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# The A to Z of mathematics a basic guide - Sidebotham T.H.

Sidebotham T.H. The A to Z of mathematics a basic guide - Wiley publishing , 2002. - 489 p.
ISBN 0-471-15045-2
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Improper fractions can be converted into mixed numbers, and the process is explained in the following example.
Example 2. Write the improper fraction as a mixed number.
Solution. Write fas
3 3 3 3 1 1
3 + 3 + 3 + 3 + 3 - 1 +1 +1 +1 + 3
which is a mixed number
206 FRACTIONS
Alternatively, 3 can divide 13, in the way shown in figure b. The remainder 1 is written as a fraction of the divisor 3 to give 4|.
4 remainder 1 3)13
(b)
Mixed numbers can also be changed into improper fractions, as explained in the following example.
Example 3. Write the mixed number 3| as an improper fraction.
Solution. Write 31 as
2 5 5 5 2
1+1+1+5“5+5+5+5
= ^ which is an improper fraction
Alternatively, a quick way is to multiply the whole number part, 3, by the denominator, 5, to get 15, add the numerator 2 to get 17, and put the 17 over the denominator 5 to get 17/5.
Adding and Subtracting Fractions When fractions have the same denominators they can easily be added or subtracted.
2 3 2+3 7 3 7-3
^ = — and 0-0 = ^-
4
8
1 4
- Canceling the fraction -
2 8
When fractions have different denominators they cannot be added or subtracted until they have both been rewritten with the same denominator, called a common denominator. To do this we use equivalent fractions (this topic should be understood before proceeding with adding and subtracting fractions).
Example 4. Work out | + |.
Solution. Write down a few equivalent fractions to each of the fractions being added.
FRACTIONS 207
From each list select the two fractions that have the same denominator, and then add them together:
5 3 _ 20 9
6 + 8 “ 24 + 24
Fractions can be added when their denominators are the same.
Writing the improper fraction as a mixed number
This method can be streamlined by identifying the lowest common denominator as the lowest common multiple of the two original denominators, 6 and 8. In this case the lowest common multiple of 6 and 8 is 24. This method is used in the next example.
Example 5. Work out 2^ - 1^.
Solution. First write each fraction as an improper fraction:
23 19
10 ~ 15
The lowest common denominator of 10 and 15 is 30, and using equivalent fractions, write each fraction with a denominator of 30:
23 19 23 3
— - - — x
10 15 10 3
69 38
" 30 ~~ 30
69 -38
30
31
30
= 1 — Writing the improper fraction as
^ a mixed number
For multiplying and dividing fractions, see the entry Multiplying Fractions. Some rulers are graduated in fractions, as in the next example.
29
24
208 FRACTIONS
Example 6. The drawing in figure c is of part of a ruler. Write down the reading to which the arrow points.
i
| i i i | i i i | i i i | i i i |
3 4 5
(c)
Solution. Each unit of length from 3 to 4 and from 4 to 5 is divided up into 8 equal
parts, so each part is |. The arrow points to a reading of 4|, which can be written as
41 by canceling down the |.
Example 7. What fraction of the figure is shaded?
Solution. The large triangle is made up of eight small triangles like the one pulled out of figure d. Three of these triangles are shaded, so the fraction of the shape that is shaded is |.
Example 8. David takes his family on a trip. Of the journey, | is spent flying and | is spent on a train. The rest of the journey is made by cab. What fraction of the journey is made by cab?
Solution. The fraction spent flying and on the train is
3 19 2
- + - = — —- Writing each fraction with a lowest common
4 6 12 12 denominator of 12
9 + 2 “ 12
_ 11
“ 12
This fraction is now subtracted from 1 to obtain the fraction of the journey spent in the cab:
11 _ 12 11
1 “ 12 “ 12 “ 12
1
“ 12
The fraction of the journey made by cab is
FREQUENCY CURVE 209
Fractions can be added, subtracted, multiplied, and divided using a scientific calculator, and you are referred to your calculator handbook.
References: Algebraic Fractions, Canceling, Equivalent Fractions, Integers, Percentage, Ratio.
FREQUENCY
In statistics, frequency refers to the number of times an event occurs in an experiment. When we collect together the frequencies of all the events in an experiment and record them in a table we have a frequency table, or a frequency distribution. These ideas are illustrated in the following example, in which a tally column is used to count the number of times the event occurs.
Example. There are 28 houses on Washington Street and the number of people living in each house is recorded. The results are
0, 2, 1, 1, 4, 2, 3, 3, 2, 3, 0, 4, 2, 0, 3, 3, 3, 2, 3, 1, 2, 3, 5, 3, 4, 0, 3, 3
Draw up a frequency table for these data.
Solution. A frequency table is drawn up and the tally and frequency columns filled in. In the tally column a short ‘stick’ is drawn to keep a record of each time a number is counted. Each time a fifth number is counted a sloping stick is drawn to form units of five sticks. In this way the tally can be easily counted to get the frequency total. It is usual to call the variable x, in this case the number of people, and the frequency/.
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