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Transcript
Arithmetic (Part 1)
We are going conduct a revision of basic arithmetic skills which are essential for any
statistics and economics program. The topics will include:
1.
2.
3.
4.
5.
6.
7.
Order of operations
Fractions
Decimals
Percentages
Signed numbers
Exponents
Inequalities
ORDER OF OPERATIONS
Symbols for common operations used in this Workbook:
+
Addition (plus)

Subtraction (minus)
 or /
Division
* or 
Multiplication

Root
RULE 1
Work from left to right when performing calculations.
40  5 + 7 – 20
= 35 + 7 – 20
= 42  20
[40  5 = 35]
[35 + 7 = 42]
= 22
RULE 2
Always perform division and multiplication first before addition and subtraction.
Example
7 + 500  10 – 24
= 7 + 50 – 24
[500  10 = 50 RULE 2]
= 57 – 24
[RULE 1]
= 33
RULE 3
Any operations contained in brackets are performed first.
Example
20 * (7  4)
= 20 * 3
[7  4 = 3
RULE 3]
1
FRACTIONS
WHAT IS A FRACTION?
Before we use fractions in performing calculations we need to understand what a fraction is.
Fraction can be thought of as a portion of the whole. The bar below is divided into 5 equal
parts.
2
5
1
5
The light shaded area on the left represents one-fifth (
area on the left represents two-fifths (
1
) of the bar and the dark shaded
5
2
) of the bar. Applying this idea of a fraction being a
5
part of the whole, we can do addition, subtraction, multiplication and division of fractions.
SIMPLIFYING FRACTIONS
This technique allows us to obtain the simplest form of a fraction. This is done by finding a
common factor of the numerator and denominator, and then dividing them by that same
number. Before we look at the method of simplifying a fraction, you may want to refresh
the idea of factor and common factor.
A factor is a whole number (except 1) by which a larger number can be exactly divided. Put
it in a slightly different way, a number is a factor of (say) 12, if it can wholly divide 12. For
example, 4 is a factor of 12 since 4 can wholly divide 12. The number 12 has five factors: 2,
3, 4, 6 and 12.
As another example, 8 has three factors: 2, 4 and 8.
A common factor, as the name suggests, is a factor that is common to two (or more) larger
numbers. 12 and 8 have two common factors: 2 and 4. And the highest common factor
(HCF) of 12 and 8 is 4.
In the example below, 24 and 36 have several common factors, but the HCF is 12. We can
simplify the fraction by dividing the numerator and denominator by their HCF, as shown in
the example below.
Example
24
36
24  12

36  12
2

3
We divide the numerator and
the denominator by 12.
2
EQUIVALENT FRACTIONS
Fractions that are equal in value are called equivalent fractions. For example, the following
fractions are equivalent:
2 3 5 10
,
,
,
6 9 15 30
1
3
These fractions are equivalent because they can all be reduced to the simplest form: . To
obtain an equivalent fraction we simply multiply both the denominator and numerator of a
fraction by a whole number.
Example
We can obtain 2 equivalent fractions to
So,
3
as shown below:
7
(1)
3* 2 6

7 * 2 14
[Both 3 and 7 are multiplied by 2.]
(2)
3*5 15

7 *5 35
[Both 3 and 7 are multiplied by 5.]
3 6
15
,
and
are equivalent fractions.
7 14
35
ADDING AND SUBCTRACTING FRACTIONS WITH SAME DENOMINATOR
If you have fractions with the same denominator, simply add or subtract numerators.
Example
(a)
1 2 1 2 3
 

5 5
5
5
ADDING AND SUBCTRACTING WITH DIFFERENT DENOMINATORS
When fractions have different denominators, before adding or subtracting, convert them
into equivalent fractions with common denominator.
Examples
(a)
1 1

2 3
1* 3 1* 2


2 *3 3* 2
3
3 2

6 6
3 2
5


6
6

[
1
3
1
2
is converted to , and to .]
2
6
3
6
Note: The second step can be omitted if you can work out equivalent fractions in
your head.
5 1
5* 4 1* 7
 

7 4 7*4 4*7
(b)

