Maths Test!

OK, so I haven\’t actually done this test.

But I know how I\’d do each of the questions with pen and paper, or with mental arithmetic. Most of it\’s pretty simple.

And percentages are easy to anyone who has ever worked as a waiter (calculating that tip!).

Except for questions 13 and 14. Haven\’t a clue. I would get all of the others right, no doubt about it. (Well, alright, I might make an error which can happen to anyone but I know what\’s going on at least.) But those two, wouldn\’t even know where to start.

I sorta, roughly, know what a log is. But how to manipulate them or calculate with them? Not even a chocolate button of an idea. I assume we must have done them at some point on the way to a maths A level and an economics degree but it certainly didn\’t stick.

57 thoughts on “Maths Test!”

  1. 13, 14 – By inspection these are log base 10 (log 3 ~ 0.5) so log 40 is log 10 + log 4 = 1 + 0.6020; log 0.12 = log 3 + log 4 – log 100 = 1.0791 – 2 = (-1).0791 (with the -1 conventionally written as 1 with a bar above it, if I remember my ‘A level maths from the mid ’70s correctly — which was about the last time I ever did arithmetic with logarithms)

  2. Tim, you add logs to multiply, log 10 is 1 assuming we are in base 10, so to multiply by 10 you add 1 to the log of what you are multiplying and to divide by 10 you subract 1 so log 40 is 1.6 something and log 0.12 is 0.1 something

  3. I certainly did logs – both natural and base 10 – as part of Higher and engineering degree maths. We no longer used them to calculate but partial or full log-axis graphs were still quite common (and useful) so you did need the basic understanding of them. But I’m currently helping my son with his Standard grade preparation and my daughter with her GRE (US post-grad entry test) maths – nary a mention of logs in either.

    And log to the base 2 is quite important for computing …

  4. Seems a weirdly balanced test: 12 questions on straightforward arithmetic; one on algebra; two on logs. What could be the rationale for that?

  5. 13. If log 3 = 0.4771 and log 4 = 0.6020, log 40 = ?

    Assuming base10 logs, 1.6020. For a base z log, log(zx) = 1 + log(x)

    14. If log 3 = 0.4771 and log 4 = 0.6020, log 0.12 = ?

    Just add the two together. the log(xy) = log(x) + log(y).

  6. Nothing there that I wouldn’t be comfortable with (A at GCSE, a year early in 1989, ungraded for AS Maths in 1990 and D at A Level in 1992) – with the exception of the log questions at 13 and 14. I think they’d long left the syllabus with the advent of cheap and accurate electronic calculators. I’m not completely ignorant of the use of logarithms, but they’re not something I’ve had cause to use in a calculation.

    My Dad had log tables which he used to use for all sorts of things (well, he was supposed to, but I’m not sure he was a maths whiz either), which fits in with the “50 year old test” aspect of the story.

  7. I’m old enough to know how carpenters actually used rules marked in eighths and sixteenths, and how they woukd divide lengths using fractions.

    So I’m willing to place a bet that candidates are getting the right answer to question 12, yet are being marked as incorrect by Williams.

    The mnemonic BIDMAS plus a policy of working from left to right tells the candidate that the mathematically correct answer is:

    5/16 / 1/8
    = 5 / (16 x 8)
    = 5 / 128
    = 0.039 approx.

    But in the fifties, that extra spacing in the question _was_ significant, and the question that is _implied_ is:

    (5/16) / (1/8)

    Which is a much more trivial “two and a half”.

    So yes, Williams, maths is not still maths, you are probably excluding oerfectly good candidates, and you _should_ update your questions.

  8. Needless to say that smiley above is an artifact of this website and that line was supposed to be
    = 5 / ( 16 x 8 )

  9. On the other hand, you might hope candidates would have some ability to use common sense, which – in a test where the other answers have been integers or 0.5s – would suggest the technically incorrect but much easier and hinted-at-by-weird-spacing answer…

  10. Logs had gone from the maths A level syllabus when I did it (’89; and it wasn’t a GCSE thing, it wasn’t in O-level in ’86 either).

