# Harvard Entrance Exam 1869

Discussion in 'Mathematics' started by Polecat, Dec 19, 2011.

1. ### Polecat

Makes interesting reading. I've got some entrance papers for Oxbridge colleges from about the same period, and they are comparable. The maths looks to be pretty useful stuff and not to be scoffed at. Nothing like those dreadful STEP papers that some kids are forced to take now.

Sarcasm?

3. ### pipipiNew commenter

Thanks Karvol!

4. ### Polecat

I find the History and Geography paper the most interesting, and challenging!
I like the advice given to candidates in the mathematics section about how to set out their work.
I'm wondering how the 'Applicants for Advanced Standing' were supposed to do Arithmetic Q4 given that this was not in the Logarithms and Trigonometry section. I know of classical methods to find approximations to cube roots, but would these have been known to such applicants?

5. ### Null

0.0093^(1/3)

=1/100 * (21^3+39) ^(1/3)

= 21 / 100 *( 1+ 13 / 3087 ) ^ (1/3)

Which is near enough

= 21 / 100 * (1 + 42 / 10000 ) ^ ( 1 / 3 )

and a couple of terms of the binomial expansion suffice to give 0.210294.

This would probably be a fail as I can't guarantee the accuracy.

Does anyone know what sort of method they would have used?

6. ### DMNew commenter

Two applications of Newton's method with a first guess of 0.2 would give you the answer to 5 s.f. and the arithmetic isn't ridiculous.
x1 = 0.2
x2 = (0.0093*25 + 0.4)/3 = 253/1200
x3 = (93*144/64009 + 253/600)/3 = 0.2102957483

7. ### Polecat

OK, I can see that 9300 = 21^3 + 39, as a first step in a trial and
improvement approach.
Then I suppose 13/3087 is approximately 42/10000 by long division.
But, as you say Null, the accuracy to 5 dps is an issue.
I can think of Heron's method and Newton's, but neither look good in this situation.
Must be missing something.

8. ### Polecat

Yes DM, but sadly the final sf or dp is wrong. It needs one more iteration.

9. ### DMNew commenter

I didn't read the question so didn't realise it specified a certain accuracy.
I suspect the candidates were supposed to use this algorithm then as it works to whatever accuracy you need.

10. ### DMNew commenter

To follow the algorithm you will probably find it easier to find the cube root of 9800 and then divide the answer by 100.

Why?

12. ### DMNew commenter

Because it says "Mark off triples of digits, starting from the decimal point and working left." I thought this might defeat some people who wanted to give it a go.

13. ### Polecat

Co blimey, luv a duck, you must be joking.
I think they must have had access to 7 figure log tables, like we all used to have in the old days.

14. ### DMNew commenter

My memory of 1869 is a little hazy.

15. ### Polecat

but mine is as clear as the day is young.
Yes we did have 7 figure tables, blast you ...