Square Measuring the area and perimeter of a square is basi- cally the same as a rectangle, only the length and width are the same measurement. Area: side × side Perimeter: side × 4 Triangle Remember that a triangle is half a rectangle. Area: ᎏ 1 2 ᎏ b × h. Multiply the height and the base. Since the triangle is half, divide by two. Note: The height of a triangle is not always one of the sides. For example, in triangle ABC which fol- lows, side AB is not the height, BD is the height. AC is the base. To find the area, ignore all the numbers but the base and the height. The base can be found by adding 4 and 8: 4 + 8 = 12. The height is 5. ᎏ 1 2 ᎏ × 5 × 12 = ᎏ 6 2 0 ᎏ or 30. Perimeter: Add the sides all around: 12 + 6 + 9 = 27 Circle The diameter of a circle goes from one point on the circle, through the middle, and all the way across to another point on the circle. The radius (r) is half of the diameter. When working with π, consider the follow- ing: The symbol π is usually found in the answers so you don’t have to worry about converting it to a num- ber. But if π is not found in the answers, and the ques- tion calls for an approximate answer, substitute 3 for π. The question may tell you to use ᎏ 2 7 2 ᎏ or 3.14. Area: πr 2 . Square the radius and look at the answers. If π is not found in the answers, multiply by 3. In the example above, A = π4 2 or 16π. Circumference: 2πr. Circumference is to a circle what perimeter is to a rectangle. Multiply the radius by 2 and look for the answers. If π is not in the answer choices, multiply by 3. In the example above, the circumference is 2π4, or 8π. Other Areas Cut the figure into pieces, find the area of each, and add. If you’re asked to find the area of a figure with a piece cut out of it, find the area of the whole figure, find the area of the piece, and subtract. Other Perimeters For any perimeter, just add the outside lengths all the way around. Practice 8. Find the area of a circle with a diameter of 6. a. 36π b. 24π c. 16π d. 9π e. 6π –CBEST MINI-COURSE– 108 9. What is the area of the triangle above? a. 48 b. 9 c. 12 d. 18 e. 24 10. A box measured 5′′ wide, ᎏ 1 2 ᎏ ′ long, and 4′′ high. How many one-inch cube candies could fit in the box? a. 10 candies b. 60 candies c. 90 candies d. 120 candies e. 150 candies Answers 8. d. If the diameter is 6, the radius is 3. A = π(3) 2 = 9π. 9. c. 8 is the base, b, and 3 is the height, h. ᎏ 1 2 ᎏ × 8 × 3 = 12. 10. d. The words “one-inch cube” are there to throw you off; volume is always measured in one unit cubic space. Since ᎏ 1 2 ᎏ foot = 6 inches: 6 × 4 × 5 = 120. Math 6: Ratios, Proportions, and Percents Ratios and proportions, along with their cousins, per- cents, are reportedly on every CBEST. A good under- standing of these topics can help you pick up valuable points on the math section of the test. The Three-Step Ratio The three-step ratio asks for the ratio of one quantity to another. Sample Three-Step Ratio Question Use the three steps above to help you work out the fol- lowing problem. 1. Which of the following expresses the ratio of 2 yards to 6 inches? a. 1:3 b. 3:1 c. 1:12 d. 9:1 e. 12:1 Answer 1. One yard is 36 inches, so 2 yards is 72 inches. Thus, the ratio becomes 72:6. (The quantities can also be put in yards.) 2. This ratio can be expressed as ᎏ 7 6 2 ᎏ or 72:6. In this problem, 72:6 is the form that is used in the answers. 3. Since the answer is not there, reduce. 72 inches: 6 inches = 12:1. The answer is e. Notice that choice c, 1:12, is backwards, and therefore incorrect. Three Success Steps for Three-Step Ratios 1. Put the quantities in the same units of meas- urement (inches, yards, seconds, etc.). 2. Put the quantities in order and in the form given by the answer choices. 3. If the answer you come up with isn’t a choice, reduce. –CBEST MINI-COURSE– 109 Practice Try the three steps on the following problems. 