    However they hadn’t changed the exam instructions, so we were each solemnly handed a set of log tables that we had no idea how to use (or even what they might be used for, other than a bad attack of exam nerves).

  11. You can definitely tell the generation that grew up with calculators cos we’re useless at mental arthritic.

  12. Ah yes, the Barmaid Cup: betting in advance whether someone serving drinks will do mental arithmetic or rely on the register. It was fun ten years ago. Now the likelihood of anyone adding anything up in their head is roughly zero. Unless they deal in drugs.

  13. We’re using base 10 logs.

    log(ab) = log a + log b
    log(a/b) = log a – log b
    log 10 = 1

    log 3 = 0.4771
    log 4 = 0.6020

    log 40 = ?
    40 = 4 * 10
    log(40) = log 4 + log 10 = 0.6020 + 1 = 1.6020

    log(0.12) = ?
    0.12 = 3 * 4 / 10 / 10
    log(0.12) = log 3 + log 4 – log 10 – log 10
    = 0.4771 + 0.6020 – 1 – 1
    = -0.9209

  14. Peter: reminded of the old joke about the drugs case judge and prosecution lawyer conferring unproductively about the number of grammes in an ounce, to which the defendant immediately pipes up “28.3, your honour”.

  15. The Sage has it, at comment #2, on the logs. The key is sufficient familiarity with the actual logs (to notice they are to base 10). Sufficiently aged persons have this through the (required frequent) use of tables of logarithms (turning multiplication into addition); IIRC from around age 14 or 15.

    I agree with Martin Audley’s #11 criticism of Q12: 5/16 / 1/8 = ? As well as the obvious ambiguity, the arrival of computer programming means that this question is even less well posed. To an oldie like me (and presumably Martin too) who also programs computers, such a question (in which space is syntactically significant, and also semantically different in multiple legal interpretations) is most unhelpful. However, to oldies, it is also obvious that the question must be about cancellation within expressions of fractions. And why would the ‘1’ be there if the interpretation were 5/(16*1*8)?

    I am disappointed to read that Tim can no longer remember what are logarithms, though I suppose it helps that he can remember that he once knew. At least that way he probably won’t go around writing that they are irrelevant. They are, of course, used in measurements quite a lot, like in the Decibels (dB) commonly use for sound level measurement and electronics. Though, of course, most journalists forget the need for a reference level; eg as given in dBA (spectrally weighted to human hearing/pain and to a physical reference level of 20uPa) or dBm (relative to 1mW).

    Moving back from the suitability of the tests to engineering apprenticeships, part of the problem is almost certainly that those who could pass the test with adequacy are now all going off to university for 2-year or 3-years in further full-time education: God help us but it is so.

    I wonder if Williams F1 is offering any significant part-time study support, or even the possibility of a full-time 3-year bursary. I had a look at their careers web pages: and found no mention of that. Also, though they do mention technical staff with long careers with them, the apprenticeship scheme does not feature at all strongly on the main recruitment web page. Its only mention is in the invitation to apply (before 30th November) for one week of ‘taster’ work experience, and not hear back until February. There is no mention of an undergraduate scheme. Given the test content, one must wonder whether these jobs they cannot fill are for apprentice engineers or for apprentice technicians. Surely anyone with aspirations to be an engineer (in the true sense) would be surprised and insulted to be required to complete such a low-level test: it would define the job on offer as well below what they were looking for.

    And, of course, one must also wonder how good is the pay (and other terms), when the firm is having difficult recruiting.

    Best regards

  16. “Logs had gone from the maths A level syllabus when I did it (’89….”
    Wonder if that coincides with the cohort who continually fuck up on the orders of magnitude?
    Long time since I was taught logs, rather than used them, but didn’t one do a separate sum to position the decimal point?
    And oh did Martin Audley’s calculation ring a bell. Started working life with share prices in shillings/pence. Prices were in increments of base 8. Hardly needed a calculator for doing those.

