Basic Math Exercises: Mastering Order of Operations

BASIC MATH EXERCISES

Section A Order of Operations You will first learn about order of operations. In mathematics, some operations take prece- dence over others (i.e., some operations should be performed before others). If you perform them in the wrong order, you might get wrong answers. Study the following material on order of operations, and then complete the exercise for Sec- tion A. With your professor's help, determine if any of your answers are incorrect. If so, go back and restudy the relevant material in this section.

Section A Order of Operations In statistics, you will use the four basic operations: addition, subtraction, multiplication, and division. Keep in mind that: (3)(7) means "3 multiplied by 7." 6/2 means "6 divided by 2." The result of addition is known as the sum. The result of multiplication is known as the product. The result of subtraction is known as the difference. The result of division is known as the quotient. ✓ Rule One: When there are parentheses, perform the operations inside the parentheses first. Here are three exampl.es: (4)(2 + 1) =? Thus, (4)(3) = 12 (4-3)/(5-4)=? Thus, 1/1 = I (9)(7 + 3 - 2) =? Thus, (9)(10 - 2) =? and (9)(8) = 72 ✓ Rule Two: Unless parentheses indicate otherwise, multiply and divide before adding and sub- tracting. In these examples, you must multiply before adding: 5 + (3)(2) =? Thus, 5 + 6 = 11

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8+(5)(10)+1=? Thus, 8 + 50 + I = 59 In these examples, you must divide before subtracting: 6- 2/2 =? Thus, 6- I = 5 20 - 36/6 -5 =? Thus, 20 - 6 - 5 = 9 In these examples, you must multiply and divide before adding and subtracting: (5)(2) - 1 + 4/2 = ? Thus, IO - 1 + 2 = 11 11 +(10)(3)-4/2=? Thus, 11 + 30 - 2 = 39 ✓ Rule Three: If there are both parentheses and brackets, solve first within the parentheses, then within the brackets, and then perform any remaining operations. Study these examples: [(2)(2) -(5 -3)][7 - I]=? Thus, [4 -2][6] =? and [2][6] = 12 [(25)(10 -5)]/5 =? Thus, [(25)(5)]/5 =? and 125/5 = 25

Exercise for Section A I . ( l 0)(3 + 5)= ? 2. (5+ 4)/(2+ I)=? 3.(6 - 5 + 2)(5) =? 4.8+(5)(4)=? 5. ( I 0)( 11) - 1 = ? 6. 5 + 12/4 = ? 7.10+(2)(5)-5=? 8. 25 - (9)(2) + 3 =? 9. [ (4 + 7)(3 - 1) ][ 8 - 3 ] = ? 10. [(3 + 5) + (1)(2) ]/ 2 =? 11. The result of multiplication is known as the A. product. B. quotient. C.sum. D. difference. 12. The result of addition is known as the A. product. B. quotient. C. sum. D. difference. 13. (4 + 6)(11)=? 14. (7 - I + 2)(4) =? 15. 20/(5 + 5) =? 16. 9 + 8/2 =? 17. (12)(12)-3 =? 18. 9+(4)(8)=? 19. 15/3 + 5 - 6/2 = ?

20.19+(2)( 6 ) -4= ? 21. [(5 - 3)+( 2)(3)]/8 =? 22. [(9 + 3) - (1)(3)][6- 2 ] =? 23. The result of division is known as the A. product. B. quotient. C. sum. D. difference. 24. The result of subtraction is known as the A. product. B. quotient. C. sum. D. difference. 25. (8)(3) + 7 - 12/4 =?

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Section B Squares and Square Roots The second essential is the knowledge of squares and square roots. Working with these is very uncommon in everyday life, so if it has been a long time since you have taken a math class, your knowledge of squares and square roots will probably be rusty. In contrast, squares and square roots are widely used in the world of statistics. Thus, you would be well advised to master them now before you begin your study of statistics.

