Accuplacer Self Study Manual

Transcription

1 Pursue challenging learning goals Know when and how to ask for help Believe that learning is its own reward Convert fear into motivation See mistakes as opportunities for growth Appreciate the value of self-assessment Use prior learning as a problem-solving tool Celebrate success Be Independent Learner Accuplacer Self Study Manual Basic Arithmetic Skills Study Packet / Student Preparation Study Guide Created by Aleksandr Rusinov and Barrington Scott Canal Street, New York, NY , Math Specialist, Learning Enhancement Center, ext. 2446, arusinov@mcny.edu

2 2 Preface This manual is designed for students who struggle with basic mathematical concepts and need to take Accuplacer test examination. The manual includes four parts. First part contains detailed explanation on fractions, adding and subtracting fractions, multiplying and dividing fractions, reducing fractions, improper fractions and mixed numbers, fraction equivalency, decimals, computing with decimals, percents, solving with percents, practice, word problems, four steps to problem solving. Second part gives students a lot of problem practice to work on. Third part provides problem solution review given in section 2. And the last part is designed as real test exam practice.

3 3 Contents Preface... 2 PART 1: READING PACKET... 6 What s the Fraction?... 6 Adding and Subtracting with Fractions... 7 Reducing Fractions... 8 Multiply and Divide Fractions... 8 Improper Fractions and Mixed Numbers... 9 METHOD FOR FINDING FACTORS!... 9 Fraction Equivalency DECIMALS Computing with Decimals Adding and subtracting TURNING A FRACTION INTO A DECIMAL Multiplying Dividing PERCENTS SOLVE!!! Four Steps to Problem Solving PRACTICE!!! WORD PROBLEMS PART 2: PRACTICE PACKET FRACTION / DECIMAL FRACTION ADDITION SUBTRACTION OF FRACTION FRACTION MULTIPLICATION FRACTION / DECIMALS FRACTION / DECIMALS DECIMALS Addition DECIMALS Subtraction... 23

4 4 DECIMALS Division DECIMALS Multiplication CONVERTING PERCENTAGES TO FRACTIONS CONVERTING FRACTIONS TO PERCENTAGES PERCENTAGES PERCENTAGES PERCENTAGES TIMES CHANGE WORD PROBLEMS Finding Missing Numbers WORD PROBLEMS Fraction Addition WORD PROBLEMS Fraction Subtraction WORD PROBLEMS Fraction Multiplication PART 3: PROBLEM SOLUTION REVIEW FRACTION / DECIMAL FRACTION ADDITION FRACTION SUBTRACTION FRACTION MULTIPLICATION FRACTION / DECIMALS FRACTION / DECIMALS DECIMALS ADDITION DECIMALS SUBTRACTION DECIMALS DIVISION DECIMALS MULTIPLICATION CONVERTING PERCENTAGES TO FRACTIONS CONVERTING FRACTIONS TO PERCENTAGES PERCENTAGES

5 5 PERCENTAGES PERCENTAGES TIMES CHANGE WORD PROBLEMS Finding Missing Numbers WORD PROBLEMS Fraction Addition WORD PROBLEMS Fraction Subtraction WORD PROBLEMS Fraction Multiplication PART 4: PRACTICE EXAMINATION PACKET GLOSSARY OF TERMS WEBSITES OF INTEREST... 58

6 6 PART 1: READING PACKET A fraction is a piece of something, anything. For example, if I have a pizza and I cut it into 8 slices, each slice represents a fraction of the whole pizza pie. The whole pie is represented by the total number of pieces that make up a whole pie, in this instance 8. The fractional part of a fraction is the number of pieces out of the whole that you are concerned with. The whole part is the number of pieces that would total one complete set of pieces, or a whole. These two parts of the fraction are also referred to as the numerator and the denominator. The numerator is the fractional part and can be found at the top of the fraction. The denominator is the whole thing, and can be found at the bottom. Let s take a look at this example. Here I have 7 triangles. As you can see, 2 of them have smiley faces on them. The total number of triangles did not increase or decrease. However, the numerator becomes 3, because the total number of triangles with smiley faces did increase. Notes: What s the Fraction? In the Space below, write a fraction that represents the colored part of the figures to your right If I wanted to represent the number of triangles with smiley faces out of the total number of triangles as a fraction I would start like this. The total number of triangles is 7, so 7 is the denominator. It represents the whole number of triangles. The number of triangles that have smiley faces on them is 2, so 2 is the numerator. The fraction would then be 2/7 3. Now, let s look at what happens if the number of triangles with smiley faces grows to 3, but the total number of triangles remains at 7, what is the new fraction? If you answered 3/7 you d be correct. The denominator remains as 7 because the total number of triangles did not increase or decrease. However, the numerator becomes 3, because the total number of triangles with smiley faces did increase. Notes:

7 7 Adding and Subtracting with Fractions Adding and subtracting fractions is easy, so long as the fractions have the same denominator. You simply add or subtract the numerators, and place the result over the denominator. See example: EXAMPLE: When the fractions 1 have 2 different 1+2 denominators, 3 you have to find the lowest common denominator. Why?? BECAUSE: As we have reviewed so far, a fraction is a part of a whole number. The number of parts that the whole is divided into is the denominator and the portion you have, or are concerned with is the numerator. When you are adding these two values together, we have to make sure that we are adding equivalent values. For example, say you and your best friend each order your own pizza pie. You cut your pizza into 8 slices and eat 3 slices. Your friend cuts his pizza into 6 slices and eats 2 slices. You then both take your leftovers and combine them. How much pizza do you have? First, let s begin by looking at the amount of pizza you each ate. You ate 3 slices and your friend ate 2, so, just looking at those numbers it would seem that you ate more pizza. However, you ate 3 slices out of a pizza cut into 8 slices, or rather you ate 3/8 of a pizza. Your friend ate 2 slices of a pizza cut into 6 slices, or rather 2/6 of a pizza. Therefore it is not immediately clear who ate more pizza. If you ate 3/8 and your friend ate 2/6, let s find a common denominator for 6 and 8 to create equivalent fractions and add them together. What are the multiples of 6 and 8? Some multiples of 6 are 12, 18, 24, 30 and so on. The multiples of 8 are 16, 24, 32, and so on. As you can see, they both have the number 24 in common, and it is the smallest number that they have in common (Lowest Common Multiple). Therefore we can use it for a common denominator. Now, we find the equivalent fractions by transforming each fraction into a new one with the denominator of 24. To do this, I need to consider what I would multiple each original denominator by in order to get 24, then multiply the top by the same number 3? 3*39 9 So, you see, you ate 1/24 more pizza then your friend * Add these up and you see that the two of you together ate 17/24 of one pizza 2? 2* * Now, we find how much pizza remains using the same principle. Since you ate 3/8 or pizza, 5/8 remain and since your friend ate 2/6 of pizza, 4/6 remain. 5? 5* /24 slices of pizza remain. Reduced down to lowest terms, * that is 1 pizza and 7/24* of a pizza. The method for reducing a fraction will be discussed in greater detail in 4? 4* the next section * *Notice that 7/24 and the 17/24 you calculated above equal 24/24, or one full pie. Therefore, what you ate combined with what is left over equals the original amount: two full pizzas. To find all the multiples of one or more numbers is to create this simple chart In the left column, list numbers 1, 2, 3, etc. In the right, list the product of the number you are trying to find multiples of and the corresponding number. From the example above, the numbers we were using above were 8 and 6. You can continue as far as you need.

8 8 (* 8) (* 6) You can see that we identified the lowest common multiple using this method! ***Subtracting fractions you do the same, except you subtract one numerator from the other and place the result over the denominator. Notes: Multiply and Divide Fractions Multiply: The rule for multiplication is easy; you multiply the numerator by the other numerator and the denominator by the other denominator. The product of each becomes your new numerator and denominator. For example, if you multiply 4/5 by 2/3, you multiply 4*2 to get the numerator and 5*3 to get the denominator, leaving you with 8/15, which is already in lowest terms. Divide: The rule for dividing is also easy. You actually multiply the first fraction by the reciprocal of the divisor (the fraction that is dividing the other fraction). You find the reciprocal by switching the numerator with the denominator. For example, if you divide 2/7 by 8/10 1, you actually multiply 2/7 by 10/8, or 2*10 and 7*8, leaving you with 20/56. Reduced to lowest terms that would be 5/ Notes: 1 Since 2 / 7 is being divided by 8 / 10, 8 / 10 is the divisor. Reducing Fractions Earlier in this study guide we reviewed the Lowest Common Multiple (LCM) when creating common denominators, for fraction reduction we need to examine the Greatest Common Factor (GCF). The greatest common factor is the largest number that can evenly divide two numbers. If the numerator and denominator of a fraction have a common factor that is the number 2 or higher, it is not in its lowest form. If the only common factor is the number 1 then it is in lowest (or simplest) terms, something that is also called relatively prime. Let s look at a few examples. Take the fraction 10 / 15. The numerator (10) and the denominator (15) have a common factor of

