Puzzle Time: The Struggling Classroom - A stick-figure based cartoon where a teacher is standing in front of a class room board, where the following information is laid out: M -> 85%, P -> 75%, G -> 80%, and B -> 70%. Besides these scores is the big letter "F". In front of the board are three students who seem depressed. The little girl in the centre seems to be crying.

Probability is one of those branches of Mathematics that is counterintuitive by its very nature. So, even simple puzzles involving probability can be surprisingly deceptive. Considering this caveat, let us begin.

Our puzzle originates from a concerning situation playing out in a ninth-grade classroom. The students had recently taken an exam and the examiners have just published the scores.

To everyone’s dismay, 85% of all students have failed Mathematics! Unfortunately, this bad turn of events is not a one-off either.

75% of all students have failed Physics, 80% of all students have failed Geography, and 70% of all students have failed Botany.

Given this sad situation, your challenge is to answer the following question:

What is the minimum percentage of all students that have failed all four subjects (Mathematics, Physics, Geography, and Botany)?

Do you think you can crack this puzzle?

Spoiler Alert

Beyond this section, I will be explicitly discussing the solution to this puzzle.

So, if you intend to solve this puzzle on your own, I recommend you pause reading at this point and give it a try. Once you have a solution, you may continue reading and comparing approaches.

Setting Up the Puzzle

We have four subjects in total, and we also have the total percentage of students who have failed each subject:

1. Botany — 70% of all students have failed

2. Physics — 75% of all students have failed

3. Geography — 80% of all students have failed

4. Mathematics — 85% of all students have failed

I have rearranged the subjects in the ascending order of student-failures for convenience here.

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Now, the trick to solving this problem is to first list the percentage of all students who have not failed (or have passed) each subject as follows:

1. Botany — 30% of all students have passed

2. Physics — 25% of all students have passed

3. Geography — 20% of all students have passed

4. Mathematics — 15% of all students have passed

Given this arrangement, we are just one step away from solving this puzzle. Can you think of a way to use this information?

The Solution to the Struggling Classroom Puzzle

Our goal is to find the minimum percentage of all students who have failed all four subjects.

Let us consider Botany first. If 30% of all students have passed Botany, these 30% will not be in our minimum percentage of all students who have failed all 4 subjects.

Since these 30% have passed Botany, they have NOT failed at least 1 of the four subjects. If we subtract this 30% from the total of 100%, we will arrive at a minimum of 70% of all students who could have failed all 4 subjects. But we still need to consider this scenario for the other three subjects.

The struggling classroom — Illustrative art created by the author

Similarl to Botany, we can subtract the percentage of students who have passed each subject individually from the total of 100% to arrive at the answer as follows:

Minimum percentage of students who have failed all 4 subjects

= 100% – 30% – 25% – 20% – 15% = 100% – 90% = 10%

There you go! This answer may come across as counterintuitive if you are new to probability theory. But the minimum percentage of all students who have failed all four subjects is merely 10%.

I hope you had fun solving this puzzle. If you are into puzzles like these, be sure to check back here in the future!

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Further reading that might interest you: 

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