quadratic equation and meets the positivity condition.

Common Pitfalls:
A common misinterpretation is to write 21n - n^2 = 100 instead of n^2 - 21n = 100, which reverses the relationship and leads to wrong roots.Some learners forget to check the sign condition and keep the negative solution even when the problem asks specifically for a positive number.Errors in basic multiplication such as computing 25^2 incorrectly can also cause incorrect final answers.

Final Answer:
The required positive number is 25.