9 9 5. Dividing both the numerator and the denominator by the greatest common factor yields the following : The answer is 2. 15: The answer is 3 The fraction, 2 / 3, is in lowest terms because the only common factor of 2 and 3 is 1. Another way to think of reducing fractions is to think of creating a fraction that is equivalent to the current fraction you have. This will be covered in greater detail at the end of the fraction lesson. PRACTICE!! Create fractions from the shaded parts of each figure, and then reduce that fraction to lowest terms Improper Fractions and Mixed Numbers An improper fraction is a fraction greater than 1. What does that mean? Basically, it s a fraction whose numerator is larger than its denominator. The fraction 5 / 3 is improper. Since the denominator is the number 3, then a numerator totaling 3 or more is no longer just a fraction. A fraction is a part of a whole, and if you have at least one whole, you can change the improper fraction to a mixed number. A mixed number is a whole number combined with a proper fraction. Take our previous example, 5 / 3, if you need 3 of something to make 1 whole, then you have 1 whole, and 2 / 3 left over, or 1 & 2 / 3. THINK ABOUT IT! METHOD FOR FINDING FACTORS! Like above, you can go through number 1, 2, 3 etc. When a number doesn t work, (EX: 15 cannot be divided by 2), simply skip it.? x? ? x? You can see how this method is used to find the Greatest Common Factor (GCF) Notes: If you have a box with 10 compartments in it to hold tennis balls, if you have 14 tennis balls, you have an improper fraction. You have 14 / 10. Therefore, you have one (1) full box, and 4 / 10 left over, 1 & 4 / 10. You can also reduce this to 1 & 2 / 5 (refer back to reducing fractions on page 7) You convert an improper fraction into a mixed number by dividing the numerator by the denominator and then use the remainder to create a new fraction. For example if you have the improper fraction, 47 / 15, when you divide 47 by 15 you get 3 with a remainder of 2. Therefore, you have the mixed number 3 & 2 / 15. Notes:

10 10 Fraction Equivalency Now, let s look at this pie. As you can see it has been divided into 6 pieces. Each piece of the pie represents 1 / 6 of the total. However, if we look at this pie, we can see that it is the same size as the first, only it has been divided into 4 pieces. Each piece of the pie represents ¼ of the total. In the first pie, one side of the pie held 3 pieces. In the second, one side of the pie held just two pieces. This is important, because it shows us equivalency. Not in so far as one side of the pie that holds two is the same as the side holding three, but in that the total number of pieces the pie is cut into is changed accordingly as well. In the pie that has two slices on one side, there are four total pieces, in the second pie where one half of the pie is cut into three pieces; the whole pie is cut into six. 2 / 4 is equivalent to 3 / 6. In other words the relationship between the numerator and denominator remain the same as the number of one part of the fraction increases and decreases. ***When you change either the top or the bottom of a fraction in anyway, you must perform the same operation to the other part, to maintain equivalency. This is important as well in making estimations using ratios. Let s say the ratio of men to women in New York 1 is 2:1, then that means there are two men for every one woman. As you increase the value of one portion of that ratio, you must equally increase the value of the other ****Fraction equivalency is a lot like ratio, in that one fraction is equivalent to another so long as they maintain the same proportions, and ratios maintain their equivalency if they also maintain the same proportions. Review Corner: If the ratio of chocolate candies to strawberry candies is 3:1, and you have 36 chocolate candies, how many strawberry candies do you have? How did you solve that? 1 Hypothetical example, not based on evidence DECIMALS A decimal is another kind of fraction, except it is to a power of ten. What does that mean??!?? It means that a decimal is a fraction with a denominator that is 10,, 1,000 and so on (powers of TEN). But you do not necessarily need a denominator already as a power of 10; you can convert any fraction into a decimal as you will see in the following section. The best way to think about a decimal is to think of money. If you have $1.25 you have one full dollar ( pennies) and 25 cents towards another dollar (25 pennies). In its current form it is represented as a decimal. However, we could translate it into a fraction. The number in front of the decimal represents the amount of whole numbers you have (in this case 1 whole dollar) and the numbers following the decimal represent the fractional part, 25 pennies out of a total of pennies that would make a new dollar, or 25 /. This can be reduced to ¼ (see page 7 if you need to review reducing fractions).

