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[there are two earlier stories in this saga [see Index], which should logically be read first]



FRACTIONS AND PERCENTAGES ...



Miss Walker sat down stiffly at the desk. Bart looked at her, curiously. "Have you been out rollerblading all weekend again?"



"Mmmmm," said Miss Walker enigmatically, putting a pile of books on the desk.



She looked across at Bart. "We need to learn about percentages today," she said. "Any idea what those are?"



Bart thought for a moment, and then said: "A bit like fractions?"



"Yes," said Miss Walker, "in a way, they are. Any percentage less than one hundred per cent means some sort of fraction of the whole thing - which itself is one hundred per cent. Do you remember the apple, and how we cut it into 24 pieces?"



Bart nodded.



"Well," said Miss Walker, "imagine that same apple cut into one hundred pieces. One of those pieces would be 'one per cent', or one hundredth of the total apple."



Bart looked at her, quizzically. "The figure 100 has two zeroes in it, right?" he said.



"Yes," agreed Miss Walker.



"And when you see a percentage sign, that symbol has a little-zero-at-the-top, followed by a forward slash followed by a little-zero-at-the-bottom, but they're all squashed up," Bart continued.



"Yes," said Miss Walker, slowly.



"Well then," said Bart, triumphantly, "that percentage symbol means 'one hundredth' and that's why it has the two little zeroes, to remind you!"



Miss Walker chuckled. "You may well be right, Bart! Personally I've never actually known *where* that symbol came from, but you're right, it *does* mean a hundredth (so the figure in front of it tells you how many hundredths you have) and that's probably as good a way of remembering it, as any I've come across!"



She looked at him again. "So how would you actually write one hundredth, as a percentage?"



"One per cent," said Bart, confidently.



"Write that down for me please," said Miss Walker, passing him a large sheet of paper and a pen.



Bart wrote:



1%



"Excellent," said Miss Walker. "Now write down one hundredth for me, as a fraction."



Bart wrote:


1
-
100


"First class," said Miss Walker. "So now you have the beginnings of the method we use to convert percentages to fractions, or fractions to percentages. Ready for the next step?"



Bart nodded.



"Okay, how would you write two hundredths, as a percentage?"



Bart wrote:



2%



"Correct," said Miss Walker. "Now write down two hundredths as a fraction."



Bart wrote:


2
-
100


He sat for a moment, thinking, and then he looked up at her. "We can divide the top figure by two and the bottom figure by two as well, can't we?" he said.



"Mmmm," said Miss Walker, taking the sheet of paper and putting a large "equals" sign next to what Bart had just written. She eyed him expectantly.



He smiled. "One over 50," he said. "Two hundredths is equal to one fiftieth ..... so two per cent equals one fiftieth!"



Miss Walker wrote it down.


2
-
100
= 1
-
50


2% = 2
-
100
= 1
-
50


"Well done, Bart," she said. "You've grasped the basics *very* well!"



She drew some lines across a new sheet of paper, and drew some boxes as well. "Let's do one per cent through to ten per cent." She passed the piece of paper to Bart.



Bart studied it. "One box next to 1 per cent," he said. "We've already done that one, that's one over one hundred." He filled in the box.


1% = 1
-
100



Next to the two per cent written on the left, were two boxes. Bart wrote in two over 100, and one over 50, as before.


2% = 2
-
100
= 1
-
50


Miss Walker nodded, and watched him look at the next line. There was only one box next to the three per cent. Bart hesitated for a moment, then wrote in three over 100.


3% = 3
-
100


He looked up at her. "Just one box, because three over one hundred can't be divided by the same number, top and bottom, so it doesn't reduce down to a simpler fraction?"



"Gosh," said Miss Walker, "you *are* doing well."



Bart grinned.



He looked at the next line. There were *three* boxes. "Four per cent," he said, "equals four over 100." He thought for a moment. "Equals two over 50."



Miss Walker sat there silently, and watched him.



"Equals one over 25!" said Bart.



"Exactly!" said Miss Walker. "Well done, Bart!"



Bart wrote it down. Four per cent equals four over 100, equals two over 50, equals one over 25.


4% = 4
-
100
= 2
-
50
= 1
-
25


"So far, so good," said Miss Walker, smiling.



Bart looked at the next line. There were two boxes. "Five per cent," he said, "equals five over 100." He paused, briefly. "Can't divide five by two."



"No," agreed Miss Walker, "you can't."



"But," said Bart, "I *can* divide five by five."



"Yes," said Miss Walker, "but you must only divide it by five if you can *also* divide 100, at the bottom, by five. Is that possible?"



Bart grinned again. "Yes!



"Five over 100 is equal to one over 20." He wrote it down.


