﻿ ﻿Fractions, Decimals, and Percents - Content and Strategy Review - Math Workout for the GMAT

## Part III Content and Strategy Review

### Chapter 3 Fractions, Decimals, and Percents

FRACTIONS

Fractions are used to express division. The top number (the numerator) is divided by the bottom number (the denominator). Just think of the fraction bar as a division symbol. For example, means the same thing as 12 ÷ 3, or 4.

Fractions also represent part of a whole. Simply put the part over the whole and you have the fraction. This approach is very useful for word problems that involve fractions. For example, suppose Oscar has a marble collection containing 2 blue marbles, 5 green marbles, and 8 red marbles. His total collection is 2 + 5 + 8 = 15 marbles. To find the fraction of his collection that is blue, put the part (2 blue marbles) over the whole (15 marbles total) and you find that of Oscar’s marbles are blue.

Reducing Fractions

A common mistake in solving fraction problems is failing to reduce the answer. If you solve the problem but don’t see your answer among the choices, check to see whether you need to reduce the fraction.

To reduce a fraction, divide the numerator and the denominator by the same number. Continue until there are no other numbers that will divide into both parts of the fraction.

Look at Oscar’s marble collection again. Suppose you need to know what fraction of his marbles are green. Putting part over whole, you find that of the marbles are green. You look at the answers, but you don’t see anywhere. Did you miss something? No. You just need to reduce the answer. Both 5 and 15 are divisible by 5. Divide each of them by 5 and becomes . You can’t divide any further, so you’re done.

A simple way to add or subtract fractions with different denominators is by using “the Bowtie.” First, multiply the denominator of the second fraction by the numerator of the first fraction and place the result above the left numerator. Then, multiply the denominator of the first fraction by the numerator of the second fraction and place the result above the right numerator. Next, multiply the denominators of the fractions to get the denominator of the answer. Last, add (or subtract) the numerators to get the numerator of the answer. It’s easier than it sounds; look at this example:

Multiply 6 by 3 to get 18 and put that above the left numerator. Multiply 4 by 5 to get 20 and put that above the right numerator. Next, multiply 4 by 6 to get 24, the denominator. The criss-cross shape gives the Bowtie its name. Putting it all together, you have 18 + 20 in the numerator and 24 in the denominator. That’s , which reduces to .

Subtraction undergoes exactly the same treatment; you just subtract the numerators instead of adding them. For example, try . For the numerators, you get 5 × 2 = 10 and 3 × 1 = 3. For the denominators, you get 3 × 5 = 15. Put it all together and you get = .

Comparing Fractions

You can also use the Bowtie to compare fractions. Suppose you need to compare and so that you can determine which is greater. Write down the fractions and use the Bowtie as if you were going to add them. However, you should stop before you reach the third step (multiplying the denominators). You should have something like this.

The 35, which is over , is bigger than 33, which is over , so is the greater of the two fractions.

Multiplying Fractions

Multiplying fractions is rather straightforward. Multiply the numerators to get the numerator of the answer, and multiply the denominators to get the denominator of the answer. For example, × = = . Then reduce if necessary.

What if you’re multiplying a fraction and a whole number? Just turn the whole number into a fraction by putting a 1 in the denominator. Then multiply normally. For example, 3× becomes × = .

Sometimes it’s helpful if you cancel the numbers before you multiply. This will make the calculation simpler. Canceling is virtually the same thing as reducing, but you do it before the calculation, not after. You can cancel any number in the numerator with any number in the denominator. Just divide them both by the same number. For example, × becomes × because you can divide both the 4 and the 8 by 4. Note that the answer is exactly the same regardless of whether you cancel before you multiply or reduce after you multiply. Just use whichever is more comfortable for you.

Dividing Fractions

To divide fractions, multiply the first fraction by the reciprocal of the second fraction (the one you’re dividing by). Just flip over the second fraction and multiply. For example, ÷ becomes ×. Multiply it to get .

Sometimes you’ll see a fraction composed of two fractions, such as . Just remember that you’re dividing the numerator by the denominator. Rewritten, this becomes ÷ or ×. Once you’ve written it as a multiplication problem you can cancel. It’s easier to do this after, rather than before, converting it to a multiplication problem. Canceling, you get × = .