Section B Squares and Square Roots In statistics, you will be squaring many numbers. To square a number, multiply it by itself, as in this example: Note that in the example, 4 is the base and 2 is the exponent. CALCULATOR HINT To square a number using a calculator, you only have to enter the number once. For example, on your calculator: I. Press 4. 2. Press the 1:1u1tiplication sign(x). 3. Press the equals sign(=). 4. You should see 16 displayed, which is the square of 4. Calculating a square root is the opposite of squaring. Thus, the square root of 16 is 4. To cal- culate a square root on a calculator, enter the number and then press the square root sign ( ✓ ). The formal name of the square root sign is the radical sign. Notice that -✓16 is read the square root of 16, which is 4. When a square root sign (i.e., a radical sign) appears in a formula, it has the same effect as parentheses in the order of operations. That is, everything under a radical sign must be solved and the square root calculated before any operations on its value are performed. For example, the following tells us to multiply 4 by the square root of the sum under the radical sign. 4 ✓ 20 + 5 =?

Because of the radical sign, you must add 20 and 5 and take the square root of the sum before multiplying by 4: Thus, 4 ✓25 =? and (4)(5) = 20 Notice that 4 2 is read/our squared, while ✓4 is read the square root of 4. Exercise for Section B 1. 14 2 is read A. 14 squared. B. the square root of 14. C. double 14. 2. The "14" in · Question l is known as the A. base. B. exponent. C. radical sign. 3. ✓9 is read as A. 9 squared. B. the square root of 9. C. half of 9. 4. The ✓ sign is known as the A. base. B. exponent. C. radical sign. 5. What is the square root of 144? 6. What is the square of l 00? 7. ✓ 225 =? 9. 6 .J l 00 - 19 = ?

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10. ✓32 4 + 2(11 -2) =? 11. Ji9 is read A. 49 squared. B. the square root of 49. C. double 49. 12. In 9 2 , the "2" is known as the A. base. B. exponent. C. radical sign. 13. 67 2 is read as A. 67 squared. B. half of 67. C. the square root of 67. 14. The "2" in Question 13 is known as the A. base. B. exponent. C. radical sign. 15. What is the square root of 81 ? 16. What is the square of 60? 17. ✓169 =? 19.4 ✓ 55-6=? 20. ✓ 250 + 6 /(3 + 5) =?

Section C Negatives Besides activities involving banking and weather, most people do not work with nega- tives in their everyday lives. However, negatives are widely found in statistics.

Section C Negatives We frequently use negative numbers in statistics. However, because we mainly use positive numbers in everyday situations, your knowledge of negative numbers and how to work with them may be rusty, so we will sta1i with the basics. First, you may recall that a number line looks like this: -5 --4 -3 -2 -1 0 + 1 +2 +3 +4 +5 Negative numbers are to the left of zero. When a number is shown without a sign, it is understood to be positive. Thus, "3" is under- stood to be "+3." When a number is negative, its sign is always shown. When working with negatives, follow these rules: ✓ Rule One: When you multiply or divide two numbers with different signs, the result is always negative. Here are two examples: When you multiply a positive number by a negative number, the result (product) is negative. Thus: (5)(-2) = -10 and (-5)(2) = -10 When you divide using a positive and a negative number, the result ( quotient) is negative. Thus: 10/-2=-5 and -10/2=-5 ✓ Rule Two: When you multiply or divide two numbers with the same sign, the result is al- ways positive, as in these examples:

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- 50 / - 5 = 10 (-3)(-9) = 27 and 50/10 = 5 and (3)(9) = 27 ✓ Rule Three: When you add a set of numbers, all of which are negative, the result (sum) is negative, as in these examples: -1 + (--4) + (-3) = -8 -10 + (-1 1) = - 2 1 ✓ Rule Four: When you add a positive number and a negative number, temporarily ignore the signs and subtract the smaller number from the larger number. Then assign to the sum the sign of the larger number. Study this example: 3 + (-7) =? First, ignoring signs, 7 - 3 = 4. Because 7 is larger than 3 and because the 7 is negative, the sum is negative. Thus, the answer is --4. To understand this example, it helps if you refer to the num- ber line at the beginning of this section. If you start at + 3 and add to that 7 points in the negative direction ( moving left from + 3 seven points), you come to -4, which is the answer. Here is another example: 15 + (-3) =? First, subtract the smaller number from the larger number: 15 - 3 = 12 . The answer is a posi- tive 12 because the larger number is positive. ✓ Rule Five: If there are some positive and some negative numbers, all of which are to be summed, first add all the positive numbers to get their sum. Then add all the negative numbers using Rule Three to get their sum. Then add the two sums using Rule Four. For example: . .4 + (-2) + 5 + 6 + 6 + (-1) + 7 + (-3) =? First, sum the positive numbers: 4 + 5 + 6 + 6 + 7 = 28 Then, sum the negative numbers using Rule Three:

-2 + ( - 1) + (-3) =-6 Then, sum the two numbers using Rule Four: 28-6 = 22 The answer is +22 because the larger number is positive. ✓ Rule Six: When you subtract a negative from a negative, the number to be subtracted be- comes a positive. Temporarily ignore the signs and subtract. Then assign the sign of the larger number. For example: -10-(-2)=? The 2 becomes positive. -10+ 2=? Ignoring the signs,10 - 2 = 8. Because 10 is larger than 2 and 10 was originally negative, the answer is negative. Thus, the answer is -8. Note that when , we ignore the sign of a number, we are using its absolute value-its value without regard to the sign. For example, +8 and -8 have the same absolute value; if you take away their signs, they are the same. ✓ Rule Seven: When you subtract a negative from a positive, the negative number becomes a positive. Add the two numbers to get the difference. For example: 5-(-4)=? Thus,5+ 4=9. At first, this rule may be confusing. It helps to think about it as though it were a double nega- tive (for example, - -4) in English. If someone says, "We are not (negative) sure that we are not (negative) going," that person is making the positive statement that he/she, in fact, might be go- ing. Another way to look at Rule Seven is to consider the following number line. It illustrates that the difference between +5 and -4 is 9. (Remember that when you subtract, you are getting a dif- ference.)

--4 -3 -2 -1 0 +l +2 +3 +4 +5 ✓ Rule Eight: When you subtract a positive from a negative, temporarily ignore the signs and add. The result is negative. For example: -8-7 = ? First, 8 + 7 = 15 The answer is -15. CACULA TOR HINT Most calculators have a± button. Use this when you enter negative numbers. Here is an example: To solve (5)(-2) =? 1. Press 5. 2. Press the multiplication sign (x). 3. Press 2 ( enter it as positive). 4. Press the± button (this will make the 2 negative). 5. Press = (the equals sign). 6. The display should show -10, which is the answer. Pressing the ± button informs the calculator that the most recently entered number is a negative number.

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Exercise · for Section C Suggestion: Before you solve these problems with a calculator, mentally estimate whether the answer to each question wiB be positive or negative. 1. (-8)(7) = ? 2. (4)(-50)=? 3. 49/-7 =? 4.-121/11 =? 5. -36/-6 =? 6. (-12)(-10) =? 7. -5 +(-9) + (-6) =? 8. 5 +(-9) =? 9. -12 + 8 =? 10.-9+8+(-3)+ 1 =? 11. 3 +(-5) + 12 + (-2) =? 12. -29-(-14) =? 13. -4 7 -(-1 7) =? 14. 10-(-9)=? 15.12 -(-21)=? 16. -20 - 18 = ? 17. The absolute value of a number is its value A. when it is negative. B. without regard to its sign. C. with its sign. 18. (-9)(12) =?