  17. Only ever come accross logs during Chemistry A-Level, and that only involved pressing the ‘log’ button on a calculator.

    On the other hand, I’m good with logs that come from trees and logs that go down toilets.

  18. The evaluation of logarithms via tables or alide rule may not be vastly relevant any, but the ability to manipulate the algebraically would still be called for, particularly when dealling with calculus. The test is a fairly simple one to see if you can handle the concepts; I am embarrassed to say that I was too rusty to get it without looking it up.

    n. And the Lord said unto the animals,” go forth and multiply!”
    n+1. And all the beasts on the land and fish in the sea and birds in the air did so with gusto, save one pair.
    n+2. And the Lord said unto the basts hanging back, “Wherefore have you not gone forth to multiply?”
    n+3. And the animals said unto Him, “We can’t Lord, we’re adders.”
    n+4. The Lord smiled and said unto them, “Ah yes, adders. Fear not, I have made special provision for you. You must multiply on log tables.”

  19. Nigel Sedgwick:

    There are two reasons why you might use logs base 10: one of them is if you don’t have a calculator or computer (which is why we teach them less these days) and have log tables instead. The other is if you are plotting a graph with a log scale, in which case a certain amount of familiarity with logs base 10 may be useful but for which you don’t really need to know how to manipulate logs as in the question.

    However, when people did use log tables instead of calculators, log of base 10 were likely to be the only kinds of logs that most people ever saw. Therefore in such days it was reasonable for it to assumed that logs were base 10, as the person who set this question does. As a couple of earlier commenters have said, you can (sort of) deduce from the question that logs base 10 have been used, but it isn’t really clear. My assumption is that the author of the question does not understand that the first question that a younger reader is going to ask is “What is the base of the log?”, which makes it a bad question.

    But it is worse. Although logs are not much used for the sorts of calculations in this question any more, logs are extremely important in calculus and elsewhere in higher mathematics. However, in such cases you use natural logs (base e) rather than log 10. So the sort of person who has used logs a log these days may assume that base e is intended, and get the question wrong.

    And there is the problem with question 12 as well. Precedence rules on multiplication and division go from left to right unless there are brackets. If you study mathematics (or computer programming) you are taught this. As Martin Audley says, people are likely getting this right and being marked wrong. Worse than that, the trouble is that the people who are best at maths are the ones who are likely to see the ambiguity of the questions, and get some of them wrong. This is not a desirable state of affairs.

    The test has the feel of being written by some sort of pedant in his sixties who is likely to be insistent on the rightness of his wrongness, too. Possibly not the sort of person you want to work for, so the test might be useful in that sense.

  20. I got through GCSE, A-Level, and Engineering maths and I’d struggle to do some of them, mainly because I’ve forgotten how to do stuff in my head or without a calculator. Knowing how to do stuff without a calculator is probably important up to GCSE, thereafter it is a waste of time*. We used calculators for all our engineering exams, it is the understanding of the problem and its application which is important at that level, not mental arithmetic. If this is what Williams are presenting to its potential engineers, it’s no wonder they’re struggling to recruit.

  21. James James @19

    “(Logarithms are the opposite of powers.
    So if 2^3 = 8, then log base 3 of 8 = 2.
    4^10 = 10000, so log base 10 of 10000 = 4.)”


    If 2^3=8 then log base 2 of 8 =3

    4^10 does not equal 10,000 but 10^4 does.

  22. Judge, you are correct. Logarithms are exponents. Everything else follows.

    My question on the test is this. After a sequence of 12 problems that involve basic arithmetic, why suddenly does the test pose 2 questions that normally do not arise until second-year algebra or pre-calculus?

    My guess is that it’s either (or both) a means of discovering which of the testees may have more advanced knowledge (I can think of logical reasons for that) or a means to keep the top scores no higher than about 85% (can’t think of a logical reason for that).