What if the division contains a fraction and a whole number? Turn the whole number into a fraction by putting a 1 in the denominator, just as you did in multiplication. For example, ÷2 becomes ÷. When you flip the second fraction and multiply, you get × = . Don’t to forget either cancel or reduce the fractions.

Mixed Fractions

A mixed fraction consists of a whole number and a fraction, such as 5. If a problem contains mixed fractions, the first thing you should do is convert them to improper fractions (improper fractions are fractions in which the numerators are larger than or equal to the denominators). Multiply the whole number (the 5 in 5) by the denominator. Then add it to the numerator. So 5 becomes = .

The Rest

Word problems involving fractions can sometimes complicate things. Look at this example.

1.Oscar has 15 marbles. If he gives of his marbles to Sally and of the rest to Mike, how many marbles does Oscar have left?

 1 3 5 7 8

Looks simple enough. One third of his marbles is 5 marbles and one fifth of his marbles is 3 marbles. So he gives away 5 to Sally and 3 to Mike, leaving 15 − 5 − 3 = 7 marbles for Oscar. A little too simple, perhaps. Remember our friend Joe Bloggs? That’s the kind of calculation he would make, and the test writers know it. (D) is the trap answer.

Look at the problem again carefully. Oscar gives of his marbles, or 5 marbles, to Sally. He gives of the rest to Mike. So that’s of the remaining 10, not of the original 15. Oscar gives only 2 marbles to Mike, leaving himself with 15 − 5 − 2 = 8 marbles. So the correct answer is (E). This is a very common pattern on the GMAT. Always be on the lookout for phrases such as “the rest” or “the remainder.” They’re there to trap the unwary.

DRILL 1

1.() × ( + ) =

 1

2.Which of the following is greater than ?

3.Ian owns a collection of 60 baseball cards. If he gives of his collection to Kevin and of the remainder to Paul, how many baseball cards does Ian have left?

 54 36 33 24 3

DECIMALS

Decimals are a way to express numbers that are not integers. The digits to the left and right of the decimal point are referred to as decimal places. Starting at the decimal point and moving left, the places are ones (or units), tens, hundreds, thousands, and so forth. To the right of the decimal point, the places are tenths, hundredths, thousandths, and so forth. This diagram shows how it all fits together for the number 10,325.0218.

Adding and subtracting decimals is fairly simple. Just line up the decimal points and put a decimal point in the same spot in your answer. You may need to add some zeroes to one of the numbers. For example, 0.0732 + 1.56 would look like this:

And 7.23 − 3.105 would look like this:

Multiplying Decimals

To multiply decimals, just ignore the decimal points and multiply the numbers. Then count the number of decimal places in the original numbers. In other words, how many digits are to the right of the decimal place in the original numbers? That’s the total number of decimal places you want in your answer.

For example, look at 1.05 × 2.2. First, ignore the decimal points and just multiply: 105 × 22 = 2310. Then, count the number of decimal places. There are two digits to the right of the decimal point in 1.05 and one digit to the right of the decimal place in 2.2. So you need three decimal places in your answer. 2310 becomes 2.310 or just 2.31. You should leave off any ending zeroes.

Dividing Decimals

To divide decimals, you must move the decimal points in both numbers. Move the decimal point all the way to the right in the second number (the one you’re dividing by) and count the number of places you moved it. Then move the decimal point in the first number (the one that’s being divided) the same number of places to the right. Add zeroes if you run out of digits. Then simply do the long division.

As an example, try 342.06 ÷ 0.0003. Move the decimal point to turn 0.0003 into 3. That’s four decimal places. Now move the decimal point in 342.06 the same number of places (four) to get 3,420,600. Notice that you had to add some zeroes to the end of the number so that you had four places. The problem is now just 3,420,600 ÷ 3. When you work the long division, you get 1,140,200.

Here’s another example: 1.175 ÷ 0.05. Move the decimal point two places to the right in each number. That changes the problem to 117.5 ÷ 5. Again, work the long division to find the answer, 23.5.