11 11 Place Value The number of digits after the decimal point will determine the number of decimal places in the number. The last digit in the number will determine the label used to write decimal numbers in words Notes: TURNING A FRACTION INTO A DECIMAL You can turn a fraction into a decimal by dividing the numerator by the denominator. Take for example the fraction ¾. If you divide 3 by 4, you will see that 3 cannot be divided by 4 with any whole number. So, you add a zero to the 3 and put a decimal in front of your answer. Four goes into thirty 7 times, with a remainder of 2. Add another zero, and 4 goes into twenty 5 times. The answer is.75 Let s take a look THINK ABOUT IT There are 4 quarters in a dollar, each worth.25 cents. If you have 3 quarters, you have.75 cents. It makes CENTS!! / Notes: Computing with Decimals Adding and subtracting When you are adding and subtracting with decimals, you have to line up the decimals. This is important because of Place value (see page 10 for review on place value). Place value is the value of a digit based on it s placement within a number. For example, in the decimals.84, the digit 8 represents 80 because it is in the tenth spot. Therefore if you are adding.84 to.7, since the 7 is in the tenth spot, it must be aligned with the 8 in the.84. The answer you will get will also become a decimal with a whole number in front, as the sum of the two exceeds (as long as you properly consider the 7 a 70). For example, if you have 84 pennies and you add 70 pennies, you will have more than one whole dollar, with change left over. Let s add! This same rule applies to subtraction a / b c <> c b a

12 12 Multiplying When you are multiplying decimals, you do not have to line up the decimals. But after you have finished multiplying the numbers, you move a decimal point in the total number of places in the decimals you multiplied. Let s look at an example 1.04 on the top, the decimal is two 0.50_ places in on the bottom, the decimal is one place in the answer, you move the decimal three places in, adding together the top number of places with the bottom Dividing When you are dividing fractions, you want to get rid of the decimal, by moving the decimal over an equal number of places in both numbers. For example, if you are dividing 4.2 by.84, you need to move the decimal over two places to get rid of both decimals. This means you will have to add a zero to the end of 4.2. Let s look! Notes: PERCENTS Percents are really easy now that we understand decimals! A percent is nothing more than a decimal multiplied by!! When you multiply a decimal by, you move the decimal two places to the right %.72 72%.65 65% and so on. We can also change a fraction into a decimal, just perform the method for changing a fraction into a decimal (divide the numerator by the denominator) and multiply your answer by!!! A really simple way to think of a percent is a fraction with as the denominator. For example, 72% can be represented as 72 /. It can also be considered a fraction to the power of ten, except it is always out of. Knowing that a percent is both a decimal and fraction at the same time is very useful when trying to solve for an amount based on the percentage value. Let s look at three questions. 1) If 400 people are in at a wedding, and 17% of them ordered the beef entry, how many beef entries must the chef prepare? 2) If at that same wedding, you count 257 female guests, what percentage of the guests are female?

13 13 3) If 92% of the guests the bride and groom invited attended the wedding, how many people did they invite? Each of these questions can be answered using the same formula. A formula is an equation where you enter the information you know and compute to solve what you do not know. Percent percentage value Part fractional part (numerator) Whole the entire group (denominator) remains always, since a percentage is always a fraction over. Notes: percent SOLVE!!! First question, you know that there are 400 people at the wedding, and 17% ordered beef. So, plug in the values! Since you are trying to find what part of the total number of guests ordered beef, then X goes in the location of the formula that calls for PART To solve this equation, you cross multiply. 17 X 400 X X 6800 part whole Now, we use algebra to get X by itself. Since times a number is equal to 6800, then 6800 divided by equals that unknown number, and 6800 divided by equals 68. Therefore, 68 people ordered beef Second question, you know that 257 of guests are female, out of 400 total guests. What you don t know is what percent of guests that is. So, the percent is the X value, and we plug in the rest. X X X X 64.25, or we can round to 64% Third question, we know that 400 people attended the wedding, and that is 92% of the people invited. Therefore, the whole part of the formula is X X X 40,000 92X THINK ABOUT IT You are creating an equivalent fraction to 17/. To turn into 400, you multiply it by 4 Multiply 17 by 4 you get 68!!! X , or we can round to 435 people were invited.

