Oxford Entrance Exam Sort The Numbers – Mind Your Decisions

The Oxford 2023b MAT featured an interesting question asking for the smallest of five numbers. I am going to slightly modify the question. Sort the following five numbers from smallest to largest.

As usual, watch the video for a solution.

Oxford Entrance Exam Sort The Numbers

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Answer To Oxford Entrance Exam Sort The Numbers

(Pretty much all posts are transcribed quickly after I make the videos for them–please let me know if there are any typos/errors and I will correct them, thanks).

I will first show that option (a) is equal to 11/10.

So it remains to compare the options:

(a) 11/10
(b) π/2
(c) 11/9
(d) √(3/2)
(e) √(3) cos(44°)

In such a question, a good starting point is to find the smallest (or largest) number. Let’s compare option (a) to the others.

Option (a) is clearly less than (b) because:

π/2
> 3/2
= 15/10
> 11/10

Option (a) 11/10 is smaller than (c) 11/9 because the numerators are the same but (a)’s denominator is larger.

Let’s compare (a) 11/10 and (d) √(3/2). We can more easily compare the squares of the numbers.

(√(3/2))² = 3/2
(11/10)² = 121/100

But 3/2 = 150/100, which is greater than 121/100. So option (d) squared is larger than option (a) squared, and thus (d) is larger than (a).

Option (e) is only slightly larger than option (d). This is because cosine is decreasing from 0 degrees to 90 degrees, so we have:

√(3) cos(44°)
> √(3) cos(45°)
= √(3) (1/√2)
= √(3/2)

So option (e) is larger than (d) which is larger than (a).

Therefore, (a) 11/10 is the smallest number.

Find the next smallest

(b) π/2
(c) 11/9
(d) √(3/2)
(e) √(3) cos(44°)

It would appear (c) is the next smallest number. Clearly (b) is much larger at about 3.14/2 = 1.57, while (c) is about 1.22.

We know (e) is larger than (d). We can compare (c) with (d) by comparing their squares.

(√(3/2))² = 3/2
(11/9)² = 121/81

Now 3/2 = 121.5/81, so option (d) squared is slightly larger than (c) squared.

Thus (c) is the next smallest, and we can see that (d) is only slightly larger, and (e) is approximately equal to (d).

Find the next smallest

(b) π/2
(d) √(3/2)
(e) √(3) cos(44°)

Options (d) and (e) are close to 11/9, which is about 1.22, so they are smaller than (b) which is about 1.57. So (b) must be the largest.

And we have already shown (d) is less than (e). So we have:

(d) < (e) < (b)

Putting it all together

(a) 11/10
(b) π/2
(c) 11/9
(d) √(3/2)
(e) √(3) cos(44°)

We have shown the numbers from smallest to largest are:

11/10 < 11/9 < √(3/2) < √(3) cos(44°) < π/2

(a) < (c) < (d) < (e) < (b)

Special thanks this month to:

Mike Robertson
Daniel Lewis
Kyle
Lee Redden

Thanks to all supporters on Patreon and YouTube!

Reference

Oxford MAT 2023b
https://www.maths.ox.ac.uk/system/files/attachments/test2023b.pdf

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