20
7

28 28

20  7
28

13
28
[
5 5* 4
1 1* 7
=
and
=
]
7 7*4
4 4*7
[Both fractions now have same denominator.]
MULTIPLICATION OF FRACTIONS
If a fraction is multiplied by a whole number, multiply only the numerator by the number.
Simplify by cancelling as early as possible.
Example
(a)
3
3* 2
6
*2 

11
11
11
(b)
2
7
7
*8 
*8 1
60
60
15
1
7
*2
15
7*2

15
14

15

To multiply a fraction by another fraction, multiply the numerators together and multiply
the denominators together. Simplify by cancelling as early as possible.
Examples
(a)
1 3 1* 3
* 
4 5
4*5
3

20
1
(b)
3 5
3 *1
* 
20 7
4*7
4
3
1

28
4
DIVIDING FRACTIONS
When dividing fractions, invert (i.e turn over) the dividing fraction and then multiply two
fractions.
Example
2 5 2 7
  *
3 7 3 5
2*7

3*5
14

15
5
7
becomes
& the  sign is changed to * .
7
5
DECIMALS
A useful way of expressing a fraction is to convert it to a decimal. The places to the right of
the decimal point have different place values.
Place value
1st place after the decimal point
tenths (
2nd place after the decimal point
1
)
10
hundredths (
3rd place after the decimal point
thousandths (
1
1000
1
)
100
)
Below is an example of a decimal.
Decimal point
2 3 . 5 6 7
tenth
hundredth
thousandth
This number has three digits to the right of the decimal point – we say it has three decimal
places. Notice also that the above number is equivalent to:
23 
Or
23
5
6
7


10 100 1000
567
1000
5
CONVERTING A DECIMAL TO A FRACTION
From the above example we can see that:
0.5 is equivalent to five tenths or
0.06 is equivalent to
5
,
10
6
, and
100
0.007 is equivalent to
7
.
1000
Examples
The following decimals can be converted into these fractions:
(a)
0.19 =
19
100
(c)
0.25 =
25
1

