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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 mn 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 an 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 an 1 an Proof: a-n * an = a-n + n = a0 =1 Hence, an 1 an 14