DRILL 2

1. =

 0.75 1.01 7.5 10.1 75

2. =

 0.02 0.12 0.2 2 1.2

PERCENTS AND CONVERSION

Percents are just like fractions and decimals; they are just another way to express non-integer numbers. The word percent means “for each 100.” So 50% translates to “50 for each 100” or the fraction , which reduces to . You can also turn percents into decimals by moving the decimal point two places to the left. So 50% would become 0.5. After converting the percent, make the calculation using the methods you just learned for fractions and percents.

You’ve learned how to convert percents into fractions and decimals. You should also know how to convert the other way—fractions and decimals into percents. Last but not least, you should understand how to turn fractions into decimals and vice versa.

To convert decimals into percents, simply move the decimal point two places to the right. So 0.4 becomes 40% and 0.654 becomes 65.4%.

To convert fractions into percents, multiply the numerator and denominator of the fraction by the same number. Decide what multiplier is necessary to turn the denominator into 100. Then use that multiplier for both the numerator and denominator. For example, suppose you want to convert into a percent. You must first multiply the 4 by 25 to get 100, so that’s the multiplier for both the numerator and the denominator. So you get = = , which you know is the same as 75%. If you can’t easily find the multiplier for the fraction (if you have a fraction such as ), just convert the fraction to a decimal (as described below) and then change the decimal to a percent by moving the decimal point two places to the right.

Converting fractions to decimals is relatively straightforward: Just divide the numerator by the denominator. So becomes 0.125.

To convert a decimal to a fraction, first determine the place of the rightmost digit of the fraction. For example, in 0.125, the 5 is in the thousandths place. Therefore, the denominator should be 1,000. The numerator of the fraction is just the number without the decimal point. So 0.125 is the same as . Of course, you should reduce the fraction if you can, so the result is .

The following chart shows the conversions for the most common fractions, decimals, and percents. You should know these by heart so that you won’t spend precious time calculating the converted values.

Translation

In fraction and percent problems, especially word problems, you can get confused about which numbers to multiply, divide, add, or subtract. Translating the words into an equation is often helpful in starting a problem. Take the “stem” of the question (the part that contains the question word such as “what”) and translate each word using this chart.

So the question “What percent of 50 is 10?” becomes ×50=10. Then you would solve for n to get the answer, which is 20%.

You can often use translation to help you solve long word problems. Translate the question stem, usually the last sentence, to set up the equation. Replace the items described with numbers from the problem.

1.Paul owns 10 television sets, of which 2 do not work. Of the working television sets, 2 have black-and-white screens and the rest have color screens. What percent of Paul’s working television sets have color screens?

 20% 25% 40% 60% 75%

When you translate the question stem, you should get something like ×working=color. From the problem, you know that there are 10 − 2 = 8 working television sets and 8 − 2 = 6 with color screens. Plugging these numbers into the equation, you get ×8=6. Solving the equation, you get n = 75, so (E) is the correct answer.

Translation is not always the fastest way to solve a problem. For example, you could answer the question about Paul’s television sets by dividing the color televisions by the number of working televisions: = = 75% (Remember your common fraction-to-percent conversions). However, translating is a reliable way to set up a problem when you don’t immediately see the solution.

Translation can also be useful for data sufficiency word problems. Although you don’t need to calculate a numerical answer, setting up the equation can help you determine whether you have enough pieces of information to answer the question. Just set up the equation, but don’t solve it.

2.Brady, Charlie, and Daryl play on the same baseball team. The number of home runs hit by Brady in a particular season is what fraction of the total home runs hit by the three players in that season?

(1) Brady hit 12 home runs in that season.

(2) In that season, the number of home runs hit by Brady was twice the number hit by Charlie and Daryl combined.

When you translate the question stem, you get b = n × (b + c + d) or n = . Consider Statement (1). If you plug in b = 12, you get n = . You can’t solve that equation for n, so narrow your choices to (B), (C), and (E). Now consider Statement (2). You can translate that statement into b = 2(c + d). If you insert that into your equation from the question stem, you get n = or n = . You can solve that by canceling the (c + d) in both the top and bottom of the fraction. (You get n = .) So Statement (2) is sufficient and the correct answer is (B).