14 14 PRACTICE!!! 1) A survey was conducted using a sample of NYC residents that found that 72% of people use mass transit. If the total sample size they used was 600 people, how many people responded that they use mass transit? 2) If you ate 8 chocolates out of a box of 20 chocolates, how percentage of the chocolates did you eat? 3) If 560 of students vote to extend summer break at a local school, representing 80% of their schools total enrollment, how many students attend the school? Notes: WORD PROBLEMS Of the most dreaded tasks to tackle in mathematics, the word problem beats them all by a mile. Often believed to be confusing and filled with needless information meant to throw a student off track (and often they are), a word problem is not always difficult in the problem it is asking you to solve, but first determining what it is asking you. Here are some key words for examining a word problem. Difference: means subtraction Sum: means to add Quotient: is the answer to a division problem Product: is the answer to multiplication Notes: Four Steps to Problem Solving Adopted from "Science World," November 5, 1993 ( sonplans/steppro.htm ) 1. UNDERSTANDING THE PROBLEM * Can you state the problem in your own words? * What are you trying to find or do? * What are the unknowns? * What information do you obtain from the problem? * What information, if any, is missing or not needed? 2. DEVISING A PLAN The following list of strategies, although not exhaustive, is very useful. * Look for a pattern. * Examine related problems, and determine if the same technique can be applied. * Examine a simpler or special case of the problem to gain insight into the solution of the original problem. * Make a table. * Make a diagram. * Write an equation. * Use guess and check. * Work backward. * Identify a sub goal. 3. CARRYING OUT THE PLAN * Implement the strategy or strategies in step 2, and perform any necessary actions or computations. * Check each step of the plan as you proceed. This may be intuitive checking or a formal proof of each step. * Keep an accurate record of your work.

15 15 4. LOOKING BACK * Check the results in the original problem. (In some cases this will require a proof.) * Interpret the solution in terms of the original problem. Does your answer make sense? Is it reasonable? * Determine whether there is another method of finding the solution. * If possible, determine other related or more general problems for which the techniques will work. Notes:

16 16 PART 2: PRACTICE PACKET Fraction / Decimal FRACTION / DECIMAL Examples: Change the following decimals to fractions Change the following decimals to fractions 1a b c a b c / 10 [reducing fraction] 1 / 2 3a b c a b c / [reducing fraction] 1 / 4 5a b c a b c a b c a b c / [reducing fraction] 9 / 20 9a b c a b c. 0.15

17 17 FRACTION ADDITION Examples: FRACTION ADDITION Carry out the indicated operation. Then reduce the answer to the lowest terms where possible / 3/ / Or [adding whole 4 parts]

18 18 FRACTION SUBTRACTION SUBTRACTION OF FRACTION Examples: Carry out the indicated operation. Then reduce the answer to the lowest terms / [subtracting whole 2 parts] /

19 19 FRACTION MULTIPLICATION FRACTION MULTIPLICATION Examples: Perform the indicated operation and reduce answers where possible of of of 12. of of of ,

20 20 Fraction / Decimal FRACTION / DECIMALS 1 Examples: Change the following decimals to fractions or mixed numbers. Change the following decimals to fractions or mixed numbers. 1a b c / 10 [reducing 2a b c fraction] 3 1 / 5 or 16 / 5 3a b c / [reducing 4a b c. 4.5 fraction] 1 1 / 4 5a b c a b c / [reducing 7a b c fraction] 1 9 / 20 8a b c a b c a b c. 2.93

21 21 FRACTION / DECIMALS 2 Fraction / Decimal 2 Change the following fractions to decimal numbers. Examples: / a. 2a. 16 1b. 16 1c b. 1 2c a. 4a. 3 3b. 92 3c b. 31 4c / a. 1 5b. 1 5c a. 7 6b. 1 6c [reducing fraction by 225 7a. 6 7b. 40 7c ] 1 [converting to 9 8a. 6 8b. 58 8c decimal] 0.11 (1 is repeating number) 9a. 41 9b. 99 9c a. 8 10b. 5 10c

22 22 Decimals 1 DECIMALS 1 Examples: Addition 1a b a b a b a b a b a b a b a b a b a b

23 23 Decimals 2 DECIMALS 2 Examples: Subtraction 1a b a b a b a b a b a b a b a b a b a b

24 24 Decimals 3 DECIMALS 3 Division Example: 1a b a b Solution 3a b / / / / / / / a b / / / 25 5a b / Or 48 / a b a b a b a b a b

25 25 Decimals 4 DECIMALS 4 Multiplication Example: a a b b c c a a b b c c

26 26 Converting Percentages to Fractions Examples CONVERTING PERCENTAGES TO FRACTIONS Convert these percentages to fractions by putting the percentage over a denominator of. Next write it in simplest form. Leave the answer as an improper fraction if it is greater than % 2. 25% 3. 30% 12% 12 / 4/ 12 3 / % 5. 66% 6. 82% % 8. 26% 9. 75% 40 % 40 / 20/ % % % % % % 2 / % % % % % % % % %