19. (15)( - 7) = ? 20. -66/11 =? 21. 169/-13 =? 22. -12/-3 =? 23. (-15)(-14) =? 24. -8 + (-7)+ (-9) =? 25. 10+(-13)=? 26. -17 +9 =? 2 7. 5 + (- 7) + (-6) + 4 = ? 28. 8+(-2)+9+(-l)=? 29. -30-(-29) =? 30.-17-(-18)=? 3 l. 20 - (-11) = ? 32. 14 - (-18) =? 33. -40 - 11 =? 34. How does "-18 - 14=?" read? A. Minus 18 minus negative 14= ? B. Negative 18 minus negative 14 =? C. Negative 18 minus positive 14 =?

Section D Decimals and Rounding In everyday life, it is common to use to some basic fractions, such as one qua11er, one half, and two and a third (for example, in a recipe). To report precise results scientific studies decimal equivalents are usually preferred over fractions. For a simple fraction, the decimal equivalent is calculated by dividing the numerator by the denominator.

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For a simple fraction, the decimal equivalent is calculated by dividing the numerator by the denominator. As you probably recall: 3.2 is three and two-tenths. 3.02 is three and two-hundredths. 3.002 is three and two-thousandths. It is customary in statistics to report answers to the hundredths' or thousandths' place. In in-formal discussions, we sometimes refer to 3.02 as three point zero two and to 3.002 as three point zero zero two. Sometimes a number is less than one, such as: 0.56, which is zero and 56 hundredths. The number .56 is also zero and 56 hundredths; the zero is implied. It is a good idea to show the zero, however, because it helps draw your readers' attention to the decimal point and the fact that the value is less than one. Modern calculating devices have eliminated the need to keep track of decimal places when computing. However, these devices have not eliminated errors in entering the data, which lead to incorrect answers. Frequently, when you divide using a calculator, you will obtain a long string of decimal places, such as: 7.7/3 = 2.566666666666 You will want to round this. To the nearest hundredth, it rounds to 2.57; to the nearest thousandth, it rounds to 2.567. Review this terminology: 10.366 = 10.37 is an example of rounding up. 10.361 = 10.36 is an example of rounding down. A special rule for rounding off five that is often used in statistics is: All numbers ending in five or higher - as you drop the five, round up by adding a number to the previous. Example: 10.335 = 10.35 ( round up ).

All number ending in four or less - drop the four ( round down ) Example : 10.334 = 10.33 ( round down) . If you are not comfortable with this special rule, study these examples rounded to the hundredth. 113.485 = 113.49 (round up because 9 is higher than 4). 27.655 = 27.66 (round up because 6 is higher than 4). 9.1135 = 9.11 (round down because the 3 is 4 or less). 896.842 = 896.84 (round down because the 2 is 4 or less). There are many multi-step problems in statistics. That is, after you perform one computation, you will use the answer in the next computation; the second answer will be used in the third computation, and so on. At the end of each step, how many places should you round your answer to? Here are two general rules to guide you: The more decimal places you retain at the end of each step, the more accurate your answer will be. At the end of each step in a problem, retain at least one more decimal place than you will be reporting in the answer to the final step. For example, if you plan to report the answer to the final step in a problem to two decimal places, keep at least three in all steps leading to that answer. Then round the final answer to two decimal places. If you, your instructor, and your classmates keep a different number of decimal places at the end of each step in a problem, you may obtain slightly different answers. Slight differences in the hundredths' place are usually not of concern for all practical purposes- as long as they are attributable to rounding and not to an error.

Exercise for Section D 1. 5.24 is read as five and twenty-four tenths. TRUE or FALSE? 2. 16.1 is read as sixteen and one-tenth. TRUE or FALSE? 3. It is customary in statistics to report an answer to the nearest whole number. TRUE or FALSE? 4. When rounded to a whole number, 5.189 becomes 5. TRUE or FALSE? 5. The answer to 100.0812/9.8746 should be close to 10. TRUE or FALSE? 6. The answer to (19.953)(5.162) should be close to 20. TRUE or FALSE? 7. When rounded to the nearest hundredth, 6.4734 becomes 6.473. TRUE or FALSE? 8. When rounded to the nearest thousandth, 7.5766 becomes 7.577. TRUE or FALSE? 9. When rounded to the nearest tenth, 15.657 becomes 15.7. TRUE or FALSE? 10. If the special rule is applied, 10.25 becomes 10.3 when rounded to the nearest tenth. TRUE or FALSE? 11. If the special rule is applied, 111.665 becomes 111.66 when rounded to the nearest hun- dredth. TRUE or FALSE? 12. If the special rule is applied, 15.5555 becomes 15.556 when rounded to the nearest thou- sandth. TRUE or FALSE? 13. 6.12 is read as six and twelve-hundredths. TRUE or FALSE? 14. 7.101 is read as seven and one hundred and one thousandths. TRUE or FALSE? 15. It is customary in statistics to report answers to the nearest hundredth or thousandth. TRUE or FALSE? 16. When rounded to a whole number, 3.119 becomes 4. TRUE or FALSE?

