GMAT Practice | Problem Solving | Equations

GMAT Problem Solving Question | Algebraic Inequalities

A 650 level GMAT Practice question. A GMAT sample question in algebraic inequalities – problem solving

Question

In a test comprising 50 questions, a student attempts all questions. For every correct answer the student is awarded 1 mark. She will get negative marks for incorrect answers as per the following rule.

1. 0.25 negative mark for each of the first 10 incorrect answer.
2. 0.5 negative mark for each incorrect answer, from the 11th to the 20th.
3. 0.75 negative mark for each incorrect answer, from the 21st.

What is the minimum number of questions that the student should get right to get a non-negative score?

A. 22
B. 18
C. 23
D. 21
E. 17

Correct Answer : Choice B. Minimum of 18 questions correct.

Explanatory Answer

Quickest way to solve this GMAT Practice Question

The student has to get a non-negative mark.

The quickest way to solve this GMAT equations question is to back substitute answers.

Because we are finding the minimum number of questions that she should get right, let us start with the smallest number in the given set. 17 questions correct and 33 incorrect.

If she had got 17 questions correct, she will get 17 * 1 – (10 * 0.25 + 10 * 0.5 + 13 * 0.75)
i.e., she will get 17 – (2.5 + 5 + 9.75) = 17 – 17.25 = -0.25 marks.

So, if she got only 17 questions correct she will end up with a negative mark.

If she had got 18 questions correct, then she will get -0.25 + 1.75 = 1.5 mark, a non-negative mark.

Correct answer is choice B.

Alternative Approach : Framing Algebraic Inequalities

Step 1 of solving this GMAT practice question: Let us frame an inequality to represent the information given

Let the minimum number of questions that the student should get right to get a non-negative score be ‘n’.

Therefore, the student has got (50 – n) questions incorrect. From the answer options we know n is not more than 25. So, (50 – n) is more than 20.

The student will get 1 mark for every question answered correctly. So, the student will get n positive marks for n correct answers

For the first 10 incorrect answers, the student will get negative marks of 0.25 per question.
Therefore, negative marks for the first 10 questions = 2.5

For the second 10 incorrect answers, the student will get negative marks of 0.5 per question.
Therefore, negative marks for the second 10 questions = 5

For more 20 incorrect answers, the student will get negative marks of 0.75 per question.

If the student attempted (50 – n) questions incorrectly, (50 – n) – 20 = (30 – n) questions will get 0.75 negative marks per question.
Therefore, negative marks for the questions answered correctly from the 21st incorrect answer is (30 – n)*0.75

The net marks that the student gets is {n – 2.5 – 5 – 0.75(30 – n)}

Step 2 of solving this GMAT practice question: Solve the inequality

The net mark got by the student has to be non-negative. An additional condition is that n is an integer.

So, n – 2.5 – 5 – 0.75(30 – n) ≥ 0

n – 2.5 – 5 – 22.5 + 0.75n ≥ 0

1.75n – 30 ≥ 0

1.75n ≥ 30

n ≥ 17.14

The smallest integer value greater than 17 is 18
Hence, the student should answer at least 18 questions correctly to get a non-negative score in the test