27 27 Converting Fractions to Percentages CONVERTING FRACTIONS TO PERCENTAGES Convert these fractions to percentages by converting them to decimals and multiplying by. Round al your percentages to the nearest decimal place. Examples: % % % %

28 28 PERCENTAGES 1 Percentages 1 Find what numerical percentage is out of given number. Solve each problem. Examples: 1. What is 73 % of 370? Converting 73% to decimal and then multiplying by the given number Answer: What is 73 % of 370? Answer 2. What is 3% of 740? Answer 3. What is 91% of 500? Answer 4. What is 42 % of 250? Answer 2. What is 3% of 740? Converting 3% to decimal and then multiplying by the given number Answer: What is 36% of 760? Answer 6. What is 97% of 530? Answer 7. What is 20% of 180? Answer 8. What is 74% of 150? Answer 9. What is 2% of 370? Answer 10. What is 12% of 54? Answer

29 29 PERCENTAGES 2 Percentages 2 Examples: Find what percentage is out of given number that is percent part of the other number. Solve each of the problems is what % of 50? We need to find what 1% of 50 is and then divide 40 by this number. 40 ( 50 / ) 40 / 50 80% Answer: 80% is what % of 50? Answer is what % of 60? Answer 3. 2 is what % of 20? Answer is what % of 40? Answer is what % of 80? Answer 2. 2 is what % of 20? We need to find what 1% of 20 is and then divide 2 by this number. 2 ( 20 / ) 2 / 20 10% Answer: 10% is what % of 60? Answer is what percent of 40? Answer 8. 8 is what % of 80? Answer is what % of 50? Answer is what % of 50? Answer

30 30 PERCENTAGES 3 Percentages 3 Examples: Find what numerical percentage is out of given number. Find what percentage is out of given number that is percent part of the other number. Solve each of the problems is what % of 70? We need to find what 1% of 70 is and then divide 35 by this number. 35 ( 70 / ) 35 / 70 50% Answer: 50% % of what is 63? Answer is what % of 70? Answer % of what is 56? Answer 4. What is 30 % of 90? Answer 5. What is 90 % of 60? Answer % of what is 56? 56 is only 80% of the unknown number. We need to find 1% of the unknown number is and then multiply this number by a. ( 56 / 80 ) 70 Answer: is what % of 50? Answer 7. 20% of what is 18? Answer 8. % of what is 90? Answer is what % of 50? Answer 10. What is 50 % of 90? Answer

31 31 TIMES CHANGE Examples: 1. Find 27 % of 50. Solution , Answer: 27 % of is what percent of Solution 160? TIMES CHANGE Solve each problem. By matching your answers with those in the Answer Table, you will match the name of an occupation of the past with its modern equivalent. Write your answer and the modern occupation on the line next to each problem. 1. Find 27 % of 50. (Carter) 2. What percent of 80 is 44? (Doorkeeper) % of what number is 18? (Coachman) 4. Find 75 % of 24. (Apothecarist) is what percent of 160? (Cotter) % of what number is 64? (Fuller) % of 85 is what number? (Tinker) % of 75 is what number? 3 (Vizier) % of what number is 63? (Bagger) 10. What percent of 12 is 54? (Chiffonier) 140 ( 160 ) , % Answer: 87.5 % Answer Table 119 handyman, 13.5 garbage collector, 80 cleaner, 55 security guard, 50 judge, 18 pharmacist, 87.5 groundskeeper, 140 porter, 90 limousine driver, 450 secondhand clothes dealer

32 32 Finding Missing Numbers WORD PROBLEMS 1 Finding Missing Numbers Example: Use the clues to find the missing numbers. Half of this two-digit number is 3 times half of 28. The number is. Solution: To solve this problem we need to look into this problem backwards. Starting from end we find that 3 times half of 28 is Continue solving this problem 1. Half of this two-digit number is 3 times half of 28. The number is. 2. A third of this three-digit number is 5 times 20 plus 7. The number is. 3. This two-digit number is one-fourth of one-tenth of a thousand. The number is. 4. This two-digit number is 2 times 40 percent of 120. The number is. 5. This three-digit number plus 58 is half of 618 less 8. The number is. 6. This one-digit number is one-fifth of 30 minus 2 squared. The number is. 7. A fourth of this five-digit number is 5 times one-third of 99 plus half of 5,240. The number is. 8. This two-digit number is 4 times 46 divided by difference of 16 minus 75 percent of 16. The number is. 9. This three-digit number is 5 times 4 cubed. The number is. 10. This four-digit number is a fourth of 4,000 plus the difference between 20 and 10. The number is. we see that half of this twodigit number is 42. The number is 84. Answer: 84