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17. The answer to 10.2 plus 40.18129 should be · close to 50. TRUE or FALSE? 18. The answer to 24.98762/4.823423 should be close to 5. TRUE or FALSE? 19. When rounded to the nearest tenth, 8.129 becomes 8.2. TRUE or FALSE? 20. When rounded to the nearest hundredth, 121.564 becomes 121. TRUE or FALSE? 21. When rounded to the nearest thousandth, 14.47451 becomes 14.475. TRUE or FALSE? 22. If the special rule is applied, 5.545 becomes 5.54 when rounded to the nearest hundredth. TRUE or FALSE? 23. If the special rule is applied, 19 .445 becomes 19 .45 when rounded to the nearest hundredth. TRUE or FALSE?

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Section E Fractions As it turns out, it is usually much easier to perform simple math on decimal equivalents than on fractions due to the need to have common denominators for fractions. For example, it is easier to add 0.14 to 0.12 (0.14 + 0.12 = 0.26) than to add 1/7 to 1/8 because one must find a common denominator when adding fractions.

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Section E Fractions Many formulas in statistics contain fractions. In order to understand these formulas, it is es- sential that you know the basics about fractions, which are illustrated here by example: In 1/6, 1 is the numerator and 6 is the denominator. 1/6 may be represented as follows. The whole consists of six equal parts, and one of them (1/6) is shaded: I As the numerator increases, the value of the fraction increases. For example, 2/6 is greater than 1/6. Here, 2/6 of the area is shaded: As the denominator increases, the value of the fraction decreases. For example, 1/12 is less than 1/6. This represents 1/12: When performing calculations involving fractions, follow these rules: ✓ Rule One: Use your calculator to convert fractions to their decimal equivalents, and solve problems using the decimal equivalents. Suppose you have to multiply 29 by 1/4: To solve (29)( 1/4) = ? First, convert 1/4 to its decimal equivalent by dividing 1 by 4, which equals 0.25. Then multiply on your calculator: (29)(0.25) = 7.25. I I II

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✓ Rule Two: If a formula specifies operations in the numerator and/or denominator, perform these operations before converting to a decimal equivalent. l + 5 To solve --- (6)(2) First, perform the addition in the numerator and the multiplication in the denominator to get: 6 12 Then divide 6 by 12 to get 0.50. Note that in statistics, you should report decimal equivalents and not fractions as answers. Thus, you should report 0.50 and not 6/12 or 1/2. If a number consists of a whole number and a fraction, it is known as a mixed number. For example: 5 1/7 is a mixed number that consists of 5 and 1/7. Before working with mixed numbers, convert the fractional part to its decimal equivalent. For example, to convert 5 1/7 to its decimal equivalent, do this: Divide 1 by 7 = 0.14. Add it to the whole number: 5 + 0.14 = 5.14. Thus, 5 1/7 equals 5.14.

