Inequalities Word Problem | GMAT Algebra

A 625 level GMAT Practice question. A GMAT sample question in algebraic inequalities – problem solving. You can expect 1 to 3 questions testing your understanding of inequalities in the GMAT Quantitative Reasoning section. This question is a word problem in inequalities.

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. She has to get a minimum of 18 questions right to get a non-negative score in the test.

Explanatory Answer

Method 1: The Quickest One | Back substituting answer options

For at least a handful of GMAT problem solving questions in the GMAT exam, the quickest and the most intuitive way of solving the question will be to back substitute answer options and check the option that meets the set criteria. Let us apply that method to solving this GMAT Inequalities practice question.

Stated Condition: The student has to get a non-negative mark. i.e., zero or any positive mark is accepted.

What is to be found? The least number of questions that she has to get right to achieve this objective.

What is the first option that we should plugin? Because we are finding the minimum number of questions that she should get right, let us start with the smallest number among the five options.

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)

The negative marks are 0.25 for the first 10, 0.5 for the next 10. The remaining, i.e., 13 incorrect answers, the negative mark is 0.75 marks per question.

So, 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.

Read on for the second method to solve this question.

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Method 2: Framing an Inequality to solve the question

The crux of getting the answer to a GMAT word problem right lies in correctly decoding what is given in words and expressing it rightly as a mathematical expression. We will be better off spending a disproportionately large amount of time in this step. Else, we will end up solving a wrong expression resulting in an incorrect answer.

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. Let’s use some intelligence from the answer options as we navigate through this step.

From the answer options we know n is not more than 25. So, (50 – n) is more than 20. So, she will get all 3 levels of negative marks.

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.

The student has attempted (50 – n) questions incorrectly.
Leaving the first 20 incorrect questions, she has attempted another (50 – n) – 20 = (30 – n) questions incorrectly.

These (30 – n) questions will get 0.75 negative marks each.

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)}

Setup the Inequality and 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.