33 33 Fraction Addition WORD PROBLEMS 2 Fraction Addition Example: Find the sum of5 2 15, and 2 3 Solution: 1. Jack received three shipments that weighed pounds, pounds and 37 7 pounds. What was the total weigh of the shipments? / / / Find the sum of5 2 15, and What is the total length of a ribbon needed to make ribbons of the following lengths: inches, inches, inches, 7 1 inches and inches? What is the perimeter of a triangle with sides inches, inches and inches Or

34 34 Fraction Subtraction WORD PROBLEMS 3 Fraction Subtraction Example: In a high jump competition, Larry jumped 5feet inches and his friend jumped 4feet 2 5 inches. By how many 8 inches did Larry s jump exceed his friend s jump? Solution: 1. The fraction of American adults who smoke dropped from in 1964 to 1 in How big a decline was this? 4 2. In a high jump competition, Larry jumped 5feet inches and his friend jumped 4feet 2 5 inches. By how many 8 inches did Larry s jump exceed his friend s jump? 5feet inches - 4feet inches 5feet inches - 4 4feet 21 inches 8 1 feet inches 2/ Jack bought some AT&T stocks. During the past 12 months, the highest selling value of Jack s stock was $ and the lowest value was $22 3. What was the difference 4 between the yearly high and the yearly low? Or [subtracting whole 4 8 parts first and then remaining fractions] 3 [ 3-5 ] Subtract from [ ] 3 1 8

35 35 Fraction Multiplication Example: WORD PROBLEMS 4 Fraction Multiplication A bus can carry 250 passengers to a state fair. If 7 the bus is full, how many 10 passengers is the bus carrying? 1. A bus can carry 250 passengers to a state fair. If the bus is 7 full, how many passengers is the bus carrying? 10 Solution: 2. At a county fair a pig is to be cooked for 7 2 hour for each pound. If the pig weights 42 pounds, how long will it take to cook? Answer: 175 passengers Find the area of a rectangular floor. 5 feet by 1 feet Abdul wants to rent an apartment for $1,500 per month. The real estate broker told him to deposit 5 3 of the first month s rent plus one month security. How much will Abdul deposit?

36 36 PART 3: PROBLEM SOLUTION REVIEW FRACTION / DECIMAL (p 16) Change the following decimals to fractions 1a b c a b c a b c a b c a b c a b c a b c a b c a b c a b c

37 37 FRACTION ADDITION (p 17) Carry out the indicated operation. Then reduce the answer to the lowest terms where possible / / / / / / / / / / / 3 15/ / 2/ / / 5/

38 / 3/ / 8/ / 2/ / 4/ / 4/ 1 3/

39 39 FRACTION SUBTRACTION (p 18) Carry out the indicated operation. Then reduce the answer to the lowest terms / 6 7/ / / / 5/ / / / / / / / /

40 40 FRACTION MULTIPLICATION (p 19) Perform the indicated operation and reduce answers where possible of of of of of

41 41 FRACTION / DECIMALS 1 (p 20) Change the following decimals to fractions or mixed numbers. 1a b c a b c a b c a b c a b c a b c a b c a b c a b c a b c

42 42 FRACTION / DECIMALS 2 (p 21) Change the following fractions to decimal numbers. 1a b c a b c a b c a b c a b c a b c a b c a b c a b c a b c

43 43 DECIMALS ADDITION (p 22) 1a b a b a b a b a b a b a b a b a b a b

44 44 DECIMALS SUBTRACTION (p 23) 1a b a b a b a b a b a b a b a b a b a b

45 45 DECIMALS DIVISION (p 24) 1a b a b a b a b a b a b a b a b a b a b

46 46 DECIMALS MULTIPLICATION (p 25) 1a a b b c c a a b b c c

47 47 CONVERTING PERCENTAGES TO FRACTIONS (p 26) Convert these percentages to fractions by putting the percentage over a denominator of. Next write it in simplest form. Leave the answer as an improper fraction if it is greater than % 14 2/ % 25/ % % % 2/ % 2/ % 50/ % 2/ % 25/ % 2/ % 25/ % 2/ % % % 5/ % 4/ % % 2/ % 5/ % 25/ % 20/ % 4/ % % 20/

48 48 CONVERTING FRACTIONS TO PERCENTAGES (p 27) Convert these fractions to percentages by converting them to decimals and multiplying by. Round al your percentages to the nearest decimal place % % % % 5. 5 % % % % % % % % % % % % % % % % % % % % % %

