Answer There are two requirements for the unknown number: The tens digit has to be four more than the ones digit, and the two digits have to add up to 10. The best way to solve the problem is to eliminate answers that don’t meet these two requirements. Consider the second cri- terion first. A glance at the answers shows that the dig- its in the answers a and c do not add up to 10. They can be eliminated. Next, consider the first requirement. Answer b contains a tens digit that is six, not four, more than the ones digit. Answer d has the ones digit four more than the tens, reversing the requirement. Therefore, e is the only number that correctly meets the requirements. Practice with Definitions Match the word on the left with the description or example on the right. You may want to write these def- initions on flash cards. Answers Math 2: Numbers Working Together In the last lesson, you learned the definitions of several mathematical terms. This lesson will discuss ways in which numbers work together. You will need this information to solve simple algebra and perform cer- tain arithmetic functions that will be part of some CBEST problems. Adding and Subtracting Integers The following sample questions are examples of the types of problems about adding and subtracting you may see on the CBEST. The answers are given later in this lesson. Sample Integer Questions 1. Every month, Alice’s paycheck of $1,500 goes directly into her bank account. Each month, Alice pays $800 on her mortgage payment and $500 for food and all other monthly expenses. She spends $1,650 per year on her car (insur- ance, gas, repairs, and maintenance), $500 per year for gifts, and $450 per year for property tax. What will be her bank balance at the end of a year? a. $500 b. $300 c. 0 d. −$200 e. −$400 3. e. 4. c. 5. f. 6. a. 7. b. 8. g. 9. i. 10. h. 11. k. 12. j. 13. d. __ 3. integer __ 4. whole number __ 5. zero __ 6. negative integer __ 7. positive integer __ 8. digit __ 9. < __ 10. > __ 11. ≤ __ 12. ≥ __ 13. real number a. { −3, −2, −1} b. {1, 2, 3 . . .} c. {0, 1, 2, 3 . . .} d. number set includ- ing fractions e. {. . . −3, −2, −1, 0, 1, 2, 3 . . .} f. neither negative nor positive g. one numeral in a number h. greater than i. less than j. greater than or equal to k. less than or equal to –CBEST MINI-COURSE– 93 2. A submarine is submerged 2,000 feet below the sea. An airplane directly above is flying 32,000 feet above sea level. What is the difference in their altitudes? a. 30,000 feet b. 31,000 feet c. 32,000 feet d. 33,000 feet e. 34,000 feet Adding Positive Numbers You already know that if a positive number, such as 5, is added to another positive number, such as 7, the answer will turn out to be a positive number: 12. Adding Negative Numbers Suppose you had two negative numbers to add together, such as −5 and −7. Here are two methods that make working with negative numbers easier: ■ Number line: If you were to add −5 and −7 on a number line, you would start from 0. You would then proceed back five numbers to −5. From there, you would proceed back seven more (−7), which would leave you at the answer: −12. ■ Bank account: Negative numbers put you more in debt. If you started with a balance of zero and withdrew $5, you would have a balance of −5. If you withdrew $7 more, you would be in debt $12, or −12. Adding a Negative Number to a Positive Number When adding a positive number and a negative num- ber, such as 8 + (−15), you can use the number line or the bank account approach. ■ Number line: Start from 0. Go 8 to the right and then move 15 spaces to the left. This will leave you at the correct answer: −7. ■ Bank account: You put $8 in your bank account and take out $15. Oops, you’ve overdrawn, and you’re left with a balance of −7. Now try a problem with the negative number first: −6 + 7. ■ Number line: Start from 0. Go 6 to the left since 6 is negative, then 7 to the right. You’ll end up on the answer: 1. ■ Bank account: You take $6 out of your bank account and put in $7. You have one dollar more than you had before. HOT TIP When adding a positive and negative number, simply subtract the smaller from the larger as usual and give the answer the sign (− or +) of the larger number. HOT TIP When adding two negative numbers, just add the two numbers and place a negative sign before the answer. For instance, in the problem (−3) + (−6), first think 3 + 6 = 9, and then add a negative to the answer: −9. –CBEST MINI-COURSE– 94 Subtracting Negative Numbers Two minus signs next to each other may look strange, but they actually indicate one of the simplest opera- tions; two negative signs together can be changed into one plus sign. For instance, the problem 9 − (−) 7 will give you the same answer as 9 + 7. ■ Number line: From 0, go 9 to the right. The first minus would send the 7 to the left, but another negative sign