  23. John Fembub: The problem is that it is a question from the early 1970s. It was a relatively elementary question in the days before calculators when log tables were commonly used. The concepts behind it are indeed today taught later for people studying more mathematics, and this does indeed make it possible for someone with this “advanced knowledge” to answer the question, but it is a poor test of this knowledge as makes assumptions and uses conventions that are not used when learning logs for purposes of calculus, which is why you would learn them today. Some such people are going to get the question wrong because it is unclear, and not because they can’t do it.

  24. ” Knowing how to do stuff without a calculator is probably important up to GCSE, thereafter it is a waste of time*. We used calculators for all our engineering exams, it is the understanding of the problem and its application which is important at that level, not mental arithmetic.”

    Sorry Tim N but 110% ( ;P ) disagree with you, there. I always run through any calculation in my head first before touching the idiot machine. That way I’ve usually got a good idea of the shape of the answer, at least & what the calc should look like through it’s stages. The machines there to crunch the small numbers not do the thinking.
    Relying on what the machine spat out was what bolloxed the Hubble, wasn’t it? Somebody entered metric rather than imperial values or V V? Presumably 2 1/2 times adrift. That should show up as soon as the figures are entered.

    Had a customer do that once. Measured in feet & ordered in metres. Oh, boy was she surprised when the truck arrived.

  25. I did maths A level in ’89, and I’m sure we did some logs with regard to calculus, rather than for just doing numerical calculations, which we had calculators for. But I definitely remember using the log button on my casio fx350, just can’t remember what we were using it for!

  26. BiS – agree working out dimensions and orders of magnitude is all to the good. But unless you have the misfortune to do business with American non-scientists, surely the imperial thing doesn’t arise in this day and age?

  27. So far, I’ve seen no comment here or in the Telegraph comments that the answer to question 11 is -9, not 9. Well, it seems that way to me, despite the high probability of my being about to embarrass myself:

    2(xy)=(6*-3)*2 =-36
    therefore x^2+2xy+y^2=36+(-36)+(-9)=-9

    So, are all the peeps who knocked this off in seconds and agreed with the Telegraph’s answer wrong, or am I (as I was in my teens) a complete arithmetical plonker?

  28. @ formertory
    Multiplying a minus number by a minus number results in a positive answer.
    y^2 is positive if y is a real number
    Only imaginary numbers have negative squares.

  29. ” But unless you have the misfortune to do business with American non-scientists, surely the imperial thing doesn’t arise in this day and age?”
    Or we could all switch to base 12 & make life easier for ourselves.
    Nothing wrong with Imperial. It’s what most metric preferred quantities are based on.

  30. There was a programme on Radio 4 this morning about the teaching of arithmatic in the far east called Land of the Rising Sum. I was particularly taken by the concept of National Abacus Championships, where the really advanced competitors don’t need a physical abacus because they can visualise it.

  31. Sorry Tim N but 110% ( ;P ) disagree with you, there. I always run through any calculation in my head first before touching the idiot machine. That way I’ve usually got a good idea of the shape of the answer, at least & what the calc should look like through it’s stages.

    Okay, but in an engineering application, the crunching of the numbers is normally the small bit at the end. First, you need to look at the problem, understand what you’re dealing with, identify the correct formula, ensure your parameters are correct, and right at the end, stick some numbers in. Of course you do a sense check to make sure that your bolt diameter is not coming out at 3’6″, but the real skill in engineering maths – and IMO A-level physics – was making sure your calculation is fit for purpose. As I say, the entering of real numbers comes very late in the day for most engineers, and is a very minor part.

  32. The test was “derived by Birkbeck college in the 1960s” (I think they mean “devised”). And it shows. It’s all trivial to someone who actually does maths, but it seems to be an arithmetic test, so the questions about logarithms don’t belong in it any more. (I wouldn’t be surprised if a student nowadays who does know about logarithms nevertheless doesn’t have the information that 10 is a plausible default radix.)

    On question 11, you might spot that the expression is the expansion of (x+y)^2, which makes the calculation easier.

    On question 12, I suppose Williams would like to hire people who are capable of working out that “how many eighths are there in five sixteenths” is a plausible question, whereas “what’s 5 divided by 16, divided by 1, divided by 8” is not. Alternatively, the formatting may be a lot clearer in the test as presented to candidates.