49 49 PERCENTAGES 1 (p 28) Find what numerical percentage is out of given number. Solve each problem. 11. What is 73 % of 370? Answer What is 3% of 740? Answer What is 91% of 500? Answer What is 42 % of 250? Answer What is 36% of 760? Answer What is 97% of 530? Answer What is 20% of 180? Answer What is 74% of 150? Answer What is 2% of 370? Answer What is 12% of 54? Answer

50 50 PERCENTAGES 2 (p 29) Find what percentage is out of given number that is percent part of the other number. Solve each of the problems is what % of 50? Answer % is what % of 60? Answer % is what % of 20? Answer % is what % of 40? Answer % is what % of 80? Answer is what % of 60? Answer % is what percent of 40? Answer % is what % of 80? Answer % is what % of 50? Answer % is what % of 50? Answer %

51 51 PERCENTAGES 3 (p 30) Find what numerical percentage is out of given number. Find what percentage is out of given number that is percent part of the other number. Solve each of the problems % of what is 63? Answer is what % of 70? Answer % % of what is 56? Answer What is 30 % of 90? Answer What is 90 % of 60? Answer is what % of 50? Answer % % of what is 18? Answer % of what is 90? Answer is what % of 50? Answer % 20. What is 50 % of 90? Answer

52 52 TIMES CHANGE (p 31) Solve each problem. By matching your answers with those in the Answer Table, you will match the name of an occupation of the past with its modern equivalent. Write your answer and the modern occupation on the line next to each problem. 11. Find 27 % of (Carter) 12. What percent of 80 is 44? (44 ) % (Doorkeeper) % of what number is 18? (18 ) (Coachman) 14. Find 75 % of (Apothecarist) is what percent of 160? (140 ) % (Cotter) % of what number is 64? (64 ) (Fuller) % of 85 is what number? (Tinker) % of 75 is what number? ( ) (Vizier) % of what number is 63? (63 ) (Bagger) 20. What percent of 12 is 54? (54 ) (Chiffonier) Answer Table 119 handyman, 13.5 garbage collector, 80 cleaner, 55 security guard, 50 judge, 18 pharmacist, 87.5 groundskeeper, 140 porter, 90 limousine driver, 450 secondhand clothes dealer

53 53 WORD PROBLEMS 1 (p 32) Finding Missing Numbers Use the clues to find the missing numbers. 11. Half of this two-digit number is 3 times half of 28. The number is 2 (3 14) A third of this three-digit number is 5 times 20 plus 7. The number is 3 (5 (20 + 7)) This two-digit number is one-fourth of one-tenth of a thousand. The number is (0 10) This two-digit number is 2 times 40 percent of 120. The number is 2 ( ) This three-digit number plus 58 is half of 618 less 8. The number is This one-digit number is one-fifth of 30 minus 2 squared. The number is A fourth of this five-digit number is 5 times one-third of 99 plus half of 5,240. The number is (5 (99 3) +5,240 2) 4 ( ) This two-digit number is 4 times 46 divided by difference of 16 minus 75 percent of 16. The number is (4 46) ( ) This three-digit number is 5 times 4 cubed. The number is This four-digit number is a fourth of 4,000 plus the difference between 20 and 10. The number is (20-10) 1010

54 54 WORD PROBLEMS 2 (p 33) Fraction Addition 1. Jack received three shipments that weighed pounds, pounds and 37 7 pounds. What was 8 the total weigh of the shipments? Solution: / 2/ Find the sum of5 2 15, and 2 3 Solution: / / / Or What is the total length of a ribbon needed to make ribbons of the following lengths: inches, inches, inches, inches and 2 3 inches? Solution: ( ) + ( ) ( )

55 or / / 2/ What is the perimeter of a triangle with sides inches, inches and inches Solution: / 4/ 1 2/ /

56 56 WORD PROBLEMS 3 (p 34) Fraction Subtraction 5. The fraction of American adults who smoke dropped from in 1964 to 1 4 big a decline was this? in How Solution: / / In a high jump competition, Larry jumped 5feet inches and his friend jumped 4feet inches. By how many inches did Larry s jump exceed his friend s jump? Solution: 5 feet inches 4 feet inches 1 feet inches 2/ inches 7. Jack bought some AT&T stocks. During the past 12 months, the highest selling value of Jack s stock was $ and the lowest value was $22 3. What was the difference between 4 the yearly high and the yearly low? Solution: 2/ Subtract from Solution: / /