Calculating exactly with fractions

The chapter you are starting is called Exact Values and Surds. Take the second word first. A surd is a root that will not come out whole, like the square root of two, which as a decimal runs on forever: one point four one four, never settling. Keep the root sign and the number stays exact, which is the first half of the chapter's name doing its work. Fractions carry that exactness today, multiples of pi carry it next, and then the surds themselves take over. So before the surds arrive, you need the exact-answer habit they lean on, and that habit starts with fractions.

A calculator will show you the same thing with an ordinary fraction. Type one divided by three into it. The screen fills with threes, zero point three three three, until it runs out of room. Now imagine multiplying exactly what is on that screen by three. You would hope to get one whole. You get zero point nine nine nine. A tiny sliver has gone missing. That sliver is the whole story. The moment a third became a decimal, it was cut short, because the threes never end and the screen cannot hold them all. Decimals leak. The fraction one third leaks nothing. It is the exact amount, sealed tight. That is what this video is about: keeping answers exact with fractions, and spotting the instruction words in exam questions that demand it. When a question says give your answer as a fraction, the fraction is the finish line. Not a decimal. Not a percentage.

First, the instruction words. Three phrases appear on papers again and again: exact. As a fraction. In its simplest form. Each one is an order about the form of your answer, not just its value. Nought point eight five might be the right amount, but it is not a fraction.

Let's test that straight away. A question ends: give your answer as a fraction in its simplest form. Three students finish the same working. One writes seventeen twentieths. One writes nought point eight five. One writes eighty-five percent. Which of the three obeys the instruction? Have a think. Which one is actually a fraction? The answer is seventeen twentieths. All three are the same amount, which is what makes this sneaky. But only one of them is a fraction. The other two are the right value in the wrong form, and the wrong form is exactly what the instruction rules out.

Now, the method in action. Work out three fifths plus one quarter. Give your answer as a fraction in its simplest form. The adding itself is covered in the video on adding and subtracting fractions, so we will be quick. Five and four both go into twenty. Three fifths becomes twelve twentieths. One quarter becomes five twentieths. Twelve plus five gives seventeen twentieths. Check the cancelling: seventeen is prime, so nothing divides into both seventeen and twenty. Seventeen twentieths is already in its simplest form. That is the finish line. Stop there. But watch the trap. Seventeen divided by twenty is nought point eight five, and a calculator will show you that the moment you check. Copy it down and you have swapped a sealed, exact fraction for a leaky decimal the question banned. The fraction was finished. Leave it alone.

One more, with a real-world flavour. A jug holds two thirds of a litre of juice. One sixth of a litre is poured out. Work out exactly how much juice is left, giving your answer as a fraction in its simplest form. Common denominator six. Two thirds is four sixths. Four sixths take away one sixth leaves three sixths, and cancelling three sixths gives one half. Exactly one half of a litre is left. In everyday life you would say about half a litre, and nobody would mind. The exam is different, and one word did the work: exactly. That word rules out rounding, and asks for the fraction one half on the answer line.

Time to see this through an examiner's eyes. Examiners watch students do the hard work and then give the mark away. Here is a real examiners' report: Full algebraic solutions were rarely seen but were not necessary. Many gave answers correctly as percentages, even if they had already obtained the exact answer as a fraction. Read that back. They had the exact answer as a fraction. The work was done. Then they converted it into a percentage anyway, and when the question demands a fraction, that conversion can cost the mark that was already earned. Converting a finished exact answer is not checking. It is leaking. So build the scanning habit. Before you touch the answer line, look for three phrases: exact. As a fraction. In its simplest form. Any one of them means finish on the fraction, cancel it fully, and put the pen down.

Your turn. Work out five sixths minus one third. Give your answer as a fraction, and see if you can get there without a single decimal appearing anywhere in your working. Pause the video and work it out. I'll wait. The answer is one half. One third is two sixths. Five sixths take away two sixths is three sixths, and three sixths cancels to one half. If no decimal showed up anywhere, not even to check, you have the whole habit.

Let's shrink the whole video down to three lines. Decimals leak; fractions hold the exact value. The phrases exact, as a fraction, and in its simplest form are orders about form: finish on the fraction and cancel it fully. And never trade a finished fraction for a decimal or a percentage, because the mark is earned the moment the exact answer exists.

Next in the chapter: calculating exactly with multiples of pi. Same idea, bigger stage. When circles put pi into your answer, you will learn to keep that exact too.

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