Percent Change

You will probably see at least one problem in which the value of some number increases or decreases, and you need to find the percent change. A question might also compare two numbers and ask you to state the difference in percentage terms. For both of these cases, use this formula:

Finding the actual difference is usually pretty simple; just subtract the two numbers. The key is determining which number is the original number. Sometimes the test writers will try to trick you. Just remember the following guidelines:

· If the question mentions “increase” or “greater,” you’re going from a smaller number to a larger number. So the original number is the smaller one.

· If the question says “decrease” or “less,” you’re going from a larger number to a smaller number. So the original number is the larger one.

DRILL 3

1.In a group of 20 tourists, 12 brought cameras. If one half of the tourists with cameras brought disposable cameras, what percent of all the tourists brought disposable cameras?

 12% 20% 30% 40% 60%

2.Lenny can bench press 320 pounds. Ollie can bench press 400 pounds. The weight Ollie can bench press is what percent greater than the weight Lenny can bench press?

 20% 25% 32% 40% 80%

3.The original price of a model X200 laptop computer is reduced by \$1,000 to the new price of \$2,000. What is the percentage change in the price of the X200 laptop computer?

 12% 20% 33% 40% 50%

Comprehensive Fractions, Decimals, and Percents Drill

Remember!

For Data Sufficiency problems in this book, we do not supply the answer choices. The five possible answer choices are the same every time.

1.In an engineering class that contained 50 students, the final exam consisted of two questions. Three fifths of the students answered the first question correctly. If four fifths of the remainder answered the second question correctly, how many students answered both questions incorrectly?

 4 6 10 12 24

2.Forty percent of of what is 20 ?

 400 320 100 4 1

3.After a performance review, Steve’s salary is increased by 5%. After a second performance review, Steve’s new salary is increased by 20%. This series of raises is equivalent to a single raise of

 25% 26% 27% 30% 32%

4.Of the 2,400 animals at the zoo, are primates. If the number of primates were to be reduced by , what percent of the remaining animals would then be primates?

 50% 33% 25% 20% 6.25%

5.If each of the following fractions were written as a decimal, which would have the fewest number of digits to the right of the decimal point?

6.Poetry books make up what percent of the books on Beth’s bookshelf?

(1) Of the 60 books on Beth’s bookshelf, 15 are novels.

(2) There are 12 poetry books on Beth’s bookshelf.

7. =

 −14 – 1 14

8.During one day, a door-to-door brick salesman sold three fourths of his bricks for \$0.25 each. If he had 150 bricks left at the end of the day, how much money did he collect for brick sales that day?

 \$12.50 \$37.50 \$50.00 \$112.50 \$150.00

9.In a group of 24 musicians, some are pianists and the rest are violinists. Exactly of the pianists and exactly of the violinists belong to a union. What is the least possible number of union members in the group?

 12 13 14 15 16

10.Which of the following fractions is equal to the decimal 0.375 ?

Note: Figure not drawn to scale.

11.In the triangle ABC shown above, the two shaded regions make up and of the area of the triangle. The unshaded region makes up what fractional part of the area of the triangle?

12.Justin, Max, and Paul each have a collection of marbles. Justin has 50% fewer marbles than Max has. Max has 30% more marbles than Paul has. If Paul’s collection contains 80 marbles, how many marbles does Justin’s collection contain?

 32 48 52 56 64

13.By what percent did the population of Belleville increase from 2001 to 2003 ?

(1) The population of Belleville was 72,000 people in 2003.

(2) The population of Belleville doubled from 2001 to 2003.

14.

15.If 12 is 20% of 40% of a certain number, what is the number?

 20 24 72 96 150

16.The fuel efficiency of a certain make of car was increased from 30 miles per gallon for last year’s model to 45 miles per gallon for this year’s model. By what percent was the fuel efficiency of the car increased?

 15% 33% 50% 66% 75%

17.In 2002, 30% of the students at Maxwell State University were engineering majors. The number of engineering majors at the university increased by what percent between 2002 and 2003 ?

