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March 22, 2020

Challenge Problem 16: No One Selected The Correct Answer To This Test Question

The product of 3 integers x, y, z is 192. Also z is equal to 4, and p is equal to the average of x and y. what is the minimum possible value of p?

The answer choices are:

a) 6
b) 7
c) 8
d) 9.5

Thanks to Manjeet for suggesting this problem! This appeared on the GATE 2019 exam in India.

Let’s solve this problem step by step.

The product of 3 integers x, y, z is 192. This means:

xyz = 192

Then z = 4, so we have:

xy(4) = 192
xy = 192/4
xy = 48

We then want to minimize p, which is the average of x and y. So we want to minimize:

(x + y)/2

Recall x and y are integers. Since xy = 48, we can consider the integer factors of 48, and the average of the factors. The possibilities are:

1×48 = 48
average (1 + 48)/2 = 24.5

2×24 = 48
average (2 + 24)/2 = 13

3×16 = 48
average (3 + 16)/2 = 9.5

4×12 = 48
average (4 + 12)/2 = 8

6×8 = 48
average (6 + 8)/2 = 7

It would appear the lowest average is 7. But this it not the correct answer! The question stated x and y are integers, so they can be negative values too. If we multiply both factors by -1, we still get a product of 48, but the average will get multiplied by -1 as follows:

-1×-48 = 48
average (-1 – 48)/2 = -24.5

-2×-24 = 48
average (-2 – 24)/2 = -13

-3×-16 = 48
average (-3 – 16)/2 = -9.5

-4×-12 = 48
average (-4 – 12)/2 = -8

-6×-8 = 48
average (-6 – 8)/2 = -7

Thus, the minimum value for the average is -24.5.

As this was not an answer choice, no student could have selected the correct answer. The testing agency (IIT) admitted the mistake and gave full marks to all students.

It’s an important reminder to be careful when making assumptions in mathematics! Sometimes solutions might not exist, and sometimes testing agencies can make mistakes too!

Sources

2019 GATE Exam, see question 4
http://gate.iitm.ac.in/gate2019/pdfs/QP/ME2_2019.pdf
http://gate.iitm.ac.in/gate2019/pdfs/Anskey/ME2_key_Published.pdf

In many online posts, people said 7 is the correct answer (I agree it was the intended correct answer). Do not blindly trust the first answer you find online!
https://www.quora.com/The-product-of-three-integers-X-Y-and-Z-is-192-Z-is-equal-to-4-and-P-is-equal-to-the-average-of-X-and-Y-What-is-the-minimum-possible-value-of-P
https://brainly.in/question/8091146

Mind Your Decision
https://mindyourdecisions.com/blog/2019/12/30/no-one-selected-the-correct-answer-to-this-test-question/

Yêu Tiếng Anh

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