88 1. Read the graph carefully. Read all around the graph, including the title and the key. 2. Some questions may try to trick you by leaving out numbers. If all the numbers are not given, it is a very good idea to fill in all the missing numbers on the graph. To do this, you will need to know the value of each increment. 3. Sometimes, instead of reading bars or lines, you can compare differences by using a piece of your test book- let to measure from one point to another or from the end of one bar to the end of another. Line Graphs 3. In what year is the increase in student popula- tion projected to be less than the increase in number of new homes built? a. 1998 b. 1999 c. 2000 d. 2001 e. 2002 Answer The answer is c. A look at the graph shows that during the year 2000 there was a sharper increase in the num- ber of new homes built than in student population. The line slopes up steeper there for houses than it does for student population. Percent of increase is a differ- ent question and might yield a different answer. Check Ratios, Proportions, and Percents (p. 109) on percents for details. Picture Graphs 4. How many MTAC members were there in 1990? a. 3 ᎏ 1 2 ᎏ b. 350 c. 700 d. 1,750 e. 2,250 Answer It is important to read the key at the bottom of the graph. Each piano represents 500 members. ᎏ 1 2 ᎏ a piano represents 250 members. 1990 has 3 ᎏ 1 2 ᎏ pianos. This rep- resents 1,750 members. The answer is d. Three Success Steps For Working With Graphs Circle Graphs 5. For which category was the income for the Save the Caves Foundation approximately ᎏ 1 4 ᎏ ? a. Mailings b. Sales c. Thrift Shop d. Phone e. Other Answer Look for the section that takes up approximately ᎏ 1 4 ᎏ of the circle. The answer is a. Here’s a different kind of circle graph question. 6. Which of the following could include a represen- tation of expenditures of 25% and 33%? Answer 25% is the same as ᎏ 1 4 ᎏ . 33% is close to ᎏ 1 3 ᎏ . The only answer with areas of ᎏ 1 4 ᎏ and ᎏ 1 3 ᎏ of the circle is answer d. Choice b is close, but the larger area seems a little large for ᎏ 1 3 ᎏ . e. d. c. b. a. –CBEST MINI-COURSE– 89 Oddballs Some graphs are just plain odd. 7. The circles above represent the dials on an elec- tric meter. What reading do they represent? a. 476 b. 465 c. 466 d. 486 e. 487 Answer There are no clues whatsoever on the graphs. A look at the answer choices reveals that all are three digits and that all begin with the digit 4. Maybe each circle rep- resents a digit. If so, the first dial has to represent 4. If the first dial represents 4, and the arrow is nearly half- way around (assuming the dials go clockwise), then half is probably 5. In that case, the last circle represents 5, which tells you your answer, choice b. It makes sense that the middle digit of 465 is 6, since the middle cir- cle’s arrow goes just a bit past the 5 mark. In order to correctly answer this question, it was important that you didn’t assume the graphs represent clocks. Ques- tions similar to this have been on the test before. All right, here’s a challenge. 8. Which number best represents the speed indi- cated on the speedometer above? a. 12 ᎏ 1 2 ᎏ b. 30 ᎏ 1 2 ᎏ c. 35 d. 40 ᎏ 1 2 ᎏ e. 45 Answer It is very important to label the graph in order to answer this question. If there are 11 segments between 10 and 120, each segment must represent 10 mph. The arrow is pointing halfway between 30 and 40. Halfway between 30 and 40 is 35. The answer is c.  Math 1: Words, Words, Words Many times, an otherwise simple problem may seem difficult merely because the test writers have used terms that you are not familiar with. An understand- ing of mathematical terms will enable you to under- stand the language of the problem and give you a better chance of solving that problem. This lesson presents a list of words used when speaking about numbers. Here is a sample question that will show you why knowing these words is important. Once you’ve learned the words in this sample question, you should be able to answer it.You can check your answer against the explanation given later in this lesson. –CBEST MINI-COURSE– 90 Sample Definition Question 1. If x is a whole number and y is a positive integer, for what value of x MUST x < y be true? a. −3 b. 0 c. ᎏ 1 2 ᎏ d. 3 e. Any value of x must make the statement true. Definitions Below are definitions of words you need to know to answer CBEST math questions. If some of these words are unfamiliar, put them on flash cards: the word on one side, and the definition and some examples on the other side. Flash cards are handy because you can carry them around with you to review at odd moments dur- ing the day. If you use this method, it won’t take you long to learn these words. Integer An integer is simply a number with no fraction or dec- imal attached {. . . −2, −1, 0, 1, 2 . . .}. Integers include both negative (−5) and positive (9,687) numbers. Zero is also an integer, but is considered neither negative nor positive in most mathematical texts. Positive Integer A positive integer is an integer, according to the defi- nition above, that is greater than zero. Zero is not included. Positive integers begin with 1 and continue infinitely {1, 2, 3 . . .