2. Find the ratio of 3 cups to 16 ounces. a. 2:3 b. 3:16 c. 16:3 d. 3:1 e. 3:2 3. Find the ratio of 6 feet to 20 yards. a. 10:1 b. 6:20 c. 3:10 d. 20:6 e. 1:10 4. Find the ratio of 2 pounds to 4 ounces. a. 2:4 b. 2:1 c. 1:8 d. 8:1 e. 8:5 5. In a certain class, the ratio of children who pre- ferred magenta to chartreuse was 3:4. What was the ratio of those who preferred magenta to the total students in the class? Hint: Add 3 and 4 to get the total. a. 7:3 b. 3:4 c. 4:3 d. 3:7 e. 4:7 6. In a certain factory, employees were either fore- men or assembly workers. The ratio of foremen to assembly workers was 1 to 7. What is the ratio of the assembly workers to the total number of employees? a. 1:7 b. 7:1 c. 7:8 d. 8:7 e. 2:14 Answers 2. e. 3. e. 4. d. 5. d. 6. c. The Four-Step Ratio The four-step ratio solution is used when there are two groups of numbers: the ratio set, or the small numbers; and the actual, real-life set, or the larger numbers. One of the sets will have both numbers given, and you will be asked to find one of the numbers from the other set. HOT TIP ■ You can almost do question 2 by the process of elimination. You know that 3:16 or 16:3 can’t be right because the units haven’t been converted yet. You also know that test makers like to turn the cor- rect ratio around in order to try to catch you, so a or e must be the answer. If you know that 3 cups is more than 16 ounces, you have it made. ■ In question 3, b and c are the same quantity. There can’t be two right answers, so they can be elimi- nated. Why change yards into feet? Six feet is two yards. Reducing 2:20 makes 1:10. Aren’t these fun? –CBEST MINI-COURSE– 110 Sample Four-Step Ratio Question 7. The ratio of home games won to total games played was 13 to 20. If home teams won 78 games, how many games were played? Answer This problem can be solved in four steps. 1. Notice there are two categories: home team wins and total games played. Place one category over the other in writing. ᎏ T H o o t m al e ga w m in e s s ᎏ or ᎏ H T ᎏ Note: This step is frequently omitted by test takers in order to save time, but the omission of this step causes most of the mistakes made on ratio problems. If you reversed the H and T, put- ting T on top, that is not wrong, as long as you make sure to put the total games on top on both sides of the equation. 2. In the problem above, the small ratio set is com- plete (13 to 20), and you’re being asked to find the larger, real-life set. Work with the complete set first. Decide which numbers from the complete set go with each written category. Be careful; if you set up the ratio wrong, you will most proba- bly get an answer that is one of the answer choices, but it will be the wrong answer. Notice which category is mentioned first: “The number of HOME games won to TOTAL games played . . .”Then check to see what number is first: “. . . was 13 to 20.” Thirteen is first, so 13 goes with home games; 20 goes with the total games. ᎏ H T ᎏ = ᎏ 1 2 3 0 ᎏ 3. Determine whether the remaining number in the problem best fits home wins or total games. “If home teams won 78 games” indicates that the 78 goes in the home-team row. The number of total games played isn’t given, so that spot is filled with an x. ᎏ H T ᎏ = ᎏ 1 2 3 0 ᎏ = ᎏ 7 x 8 ᎏ 4. Now cross multiply. Multiply the two numbers on opposite corners: 20 × 78. Then divide by the number that is left (13). ᎏ 20 1 × 3 78 ᎏ = ᎏ 1, 1 5 3 60 ᎏ = 120 Practice Try the four steps on the following problems. HOT TIP After you cross multiply and wind up with one fraction, you can divide a top number and the denominator by the same factor and thus avoid long computations. In the above example, 13 ÷ 13 = 1 and 78 ÷ 13 = 6. The problem would then be much simpler: 20 × 6 = 120. Four Success Steps for Four-Step Ratios 1. Label the categories of quantities in the prob- lem to illustrate exactly what you’re working with. 2. Set up the complete set in ratio form. 3. Set up the incomplete set in ratio form. 4. Cross multiply to get the missing figure. –CBEST MINI-COURSE– 111 20 1 78 13 = 120 × 6 8. On a blueprint, ᎏ 1 2 ᎏ inch equals 2 feet. If a hall is supposed to be 56 feet wide, how wide would the hall be on the blueprint? a. 1 ᎏ 1 6 ᎏ b. 4 ᎏ 2 3 ᎏ c. 9 ᎏ 1 3 ᎏ d. 14 e. 18 ᎏ 2 3 ᎏ 9. In a certain recipe, 2 cups of flour are needed to serve five people. If 20 guests were coming, how much flour would be needed? a. 50 b. 30 c. 12 d. 10 e. 8 10. A certain district needs 2 buses for every 75 students who live out of town. If there are 225 students who live out of town, how many buses are needed? a. 4 b. 6 c. 8 d. 10 e. 11 Answers 8. d. 9. e. 10. b. Percents There are only five basic types of percent problems on the CBEST. These will be explained below. As is true with most other types of problems on the CBEST, per- cent problems most often appear in word-problem format. Percents can be done by using ratios or by alge- bra. Since ratios have just been covered, this section will explain the ratio method. Percents can be fairly simple if you memorize these few relationships: ᎏ o is f ᎏ , ᎏ w p h a o rt le ᎏ , ᎏ pe 1 r 0 ce 0 nt ᎏ . –CBEST MINI-COURSE– 112 Eight Success Steps for Solving Percent Problems Feel free to skip steps whenever you don’t need them. 1. Notice the numbers. Usually you are given two numbers and are asked to find a third. Are you given the whole, the part, or both ( ᎏ w p h a o r l t e ᎏ )? Is the percent given ( ᎏ pe 1 r 0 c 0 ent ᎏ )? Is the percent large or small? Is it more or less than half? Sometimes you can estimate the answer enough to elimi- nate some alien answers. 