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Exercise for Section E 1. In 3/4, 3 is the denominator. TRUE or FALSE? 2. In 3/4, 3 is the numerator. TRUE or FALSE? 3. In 5/6, 6 is the denominator. TRUE or FALSE? 4. Other things being equal, as the numerator increases, the value of the fraction decreases. TRUE or FALSE? 5. Other things being equal, as the denominator increases, the value of the fraction decreases. TRUE or FALSE? 6. What is the decimal equivalent of 2/5? 7. What is the decimal equivalent of 1/12? 8. What is the decimal equivalent of 3/4? 9. In 7/8, 7 is the numerator. TRUE or FALSE? 10. In 7 /8, 7 is the denominator. TRUE or FALSE? 11. In 1/4, 4 is the numerator. TRUE or FALSE? 12. What is the decimal equivalent of 10/100? 13. What is the decimal equivalent of 1/3? 14. What is the decimal equivalent of 1/2? 15. What is the decimal equivalent of 7 + 3 ? (4)(6) 16. What is the decimal equivalent of 14 5/8?

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Section F Percentages Percentages are 100 times ' greater than their decimal counterparts. For example, 0.01 is the decimal equivalent of 1 %. As you can see, it is much easier to communicate using percentages than it is using decimals.

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pute the corresponding percentage: 35 divided by 79 = 0.443 x 100 = 44.3% and report that 44.3% (N = 35) are liberals. N stands for number of subjects or cases. It is desirable when reporting percentages to also report the number of cases underlying each percentage. The number of cases is an important piece of information for your reader, as these examples illustrate: Study A: 80% (N = 5) of the dentists recommend Brand X. Study B: 80% (N = 103) of the dentists recommend Brand X. Even though the percentages are the same in the studies, the larger N in Study B indicates that its results are more reliable than the results of Study A. The percentages for a group may not always sum to exactly 100%, as illustrated by the fol- lowing examp.le in Table 1. There were no computational errors. Instead, the total of 100.1 % is slightly greater than I 00% because of rounding. For example, for Method A, the precise percent- age is 18.75%. Because this was rounded to 18.8%, a slight amount of error was introduced. For most practical purposes, this error is of little consequence. Note that in the last column, the up- percase P is used, which is a symbol for percentage. Table 1 Number and Percentage of Teachers Who Prefer Each Method Mcthod N P Method A Method B Method C TOTAL 9 18 21 48 18.8% 37.5% 43.8% 100.1 %

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If you still do not feel comfortable computing percentages, consider the example of the teachers more carefully: For Method A, first divide 9 by 48 = 0.1875. Then multiply by 100, 0.1875 x 100 = 18.75%, which rounds to 18.8%. For Method B, first divide 18 by 48 = 0.3750. Then multiply by 100, 0.3750 x 100 = 37.50%, which rounds to 37.5%. For Method C, first divide 21 by 48 = 0.4375. Then multiply by 100, 0.4375 x 100 = 43.75, which rounds to 43.8%. Exercise for Section F 1. 25% stands for 25 out of 50. TRUE or FALSE? 2. The base for percentages is 100. TRUE or FALSE? 3.What percentage corresponds to 1/5? 4. What percentage corresponds to 3/1 0? 5. If 95 subjects were studied and 71 of them expressed Opinion A, what percentage of them ex- pressed Opinion A? 6. If 200 subjects were studied and 120 of them expressed a preference for Brand X, what per- centage expressed a preference for Brand X? 7. It is recommended that when you report a percentage, you should also report the correspond- ing value of N. TRUE or FALSE? 8. In statistics, the lowercase p stands for percentage. TRUE or FALSE? 9. ll¾standsfor 11 out of 100. TRUE or FALSE?

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10. By employing a common base of 100, percentages facilitate the comparison of groups of un- equal size. TRUE or FALSE? 11. What percentage corresponds to 2/5? 12. What percentage corresponds to l/100? 13. If 66 subjects were studied and 21 of them expressed Opinion X, what percentage of them expressed Opinion X? 14. If 462 subjects were studied and 99 of them expressed a preference for Brand Y, what per- centage expressed a preference for Brand Y? 15. When writing a scientific rep01i, there is no need to report Ns if percentages are being re- ported. TRUE or FALSE? 16. In statistics, the uppercase P stands for percentage. TRUE or FALSE?

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