√11⅑
This is my absolute favourite Maths question of all time. It’s hard, it’s simple, and it shows you something that all the best Maths scholarship questions do, which is this:
Knowing how to use the tools is one thing, but knowing which tool to apply to the problem at hand is something quite different.
Ready? Here we go.
Solve, without a calculator: √11⅑
Which is to say, what is the square root of eleven and one ninth?
Some of my students are as young as nine. Most of them know what a number ‘squared’ means – that you multiply it by itself, so 22 = 2 x 2 = 4. But they don’t know what the square root of a number is. And when you explain that it’s the opposite of ‘squared’, they think it means you divide the number by itself.
Finding the square root of a number without a calculator is really hard, unless it’s a square number already. 11⅑ obviously isn’t, because all square numbers are integers – whole numbers (12 = 1; 22 = 4; 32 = 9; 42 = 16). We might start with trial and error, and say that 11⅑ lies between two square numbers, 9 (= 32) and 16 (= 42). So √11⅑ must be somewhere between √9 and √16, which is to say between 3 and 4. (For what it’s worth. I do find these meanderings helpful, because when you do find a more precise answer through a less familiar method, you’ve got a rough answer to compare it to.)
(My more sophisticated students like to convert fractions (which feel rather messy and arbitrary) into decimals, which feel clean and neat. 11⅑ = 11.111 recurring, which is interesting, but not much use. Actually that’s worse than useless, because this is a question about fractions, and decimals takes us away from the right answer.)
What sort of fraction is 11⅑? It’s a mixed fraction – 11 and ⅑. A normal fraction is a ratio between two whole numbers, a/b. In fact, all rational numbers, which includes nearly all the numbers a young mathematician might encounter (pi being one example of an irrational number), can be written as a/b.
A mixed fraction looks like it’s three numbers, a + b/c, 11 + 1/9. But it’s really more useful to regard it as four numbers a/b + c/d, in this case 11/1 + 1/9.
How would 11⅑ look if it were in the form a/b? That, by the way, is what the question meant by ‘Simplify’. How can you get 11/1 + 1/9 (a/b + c/d) into the form a/b?
By making both denominators (b and d, at the bottom of the two fractions) the same. b and d are 1 and 9, so you need to multiply 1 by 9. And you need to multiply the top of the first fraction by 9 too. 11/1 x 9/9 = 99/9. Of course, 99 divided by 9 is 11, so you haven’t actually changed the fraction at all – you’ve just put it in a different, more useful format (x 9/9 is the same as x 1, which doesn’t do anything to materially change the number).
So 11/1 x 9/9 = 99/9. What was the other half of our mixed fraction, the c/d part we needed to convert the a/b part to match denominators? Oh yes, it’s 1/9. So
11⅑ = 11/1 + 1/9
= 99/9 + 1/9
= 100/9
How does that help? We still need to find the square root of this fiddly fraction, don’t we? Well, we’ve actually got very close to solving it. We’ve turned 11⅑ into a vulgar, improper, top-heavy fraction, in the form a/b, but it turns out that was exactly what we needed. Because what do you notice about a and b, 100 and 9? What was the only kind of number we can find the square root of easily, without a calculator? Square numbers, right? And what are 100 and 9? Square numbers: 100 = 102 and 9 = 32. So:
√11⅑ = √100/9
= 10/3
= 3 1/3
QED!
Actually I cheated a bit with the wording, which I changed from the way it appears in the exam – but I only did this to make the question harder for you, and to make a point that should hopefully make the exam you take easier. Which is to say that the original question asked you to Simplify √11⅑ rather than just solve. Which might have led you to the solution faster: it’s a very useful cue, the idea that you need to rearrange the format of this mixed fraction. But the point remains – sometimes you’ll need to simplify, to see things differently, before you can then solve a problem.