100
4
(b)
0.023 =
23
1000
[Divide numerator and denominator by 25.]
ADDITION AND SUBTRACTION OF DECIMALS
When adding or subtracting decimals, make sure that the decimals points are aligned.
Example
+
6
.
2
5
2
0
.
3
4
2
6
.
5
9
MULTIPLYING DECIMALS
Step 1:
Multiply decimals as if the decimals were whole numbers.
Step 2:
Count the number of decimal places of the multiplying numbers and add them up.
Step 3:
Mark off the number of decimal places in the answer equal to the total number of
decimal places in the multiplying numbers.
Example
Calculate 1 .32 * 2.1
6
x
1. 3
2.
2 6 4
1 3
2 . 7 7
2
1
0
2
2
Note that together ‘1.32’ and ‘2.1’ have 3 decimal places. So, we mark off 3 decimal places
in the answer, working from the right to the left.
DIVIDING DECIMALS
Terms to remember:
Divisor
The number doing the dividing.
Dividend
The number being divided into.
Quotient
The result of the division
To divide a number by a decimal, follow these steps: First move the decimal point in the
divisor to the right so that the divisor becomes a whole number. Then move the decimal
point in the dividend to the right the same number of places as in the divisor. Now we can
do the division.
Example
Calculate 2.156  0.04
Step 1:
Move the decimal point in 0.04 by 2 places to the right to obtain 4.
Step 2:
Move the decimal point in 2.156 to the right by 2 places; it becomes 215.6.
Step 3:
5 3 .9
4 2 1 5 .6
2 0
1 5
1 2
3 6
3 6
Answer:
53.9
ROUNDING OFF DECIMALS
When we divide 1 by 6, we will get a recurrent decimal 0.16666666666…. For most practical
purposes, it is necessary that we round the number off. We round a number to a desired
number of decimal places depending on the required accuracy of the relevant task. The
7
more decimal places you retain, the more accurate the number is. But the number with too
many decimal places will be very inconvenient to use. So, we are trading accuracy for
convenience.
The Rule
When rounding off a number, the rule is: If the first figure to be omitted is 5 or above, then
add 1 to the last figure. Here are a few examples for illustration.
Examples
(a) 16.182389
rounded off to 3 decimal places will be: 16.182. It is rounded down
because the 4th decimal place (underlined) has a value less than 5.
(b) 0.112386
rounded off to 4 decimal places will be: 0.1124. It is rounded up
because the 5th decimal place has a value greater than 5.
(c) 0.002497
rounded off to 3 decimal places will be 0.002.
SIGNIFICANT FIGURES
The difference between the actual number and the rounded value expressed as a
percentage of the actual number is usually used a measure of the rounding error. If we
compare the rounding errors in examples (a) and (c), you will observe that the rounding
error in (a) is very small. The rounded value is only 0.000389 larger the actual value which
represents an error of only 0.0024% (of 16.182389). In other word, the rounded value is
very close to its true value. But the rounding error in (c) is close to 20% and the rounded
value is very inaccurate.
Moreover, using the number of decimal places to round off numbers sometime may not
make a lot of sense. Example (a) can be put into a more meaningful context to illustrate this
point. If your supervisor asks you to measure the width of a room, would you give him the
measure in metres rounded to 3 decimal places such as 16.182 metres? It would be
unnecessarily precise.
On the other hand when we are dealing with small values such as the thickness of hair in
centimetres, whose first meaningful digit is 2 to 3 places after the decimal point, it would be
inaccurate to round the measurement to 2 decimal places. Another method of rounding
numbers is to determine the number of significant figures in a measurement. Significant
figures are simply the meaningful digits in a measurement. This method is illustrated below:
Examples
(c) 16.182389
rounded off to 3 significant figures will be: 16.2. It is rounded up as the
4th figure (counted from the first non-zero digit on the left) has a value
greater than 5. Note that in this case, rounding off the number to 3
significant figures is the same as rounding it off to 1 decimal place.
(d) 7.112386
rounded off to 4 significant figures will be: 7.112. It is rounded down
because the 5th figure (from left) has a value smaller than 5.
8
(e) 0.002497
rounded off to 3 significant figures will be 0.00250. Note the first nonzero digit in this example is ‘2’. And the 4th figure (counted from the first
non-zero digit on the left) is 7.
Note: You may have been told that the last zero(s) in a decimal can be omitted without
affecting its value. For example, 0.0025 is treated as the same as 0.00250. While it is
generally correct, when we dealing with rounded numbers, 0.0025 and 0.00250 are different
– one is correct to 2 significant figures and the other 3 significant figures. Which is more
accurate?
PERCENTAGES
Percentages can be treated like a fraction with the denominator being 100. For example 5%
is
5
100
. You must have purchased items from department store on sale; the shop wants us
to buy more by taking away 5% to 10% from the marked prices of all items in the shop.