}. Examples of positive integers are 5, 6,000 and 1,000,000. Negative Integer A negative integer is an integer that is less than zero. Zero is not included. The highest negative integer is negative one (−1). The negative integers go down infi- nitely {. . . −3, −2, −1}. Some examples of negative inte- gers are −10, −8, −1,476. The following numbers do not fit the definition of a negative integer: −4.5 (not an integer because of the decimal), 0, and 308. Zero Zero is an integer that is neither positive nor negative. Whole Numbers Whole numbers include all positive integers, as well as zero {0, 1, 2, 3 . . .}. Like integers, whole numbers do not include numbers with fractions or decimals. Digit A digit is a single number symbol. In the number 1,246, each of the four numerals is a digit. Six is the ones digit, 4 is the tens digit, 2 is the hundreds digit, and 1 is the thousands digit. Knowing place names for dig- its is important when you’re asked on a test to round to a certain digit. Rounding will be covered in the third math lesson. Real Numbers Real numbers include all numbers: negative, positive, zero, fractions, decimals, most square roots, and so on. Usually, the numbers used on the CBEST will be real numbers, unless otherwise stated. HOT TIP Negative numbers appear smaller when they are larger. To help you make sense of this concept, think of the degrees below zero on a thermometer. Three degrees below zero is hotter than 40 degrees below, so −3 is greater than −40, even though 40 appears to be a larger number. Test mak- ers like to test your grasp of this principle. –CBEST MINI-COURSE– 91 Variables Variables are letters, such as x and y, that are used to replace numbers. The letter is usually a letter of the alphabet, although occasionally, other symbols are used. When a math problem asks you to “solve for y,” that means figure out what number the letter is replac- ing. At other times, the problem requires you to work with the letters as if they were numbers. Examples of both of the above will be covered in the lesson on algebra on page 114. Reciprocal The reciprocal of a fraction is the fraction turned upside down. For example, the reciprocal of ᎏ 3 8 ᎏ is ᎏ 8 3 ᎏ and vice versa. The reciprocal of an integer is one over the integer. For example, the reciprocal of 2 (or ᎏ 2 1 ᎏ ) is ᎏ 1 2 ᎏ and vice versa. To get the reciprocal of a mixed number such as 3 ᎏ 1 2 ᎏ , first change the number to an improper fraction ( ᎏ 7 2 ᎏ ) and then turn it over ( ᎏ 2 7 ᎏ ). Numerator and Denominator The numerator of a fraction is the number on top, and the denominator is the number on the bottom. The numerator of ᎏ 6 7 ᎏ is 6 and the denominator is 7. = and  The symbol = is called an equal sign. It indicates that the values on both sides of the = are equal to each other. For example, 7 = 2 + 5.A line drawn through an equal sign (≠) indicates that the values on either side are not equal: 8 ≠ 4 + 5. Ͻ and Ͼ; Յ and Ն The symbol < means less than and the symbol > means greater than. The number on the closed side of the symbol is smaller, and the number on the open side is larger. Thus, 3 < 5 and 10 > 2. Remember: The alliga- tor eats the bigger number. The symbol ≤ means less than or equal to, and the symbol ≥ means greater than or equal to. These two symbols operate the same way as the < and >, but the added line means that it’s possible that the two sides are equal. Thus, in the equation x ≥ 3, x can represent 3 or any number greater than 3. Answer to Sample Definition Question Using the definitions above, can you solve sample question 1 from the previous page? The variable x can be any whole number including zero. The variable y can be any positive integer, which doesn’t include zero. The question reads “ for what value ofx MUST x < y be true?” Must means that x has to be less than y under all circumstances. You are being asked to replace x with a number that will be less than any positive inte- ger that replaces y. The only whole number that would make x < y true, no matter what positive integer is put in place of y, is zero. Therefore, b is the correct answer. Try another sample question. Again, the defini- tions above will be useful in solving this problem. Sample Digits Question 2. In a certain two-digit number, the tens digit is four more than the ones digit. The sum of the two digits is ten. What is the number? a. 26 b. 82 c. 40 d. 37 e. 73 –CBEST MINI-COURSE– 92 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 . 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. above? a. 12 ᎏ 1 2 ᎏ b. 30 ᎏ 1 2 ᎏ c. 35 d. 40 ᎏ 1 2 ᎏ e. 45 Answer It is very important to label the graph in order to answer this question. If there are 11 segments between 10 and 12 0, each. piano represents 500 members. ᎏ 1 2 ᎏ a piano represents 250 members. 19 90 has 3 ᎏ 1 2 ᎏ pianos. This rep- resents 1, 750 members. The answer is d. Three Success Steps For Working With Graphs Circle