Success step english 6 doc

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 Answer 1. 6,000 is the whole and 120 the part. 2. There are no pronouns, but there are words that stand for numbers. In the question at the end, the sale is 6,000 and the gift is 120. 3. The question is written out clearly: “What percent of 6,000 (sale) was 120 (gift)?” 4. “Was” is the verb. 120 is by itself on one side. It is the part, so it goes on top. 6,000 is near the of and is the whole, so it goes on the bottom. ᎏ 6 1 ,0 2 0 0 0 ᎏ 5. There is no percent so x goes over 100. ᎏ 10 x 0 ᎏ 6. The two equal each other. ᎏ 6 1 ,0 2 0 0 0 ᎏ = ᎏ 10 x 0 ᎏ 7. Solve. 8. The gift was 2% of the sale. Sample Percent Change Question A change problem is a little bit different than a basic percent problem. To solve it, just remember change goes over old: ᎏ ch o a l n d ge ᎏ . 14. The Handy Brush company made $500 million in sales this year. Last year, the company made $400 million. What was the percent increase in sales this year? Answer First of all, what was the change in sales? Yes, 100 mil- lion. You got that by subtracting the two numbers. Which number is the oldest? Last year is older than this year, so 400 is the oldest. Therefore, 100 goes over 400. ᎏ 1 4 0 0 0 0 ᎏ The percent is the unknown figure, so a variable is placed over 100 and the two are made equal to each other. Cross multiply and solve for x. The answer is 25%. Note that if you had put 100 over 500, your answer would have come out differ- ently. Sample Interest Question 15. How much interest will Jill earn if she deposits $5,000 at 3% interest for six months? Answer Interest is a percent problem with time added. The formula for interest is I = PRT. I is the interest. P is the principal, R is the rate or percent, and T is the time in years. To find the interest, you simply multiply every- thing together. Be sure to put the time in years. You may change the percent to a decimal, or place it over 100. $5,000 (principal) × 0.03 (percent) × ᎏ 1 2 ᎏ (year) = $75.  Math 7: Algebra Algebra is like a perfectly balanced scale. The object is to keep both sides balanced while isolating the part you need on one side of the scale. For example, suppose you know a novel weighs 8 ounces and you want to find out how much your thick phone book weighs. You have five novels on one side of the scale, and your phone book and two novels on the other side. They perfectly balance. By taking two novels off each side, your phone book is alone and perfectly balances with the three novels on the other side. Then you know that your phone book weighs 3 × 8, or 24 ounces. === 25 100 400 100 4 x 100 100 100 400 × 100 1 == 2 120 100 6,000 × 120 100 6,000 × 2 1 –CBEST MINI-COURSE– 114 Plugging in Numbers There are several types of algebra problems you may see on the CBEST. The first consists of a formula, per- haps one you have never seen, such as Y = t + Z − 3z. You think, “I have never seen this . . .” and you are tempted to skip it. But wait you read the question: What is Y if t = 5, Z = 12, and z = 1? All you do is plug in the numbers and do simple arithmetic. Y = t + Z − 3z Y = 5 + 12 − 3(1) = 14 Sample Question 1. Given the equation below, if t = 5 and h = 7, what is Q? Q = t 2 − 3h Answer You were right if you said 4. Q = t 2 − 3h Q = 5 2 − 3(7) Q = 25 − 21 = 4 Solving an Equation In the second type of question, you may actually be called upon to do algebra. Sample Algebra Question 2. Given the equation below, if Q = 15 and h = 1, what is the value of t? Q = t − 3h Answer First, plug in the numbers you know and do as much arithmetic as you can: Q = t − 3h 15 = t − 3(1) 15= t − 3 1. What numbers are on the same side as the vari- able? 3 2. How are the numbers and the variable connected? With a minus sign. 3. The Opposite is what? Addition. With that, add 3 to both sides to get your answer: 15 = t − 3 +3 +3 18 = t Practice Try these problems. You can probably do them in your head, but it’s a good idea to practice the algebra because the problems get harder later. 3. 3x = 21 4. 6 + x = 31 5. x − 7 = 24 6. ᎏ 3 x ᎏ = 9 7. ᎏ 1 3 ᎏ x = 5 Three Success Steps for Algebra Problems In order to make a problem less confusing, try the WHO method: 1. What numbers are on the same side as the variable? There are two sides of the equal sign, the right side and the left side. 2. How are the numbers and the variable connected? 3. The Opposite is what? The opposite of sub- traction is addition. –CBEST MINI-COURSE– 115 Answers 3. x = 7 4. x = 25 5. x = 31 6. x = 27 7. x = 15 Other Operations You Can Use The following are some other ways you can manipulate algebra on the CBEST. Square Both Sides When you’re faced with a problem like ͙x ෆ = 5, you have to get x out from under the square root sign in order to solve it. The way to do this is to square both sides of the equation. Squaring is the opposite of a square root, and cancels it. ͙x ෆ = 5 ͙x ෆ 2 = 5 2 x = 25 Take the Square Root of Both Sides If the variable is squared, take the square root of both sides. x 2 = 25 ͙x ෆ 2 = ͙25 ෆ x = 5 Flip Both Sides If the answer calls for x and the x ends up as a denom- inator, the answer is unacceptable as is, because the question called for x, not ᎏ 1 x ᎏ . If you have gotten this far in a problem, you can find the answer easily by flipping both sides. ᎏ 1 x ᎏ = ᎏ 6 7 ᎏ ᎏ 1 x ᎏ = ᎏ 7 6 ᎏ x = ᎏ 7 6 ᎏ or 1 ᎏ 1 6 ᎏ Divide by a Fraction To divide by a fraction, you take the reciprocal of the fraction and multiply. ᎏ 3 5 ᎏ x = 15 Since the reciprocal of ᎏ 3 5 ᎏ is ᎏ 5 3 ᎏ , multiply both sides by ᎏ 5 3 ᎏ : ( ᎏ 5 3 ᎏ ) ᎏ 3 5 ᎏ x = 15( ᎏ 5 3 ᎏ ) x = ᎏ 1 1 5 ᎏ ( ᎏ 5 3 ᎏ ) Reduce the fractions and multiply: Practice Solve for x: 8. x 2 = 144 9. ͙x ෆ = 7 10. ᎏ 1 x ᎏ = ᎏ 3 4 ᎏ 11. ᎏ 2 3 ᎏ x = 14 Answers 8. 