  33. Question 12 is about typography, although that may be an artifact of putting it in the Telegraph rather than the actual presentation. Without the use of proper super- and sub-script fraction symbols in your text, the correct way to indicate a fraction is with the spaces as in the example. The use of ‘/’ for division rather than ‘ / ‘ is wrong, but increasingly common.

  34. Anyone who did O Level maths and A Level physics in the sixties will remember that those numbers are indeed the base 10 logs, so everyone above is correct of course.

    My housemaster used to do log calculations on the blackboard as a party trick (numbers called out at random by us, and we didn’t try to make it easy for him), without using the book of tables.

    He was usually almost exactly right.

    He wasn’t a maths teacher, btw – Economic History.

    Sic transit…

  35. You can do Q11 the long way as ’35 above, but I think you’re supposed to recognise that

    x^2 + 2xy + y^2 = (x + y) ^2

    from which the answer 9 is immediate.

  36. Anyone who did O Level maths and A Level physics in the sixties will remember that those numbers are indeed the base 10 logs, so everyone above is correct of course.

    As well as that, the question requires you to know what the log of 10 is, and this is only trivial if you are using base 10, which implies that base 10 is being used.

  37. @50: The base of the logs has to be either 10 or e – nobody ever uses anything else. Given that you are told that the log of two numbers greater than e is less than one, you know that these aren’t logs to the base e, even if you can’t remember your log tables.

    And for all those pointing at “American non-scientists” using imperial units (which are called “English” in these parts), I am a physicist working in America, and my life is full of engineering drawings for home-brew bits of kit, all measured in thousandths of an inch.

  38. Did you hear about the three constipated mathematicians?

    The first used pencil and paper.

    The second used his slide rule.

    And the last worked it out with logs.

  39. 15/15 here but then I’m an engineer.

    RE: logs – the important thing isn’t your ability to manipulate them but to know why and where they are used. If you know why they’re needed then you’ll bother to learn how to manipulate them. Why else would you?

    If I was interviewing I’d much rather the candidate discuss the logarithmic nature of our senses. – sound, light, touch and how this might apply to a race car such as logarithmic steering wheels:

    From quantum foam to the entire universe, the world is logarithmic…
    Linear scales are the perverse abstraction.

    A misunderstanding of logs is what makes a NIMBY petition against a mobile phone mast and yet press a phone to their head. Ask them if they’d prefer to live 50 meters from a lighthouse or press a torch to their eyeball and they instinctively start to get it. It’s non-ionising radiation and bollocks either way but you get the point.

    Was interesting to watch University Challenge last night and they asked the question about a rope held tight 1 meter above the earth’s surface. Pull the rope tight to the surface and how much excess rope is left in your hand? Eight bright minds that I’m sure are all capable of basic differentiation and I’m sure know that C = 2*pi*r but were unable to comprehend when dC/dr is relevant.

  40. Michael Jennings, I believe you misrepresent me as saying that logarithms somehow constitute “advanced knowledge”

    It’s quite clear my reference to “more advanced knowledge” compares understanding of logarithms with skills in basic arithmetic required to answer the other questions.

    Your misquote reveals a carelessness that is inconsistent with mathematical thinking.

    Kevin Monk, I have seen the equivalent question posed the other way – that is, imagine a band tightly stretched around the earth at the equator; if the band were lengthened by one meter, how high above the earth’s surface would the band rise?

  41. The physicist side of me see logs and thinks natural logs (to base e), the electronic engineer sees logs base 10, the software engineer sees logs base 2. Either way, we were taught them at O-level. Internalising a few reference values like ln 2 = 0.693 is a good idea. They’re still very useful for back of the envelope or mental arithmetic questions. Also, I assume most people here are familiar with the old Law of 72 for calculating compound interest. That’s a log problem (with, admittedly, a touch of Taylor series in the Mercator expansion of ln(1 + x) ). Then there’s Benford’s Law, which is useful in forensic accounting for spotting manipulated figures in reports.

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