2. If there are pronouns in the problem, write the number to which they refer above the pronoun. 3. Find the question and underline the question word. Question words can include how much is, what is, find, etc. In longer word problems, you may have to translate the problem into a simple question you can use to find the answer. 4. Notice the verb in the question. The quantity that is by itself on one side of the verb is con- sidered the is. Place this number over the number next to the of ( ). If a question word is next to an is or of, put a variable in place of the number in that spot. If there is no is or no of, check to see whether one is implied. See whether you can rephrase the question, keep- ing the same meaning, but putting in the miss- ing two-letter word. If all else fails, check to make sure the part is over the whole. 5. Place the percent over 100. If there is no per- cent, put a variable over 100 ( ᎏ pe 1 r 0 c 0 ent ᎏ ). 6. Make the two fractions equal to each other. 7. Solve as you would a ratio. 8. Be sure to answer the question that was asked. is ᎏ of Sample Question: Finding Part of a Whole 11. There are 500 flights out of Los Angeles every hour. Five percent are international flights. How many international flights leave Los Angeles every hour? Answer 1. You are being asked to find a part of the 500 flights. The 500 flights is the whole. The percent is 5. You need to find the part. 5% is fairly small, and considering that 20% of 500 is 100, you know your answer will be less than 100. 2. The second sentence has an implied pronoun. The sentence can be rephrased “Five percent of them are international flights.” “Them” refers to the number 500. 3. The question is “How many ”Use the other sentences to reconstruct the question so it includes all the necessary information. The problem is asking “5% of 500 (them) are how many (international flights)?” The question is now conveniently set up. 4. “Are” is the verb. 500 and 5% are on the left side of the verb and “how many” is on the right side. “How many” is all by itself, so it goes on top of the ratio in the form of a variable. 500 is next to the of so it goes on the bottom. At this point, check to see that the part is over the whole. ᎏ 50 x 0 ᎏ = 5. The 5 goes over 100. ᎏ 50 x 0 ᎏᎏ 1 5 00 ᎏ 6. The two are equal to each other. ᎏ 50 x 0 ᎏ = ᎏ 1 5 00 ᎏ 7. Solve. 8. 25 international flights leave every hour. Sample Question: Finding the Whole 12. In a certain laboratory, 60%, or 12, of the mice worked a maze in less than one minute. How many mice were there in the laboratory? Answer Once again, follow the eight Success Steps to solving this problem. 1. 12 is part of the total number of mice in the lab- oratory. 60 is the percent, which is more than half. 12 must be more than half of the whole. 2. There are no pronouns. 3. The problem is asking, “60% of what number (total mice) is 12?” 4. “Is” is the verb. The 12 is all by itself on the right of the verb.“What number” is next to the of.The 12 goes on top, the variable on the bottom. ᎏ 1 x 2 ᎏ 5. The 60 goes over 100. ᎏ 1 6 0 0 0 ᎏ 6. The two fractions are equal to each other. ᎏ 1 x 2 ᎏ = ᎏ 1 6 0 0 0 ᎏ 7. Solve. 8. There were 20 mice in the laboratory. Sample Percent Question 13. Courtney sold a car for a friend for $6,000. Her friend gave her a $120 gift for helping with the sale. What percent of the sale was the gift? 3 1 == 20 12 100 60 × 12 100 60 × 4 5 500 1 5 100 == 25 × 500 5 100 × 5 –CBEST MINI-COURSE– 113 . the part is over the whole. ᎏ 50 x 0 ᎏ = 5. The 5 goes over 100. ᎏ 50 x 0 ᎏᎏ 1 5 00 ᎏ 6. The two are equal to each other. ᎏ 50 x 0 ᎏ = ᎏ 1 5 00 ᎏ 7. Solve. 8. 25 international flights leave every. the sale. What percent of the sale was the gift? 3 1 == 20 12 100 60 × 12 100 60 × 4 5 500 1 5 100 == 25 × 50 0 5 100 × 5 –CBEST MINI-COURSE– 113 . being asked to find a part of the 50 0 flights. The 50 0 flights is the whole. The percent is 5. You need to find the part. 5% is fairly small, and considering that 20% of 50 0 is 100, you know your answer