12 9. 49 10. ᎏ 4 3 ᎏ or 1 ᎏ 1 3 ᎏ 11. 21 == 25 15 5 3 x 1 ( ) 5 1 –CBEST MINI-COURSE– 116 Multi-Step Problems Now that you have mastered every algebraic trick you will need, let’s juggle them around a little by doing multi-step problems. Remember the order of opera- tions: Please Excuse My Dear Aunt Sally—Parenthe- ses, Exponents, Multiply and Divide, Add and Subtract? That order was necessary when putting numbers together. In algebra, numbers are pulled apart to isolate one variable. In general, then, it is eas- ier to reverse the order of operations—add and sub- tract, then multiply and divide, then take square roots and exponents. Here is an example: 35 = 4x − 3 In this problem, you would add the 3 to both sides first. There is nothing wrong with dividing the 4 first, but remember, you must divide the whole side like this: ᎏ 3 4 5 ᎏ = ᎏ 4x 4 − 3 ᎏ or ᎏ 3 4 5 ᎏ = ᎏ 4 4 x ᎏ − ᎏ 3 4 ᎏ As you can see, by adding first, you avoid work- ing with fractions, making much less work for yourself: 35 = 4x − 3 +3 +3 38 = 4x Then divide both sides by 4 resulting in the answer: x = ᎏ 1 2 9 ᎏ = 9 ᎏ 1 2 ᎏ Practice Try these: 12. 5y − 7 = 28 13. x 2 + 6 = 31 14. ᎏ 4 5 ᎏ x − 5 = 15 15. If a − 2b = c, what is a in terms of b and c? Hint: When a question calls for a variable in terms of other variables, manipulate the equa- tion until that variable is on a side by itself. 16. If ᎏ p 3 ᎏ + g = f, what is p in terms of g and f? Answers 12. y = 7 13. x = 5 14. x = 25 15. a = c + 2b 16. p = 3(f − g) Problems Involving Variables Sometimes you’ll find a problem on CBEST that has almost no numbers in it. Sample Variable Question 17. John has 3 more than 10 times as many students in his choir class than Janet has in her special education class. If the number of students in John’s class is v, and the number in Janet’s class is s, which of the equations below does NOT express the information above? a. v = 3 + 10s b. v − 3 = 10s c. ᎏ v 1 − 0 3 ᎏ = s d. 10s − v = −3 e. v + 3 = 10s = 38 2 4 4x 4 19 –CBEST MINI-COURSE– 117 Answer After reading question 17, you’re likely to come up with the equation in answer a. Since a is correct, it is not the right choice. Now manipulate the equation to see whether you can find an equivalent equation. If you subtract 3 from each side, answer b will result. From there, dividing both sides by 10, you come up with c. All those are equivalent equations. Choice d can be derived by using b and subtracting v from both sides. Choice e is not an equivalent and is therefore the correct answer. Distance, Rate, and Time Problems One type of problem made simpler by algebra are those involving distance, rate, and time. Your math review would not be complete unless you had at least one problem about trains leaving the station. Sample Distance Problem 18. A train left the station near your home and went at a speed of 50 miles per hour for 3 hours. How far did it travel? a. 50 miles b. 100 miles c. 150 miles d. 200 miles e. 250 miles Answer Use the three Success Steps to work through the problem. 1. D = R × T 2. D = 50 × 3 3. 50 × 3 = 150 Practice Try these: 19. How fast does a dirt bike go if it goes 60 miles every 3 hours? 20. How long does it take to go 180 miles at 60 miles per hour? Answers 19. R = 20 20. T = 3 HOT TIP Another way to look at the distance formula is When you’re working out a problem, cross out the let- ter that represents the value you need to find. What remains will tell you the operation you need to perform to get the answer: the horizontal line means divide and the vertical line means multiply. For example, if you need to find R, cross it out. You’re left with D and T. The line between them tells you to divide, so that’s how you’ll find R. This is a handy way to remember the formula, especially on tests, but use the method that makes the most sense to you. Three Success Steps for Distance, Rate, and Time Problems 1. First, write the formula. Don’t skip this step! The formula for Distance, Rate, and Time is D = R × T. Remember this by putting all the let- ters in alphabetical order and putting in the equal sign as soon as possible. Or think of the word DIRT where the I stands for is, which is always an equal sign. 2. Fill in the information. 3. Work the problem. –CBEST MINI-COURSE– 118 D R T . 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. algebra because the problems get harder later. 3. 3x = 21 4. 6 + x = 31 5. x − 7 = 24 6. ᎏ 3 x ᎏ = 9 7. ᎏ 1 3 ᎏ x = 5 Three Success Steps for Algebra Problems In order to make a problem less. problem, you can find the answer easily by flipping both sides. ᎏ 1 x ᎏ = ᎏ 6 7 ᎏ ᎏ 1 x ᎏ = ᎏ 7 6 ᎏ x = ᎏ 7 6 ᎏ or 1 ᎏ 1 6 ᎏ Divide by a Fraction To divide by a fraction, you take the reciprocal