When we say 5% of the price (or a number), we mean:
5
100
x number
Percentages are often converted into decimals. For example, 2% is 0.02. The important
point in converting percentages to decimals is to ensure that the location of the decimal
point is right.
Examples
(a)
(b)
20
, which in decimals is equal to 0.2.
100
5
0.5% is
, which in decimals is equal to 0.005
1000
20% is
PERCENTAGE CHANGES
Suppose the population increases by 3%. Then:
The increase in population = 3% * population.
The increased population = (1 + 3%) * population
9
Example
You take out a one-year loan of $20,000 from a bank. The interest rate is 8% per annum.
The interest payment at the end of a year = $20,000 * 8% = $1,600
The total repayment at the end of a year = $20,000 * (1 + 8%) = $21,600
SIGNED NUMBERS
Signed numbers are numbers with a positive (+) or () signs in front of it. We also call them
directed numbers.
Examples
100 metres above sea level
+100 m
10oC below zero
10 oC
A loss of $2 million
$2 million
ADDING SIGNED NUMBERS
Examples
(a)
(+7) + (+4) = 7 + 4 = 11
(b)
(+7 ) + (4) = 7 – 4 = 3
(c)
(7) + (+4) = 7 + 4 = 3
(d)
(7) + (4) = 7 – 4 = 11
SUBTRACTING SIGNED NUMBERS
Subtracting a negative number is equivalent to adding a positive number.
Examples
(a)
7–4=3
(b)
7 – (4) = 7 + 4 = 11
(c)
9–3=6
(d)
3 – 9 = 6
[Subtracting 4 is same as adding +4.]
[Do you know why the answer is 6? Refer to the
next page for more explanation.]
MULTIPLICATIONS AND DIVISION OF SIGNED NUMBERS
When the numbers are of different signs, the answer is negative.
When the numbers are of same signs, the answer is positive
10
Example
(a)
(+2) * (6) = 12
(b)
14  (7) = 2
(c)
(3) * ( +10) = 30
(d)
27  9 = 3
(e)
(+24) * (+2) = +48
(f)
40  5 = 8
(g)
(5) * (4) = +20
(h)
56  (8) = +7
Note: If there is no confusion, the positive sign can be omitted.
EXPONENTS
In economics and finance, we often deal with situations where a number is multiplied by
itself many times. For example:
1.02 * 1.02 * 1.02 * 1.02 *1.02 * 1.02 *1.02 * 1.02 * …
A convenient way of dealing with long expressions like this is to express them in exponential
or index form. For example: a * a * a * a can be written in more compact form:
a4 (reads ‘a to the power of 4’)
In this example, the value a is called the base, and the superscript 4 the exponent or index.
In general,
an = a * a * a *…. *a * a
(n times)
The exponent is merely the number of times we would write the base number if we wish to
write the expression in full.
Example
(a)
53 = 5 * 5 * 5
(b)
1.26 = 1.2 * 1.2 * 1.2 * 1.2 * 1.2 * 1.2
(c)
(1 + x)4 = (1 + x) * (1 + x) * (1 + x) * (1 + x)
INDEX RULES
Rule 1*
To multiply numbers with the same base, add the indices. (The plural of index is indices.)
am * an = am + n
Example
34 * 35 = 34 + 5
= 39
11
Rule 2
To divide numbers with the same base, subtract the indices.
a m mn
a
an
Example
37
 37 2  35
2
3
RULE 3
To raise the power to a power, multiply the indices.
(a m)n = a m * n
Example
(84)5
= 84 * 5
= 820
THE MEANING OF A NEGATIVE INDEX
A term having a negative index is equivalent to the reciprocal of the term with the sign of
the index changed to positive.
Example
an 
1
an
7 3 
1
73
THE MEANING OF FRACTIONAL INDEX
Fractional indices represent various roots of the number.
1
n
a n a
(nth root of a)
Examples
1
1
a2  a
The symbol
(square root of a)
a6  6 a
(6th root of a)
is called a surd or radical.
12
INEQUALITIES
An inequality is a statement that tells us the left hand side (LHS) is greater than or smaller
than the right hand side (RHS). Inequalities are commonly used in probability problems. Let
us familarise ourselves with the inequality symbols.
SYMBOLS
>
LHS greater than RHS
<
LHS smaller than RHS
≥
LHS greater than or equal to RHS
≤
LHS smaller than or equal to RHS
Examples
(a)
4 < 5
4 is less than 5
(b)
7>2
7 is greater than 2
(c)
9 < 4
9 is less than 4
(d)
X ≥ 10
X is greater than or equal to 10
USING INEQUALITIES TO DEFINE LIMITS
The last example (d) illustrates the use of inequalities to limit the values of a variable. In (d),
the value of X is 10 or above. Below we will use a few more examples to show the
applications of inequalities.
Example:
SENIOR CARD
To be eligible for the senior card a person must be 65 or older. Let X be the age of a person.
The age requirement for the senior card can be expressed as:
X ≥ 65
EURAIL YOUTH PASS
Do you know you can apply for the Eurail Youth Pass to get a substantial discount for train
tickets in Europe? But you must be 18 or above, and not older than 35. The age limit for the
Eurail Youth Pass is:
X ≥ 18 and X ≤ 35
Note: The last inequalities can also be expressed in more compact form: 18 ≤ X ≤ 35. Note
also that this set of inequalities is equivalent to:
17 < X < 36
13
CONCESSIONARY TICKETS
5
3 5  3 ?A theme park in Queensland offers concessionary tickets to people under 12 or above 60.



17 17
17
?The condition of a concessionary ticket is:
X < 12 or X > 60
ZERO EXPONENT
a0 = 1
Proof:
Since
am * an = am + n
If m = 0, then
a0 * an = a0 + n
= an
Hence,
a0
= a n / an
=1
NEGATIVE EXPONENT
an 
1
an
Proof:
a-n * an = a-n + n
= a0
=1
Hence,
an 
1
an
14