negates that operation, so 7 ends up going to the right as though there had been no negatives at all. The answer is 9 + 7 or 16. ■ Bank account: You deposit $9 in the bank. A neg- ative would be a withdrawal, but there is another negative, so do a negative withdrawal; that is, deposit another $7. You have deposited a total of $16. Answers to Sample Integer Questions 1. d. One way to find this answer is to first figure out Alice’s monthly balance—what’s left after she pays her monthly expenses: $1,500 − ($800 + $500), which equals $200. Then multiply this by 12 months to get $2,400, which is her yearly bal- ance. Her yearly spending is $2,600 ($1,650 + $500 + $450). Her yearly balance minus her yearly spending will give you the answer: $2,400 − $2,600 = −$200. 2. e. The airplane’s altitude is a positive 32,000 feet above sea level, and the submarine is a negative 2,000 below the water. Finding the difference between the two looks like this: 32,000 − (−2,000). Even though the question asked for the difference, you need to add the two numbers, giving you a total distance of 34,000 feet. Multiplying and Dividing Integers Multiplying and dividing integers is not as complicated as adding and subtracting them. Two Positives When a positive number is multiplied or divided by a positive number, the result is positive: 4 × 50 = 200 and 50 ÷ 2 = 25. Two Negatives When two negative numbers are multiplied or divided, the result is positive: −5 ×−5 = + 25 and −36 ÷−6 = + 6. Commit this rule to memory. Negative and Positive When a negative number is multiplied by a positive number, the result is negative. Multiplying −3 by 6 means that six −3’s are added together, so of course you will get a negative answer, −18. On the number line, you will go back three units six times. In your bank account, you will end up owing $18 after $3 is removed from your bank account six times. HOT TIP Multiplication can be written three different ways and divi- sion two ways. ■ Multiplication: You can write 3 × 4, 3(4), and 3 · 4. When working with a variable, such as z, 3 × z may be written 3z, without parentheses. Additionally, a negative sign directly outside a parentheses means to multiply anything inside the parentheses by negative 1. For example: −(a) = −a, and −(xy) = −xy. ■ Division: You can write 30 ÷ 4, or ᎏ 3 4 0 ᎏ . Variables in division can come in various forms. It’s important to know these different forms when taking a multiple choice exam. For example, ᎏ 1 2 ᎏ y is the same as ᎏ 2 y ᎏ , not ᎏ 2 1 y ᎏ . HOT TIP When subtracting a negative number, change the two minus signs to an addition sign. –CBEST MINI-COURSE– 95 Order of Operations You may be given a problem with more than one oper- ation, like this: 3 + 5 × 2 2 + ᎏ (2 1 + 4 5) ᎏ If you simply work from left to right, your answer will be 140, and it will be wrong. But, be assured, 140 will probably be one of the multiple choice answers. The phrase below will help you remember the correct order of operations: Please Excuse My Dear Aunt Sally 1. P stands for Parentheses. In the above example, 2 + 5 is in parentheses. That operation should be done first: 3 + 5 × 2 2 + ᎏ 1 7 4 ᎏ . 2. E stands for Exponents. Those are the numbers that are smaller in size and higher on the page than the others and indicate the number of times to multiply the number by itself. Thus, 5 3 means 5 × 5 × 5. In the problem above, 2 2 means that 2 should be multiplied by itself twice: 2 × 2 = 4. So now the problem reads 3 + 5 × 4 + ᎏ 1 7 4 ᎏ . 3. M and D stand for Multiplication and Division. The next step is to do the multiplication and division from left to right: 3 + 20 + 2. 4. A and S stand for Addition and Subtraction. The last step is to add and subtract from left to right: 25. –CBEST MINI-COURSE– 96 __ 3. negative times or divided by a negative __ 4. negative times or divided by a positive __ 5. negative plus a negative __ 6. positive minus a negative __ 7. negative minus a negative __ 8. positive plus a negative __ 9. 8 + 3 (40 −10) −9 __ 10. 2 + 5 + 6 × ᎏ 4 3 ᎏ + 69 __ 11. −97 −(−8) __ 12. 100 + −11 __ 13. −21(−4) __ 14. 21(−4) __ 15. −21(4) __ 16. −14 − 70 a. 89 b. 84 c. 56 d. −20 e. −84 f. −89 g. positive h. negative i. Change the double negative to a positive and fol- low rule j below. j. Subtract one from the other and select the sign of the largest number. Practice Match the descriptions in the first column with an appropriate answer from the second column. The answers may be used more than once. Answers 3. g. 4. h. 5. h. 6. g. 7. i. 8. j. 9. a. 10. b. 11. f. 12. a. 13. b. 14. e. 15. e. 16. e. –CBEST MINI-COURSE– 97 Math 3: Rounding, Estimation, and Decimal Equivalents The questions on the CBEST will include number rounding, estimation, and decimal equivalents. Most teachers studying for the CBEST only need a very basic brush up on these topics in order to master