5% = 5
-
100
= 1
-
20


"Excellent," said Miss Walker. "Half way there!"



Bart looked at the next figure, six per cent. There were two boxes. "Six per cent," he said, "equals six over 100." He paused again. "Equals three over 50!"



He wrote it down:


6% = 6
-
100
= 3
-
50


"Pretty good!" said Miss Walker.



"Yes," agreed Bart. "I'd never have believed, half an hour ago, that I could tell you three fiftieths was equal to six per cent!"



"It's always easier once you know what you're doing," replied Miss Walker. "See how you get on with the next four."



Bart looked at the next line. Seven per cent had only one box next to it. "Seven per cent," he said, "equals seven over 100." He wrote it down.


7% = 7
-
100


The next line had three boxes again. Bart chewed the top of his pencil, and thought for a while. "Eight per cent," he said, "equals eight over 100. Eight over 100 equals four over 50." He stopped for a moment, then said, "Four over 50 equals two over 25!"



Miss Walker nodded.



Bart wrote it down:


8% = 8
-
100
= 4
-
50
= 2
-
25


Nine per cent had only one box by it. Bart barely hesitated before writing in, confidently, nine over 100.


9% = 9
-
100


He smiled widely. "Only one left!" He looked at the last line. There were three boxes. He wrote down, slowly, 10% equals 10 over 100. Then he said, "If I divide top and bottom by two, I get 5 over 50. I can then divide top and bottom by 5 so I get 1 over 10." He wrote it down.


10% = 10
-
100
= 5
-
50
= 1
-
10


He looked at Miss Walker. "I could have done that another way, couldn't I? 10% equals ten over 100. If I divide top and bottom by 5, that gives 2 over 20 - and then dividing top and bottom by 2 gives 1 over 10." He wrote it down, underneath the others.


10% = 10
-
100
= 2
-
20
= 1
-
10


"Yes," said Miss Walker.



"Does it matter which way I go first?" asked Bart.



"No, not really," said Miss Walker, "so long as you show how you got to your final answer! You see, Bart, often there are different ways which are equally correct. A third way, which we haven't looked at yet, is that you could have divided the fraction in the first box, top and bottom, by *ten*. So you could have gone straight from ten over 100 to one over 10!" She wrote it down.


10% = 10
-
100
= 1
-
10



"Hmmm," said Bart. "So there are three ways of getting there, but only one correct answer? How do I know which method to choose, then?"



"It doesn't entirely matter," said Miss Walker. "Whilst it's possibly 'quicker' to divide by 10 straight away, when you're looking to simplify a fraction, you might not always notice that a larger number will divide without any remainder into both top and bottom. So if you start with the little numbers, like 2, 3 or 5, and you find something that "works", that's as good a place to start as any."



"What you need to bear in mind is that you want to end up with a fraction you can't simplify any further - a figure one on the top of the fraction means you've divided as much as you can, so it has to be the final answer, but you need to look carefully at fractions without a one on the top, to make sure that they can't be reduced down any further.



"If you reach your final answer by a short-cut route (eg dividing by a larger number, in this case 10, top and bottom to end up with a one on the top) simply put a line through any following boxes for that question, because these won't be necessary."



She smiled at Bart. "Don't worry too much about the details; you'll figure it out as you go along. This is one of those cases where practising on lots of examples is what you need! ..... and there are another 90 to go. Take a look at these sheets which give the "full set" from one per cent up to 100 per cent - you know how to do the first 10! and there should be enough space to put down your working for each one.



"If not, then use your eraser pen! And work out a larger number you can divide by, top and bottom, so as to use less boxes. Whatever boxes you don't use, just put a line across them so I can see you didn't need them.



"I'd stick with the "little numbers" route if I were you ..... just trying to divide top and bottom by 2 or 5, to begin with. Dividing by three won't work for these numbers because there's always 100 on the bottom of the first fraction on each line, to start with, ..... so you're limited to trying figures you know will divide, without a remainder, into 100."



"Hmmm," said Bart. "I'm not entirely sure about this!"



"You'll be fine," said Miss Walker. "Give me three fiftieths as a percentage?"



"Six per cent," replied Bart, promptly.



A wry grin spread across his face. "Point taken!"



Miss Walker smiled back. "It's really worth doing these, Bart, because learning thoroughly when you're young, how the little elements of maths actually work and fit together, will make your life a whole lot easier when you're older, especially if you decide to do advanced maths later on."



Bart laughed out loud. "Advanced maths! Me? Never!" But as he ploughed through the percentages-to-fractions worksheets at record speed that afternoon, he really wasn't altogether sure ....



© Penny Midas Rollo 28th April 2004    





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