them. If you need a more thorough review, check out some of the books listed in the “More Help” section at the end of this chapter. Rounding Numbers are made up of digits that each represent dif- ferent values according to their position in the num- ber. For instance, in the number 4,312.796, the 2 is in the ones place and equals 2 units. The 1 is in the tens place and equals 1 ten (10). The 3 is in the hundreds place and equals 3 hundreds (300). The 4 in the thou- sands place equals 4 thousands (4,000). To the right of the decimal, the 7 is in the tenths place and equals seven tenths (0.7 or ᎏ 1 7 0 ᎏ ). The 9 is in the hundredths place and equals 9 hundredths (0.09 or ᎏ 1 9 00 ᎏ ). The 6 is in the thousandths place and equals 6 thousandths (0.006 or ᎏ 10 6 00 ᎏ ). In a rounding question, you will be asked to round to the nearest tenths, hundreds, or other place. Sample Rounding Question 1. Round 4,312.986 to the nearest tenth. a. 4,310 b. 4,312.8 c. 4,312.9 d. 4,313 e. 4,312.98 Answer Find the answer by walking through the Success Steps. 1. The digit is 9, so it will either stay 9 or go to 0. 2. 8 is to the right of 9. 3. This step does not apply; 8 is not less than 5. 4. The 9 goes up one because 8 is more than 5. 5. Change 9 to 0 and change 2 to 3. The answer is d, 4,313. (Steps 6 and 7 don’t apply.) Practice Now try a few more rounding questions. 2. Round 45.789 to the nearest hundredth. 3. Round 296.45 to the nearest ten. 4. Round 345,687 to two significant digits. Answers 2. The digit you need to look at is 8; it will either stay 8 or go up to 9. The number 9 is to the right of 8; 9 is more than 5, so you change 8 to 9. The answer is 45.79. 98 1. Locate the place. If the question calls for rounding to the nearest ten, look at the tens place. Notice the place digit. Realize that the place digit will either stay the same or go up one. 2. Look at the digit to the right of the place digit—the right-hand neighbor. 3. If the right-hand neighbor is less than 5, the place digit stays the same. 4. If the right-hand neighbor is 5 or more, the place digit goes up one. 5. If the place digit is 9, and the right-hand neighbor is 5 or more, then turn the place digit to 0 and raise the left-hand neighbor up one. 6. If instead of a place, the question calls for rounding to a certain number of significant digits, count that num- ber of digits starting from the left to reach the place digit. Now start with step 1. 7. If the specified place was to the left of the decimal point, change all the digits to 0 that are to the right of the place digit. 3. The digit is 9; the 9 will either stay 9 or go to 0. The number 6 is to the right of 9; it is more than 5, so change 9 to 0 and apply step 5, raising the 2 to the left of 9 to 3. Now apply step 7, and change digits to the right of the tens place to 0. The answer is 300. 4. Here you need to apply step 6. Two places from the left is the ten thousands digit, a 4. Now apply step 1 and work through the steps: The digit is 4, so it will either stay 4 or go up to 5. Since the right-hand neighbor is 5, change the 4 to 5. Now apply step 7 and change all digits to the right of your new 5 to zero. The answer is 350,000. Estimation Estimation requires rounding numbers before adding, subtracting, multiplying, or dividing. If you are given numerical answers, you might just want to multiply the two numbers without estimation and pick the answer that is the closest. Most likely, however, the problem will be more complicated than that. Sample Estimation Question 5. 42 × 57 is closest to a. 45 × 60. b. 40 × 55. c. 40 × 50. d. 40 × 60. e. 45 × 50. Answer to Sample Estimation Question Here’s how you could use the steps to answer the sam- ple question: 1. You can round 42 down and 57 up, resulting in answer d. 2. Rounding the numbers to one significant digit yields d also. 3. Eliminate. a. Eliminated: 45 is further from 42 than is 40. b. Maybe. c. Eliminated: 50 is further from 57 than is 60. e. Eliminated: 45 is further from 42 and 50 is further from 57. 4. Check the remaining answers. The product of choice b is 2,200. The product of choice d is 2,400. The actual product is 2,394, which makes d the closest, and therefore the correct answer. Seven Success Steps for Rounding Questions . 10. > __ 11. ≤ __ 12. ≥ __ 13. real number a. { −3, 2, −1} b. {1, 2, 3 . . .} c. {0, 1, 2, 3 . . .} d. number set includ- ing fractions e. {. . . −3, 2, −1, 0, 1, 2, 3 . . .} f. neither. yearly spending will give you the answer: $2, 400 − $2, 600 = − $20 0. 2. e. The airplane’s altitude is a positive 32, 000 feet above sea level, and the submarine is a negative 2, 000 below the water. Finding. problem above, 2 2 means that 2 should be multiplied by itself twice: 2 × 2 = 4. So now the problem reads 3 + 5 × 4 + ᎏ 1 7 4 ᎏ . 3. M and D stand for Multiplication and Division. The next step is