(1) In 2003, 45% of the students at the university were engineering majors.

(2) The number of engineering majors at the university increased by 750 between 2002 and 2003.

18. =

 0.18 0.2 1.8 2 20

19.If Kim makes a \$30,000 down payment on a house, which represents 20% of the sale price of the house, how much money does Kim still owe on the house?

 \$90,000 \$120,000 \$140,000 \$150,000 \$170,000

20.If the fractions , , , and were ordered from least to greatest, the second smallest fraction in the resulting sequence would be

21.In 1991, the price of a house was 80% of its original price. In 1992, the price of the house was 60% of its original price. By what percent did the price of the house decrease from 1991 to 1992 ?

 20% 25% 33% 40% 60%

22.1+++ =

 1.539 1.5309 1.5039 1.50309 1.05039

23.Of the 400 people in an auditorium, are wearing hats. Of those, are wearing fedoras. How many people in the auditorium are not wearing fedoras?

 20 80 180 220 380

24.What fraction of the cookies in a certain bakery’s window display contains nuts?

(1) Of the cookies in the display, 22 contain nuts.

(2) Of the cookies in the display, 90% do not contain nuts.

Challenge!

Take a crack at this high-level GMAT question.

25.A process manager in a plant wishes to decrease the hours logged by his workforce by 20%, while still retaining the exact same production. If all of the workers in the workforce produce at the same constant rate, by what percent would the workforce need to increase its production?

 10% 20% 25% 33% 50%

Drill 1

1.CUse the Bowtie on the first parentheses to get = = . Bowtie the second parentheses to get + = = = . You’ll make the calculations easier if you reduce as you go. Now just multiply straight across to get × = . Choose (C).

2.EYou could use the Bowtie to compare all five answer choices to . However, you’ll save yourself some time if you first eliminate a few answers by Ballparking. You can eliminate (A) and (B) because is greater than . You can tell because 3 is more than half of 5. Similarly, is less than so it’s also less than . When you Bowtie and , you get 10 versus 9. The 10 goes with , so choose (E).

3.BIan gives ×60=12 baseball cards to Kevin, so he has 60 − 12 = 48 left. Then he gives ×48=12 baseball cards to Paul, so he has 48 − 12 = 36 left. Choose (B).

Drill 2

1.CMove the decimal point two places to the right in both the numerator and the denominator. This gives you . Do the long division and you get 7.5. Choose (C).

2.DIn the numerator, subtract the decimals by lining up the decimal points. The numerator becomes 0.12. In the denominator, multiply 2 × 3 = 6. There is one decimal place in each for a total of two. Put two decimal places in the denominator and it becomes 0.06. When you divide, you need to move the decimal point two places to the right in each number. That gives you = 2. Choose (D).

Drill 3

1.COne half of the 12 tourists with cameras have disposable cameras, so that’s ×12=6 disposable cameras. There are 20 tourists, so = of the tourists have disposable cameras. To change this fraction to a percent, multiply the top and the bottom by 10. That becomes = , or 30%. Choose (C).

2.BDivide the difference by the original amount. The difference is 400 − 320 = 80. To find the original amount, remember that “greater” means starting small and getting big, so the original is the smaller number, 320. The percent difference is , or 25%. Notice that the trap answer of 20% is there to fool you if you mistakenly divide by 400. Choose (B).

3.CUse the percent change formula. The difference is \$1,000. The original price is \$2,000 + \$1,000 = \$3,000. So the percent change is = , or 33%. Choose (C).

Comprehensive Fractions, Decimals, and Percents Drill

1.AThe number of students that answered the first question correctly is × = 30 students. That leaves 50 − 30 = 20 students who answered the first question incorrectly. Of these 20, × = 16 answered the second question correctly. That leaves 20 − 16 = 4 students who missed both questions. Choose (A).

2.AUse the translation technique to help you set up this problem. You get the equation ××n=20. Simplifying, you get n = 20. Multiply both sides by 20 to get n = 400. Choose (A).

3.BThe Joe Bloggs answer is (A), so you should be suspicious of that. It may help for you to make up a number for Steve’s initial salary, say \$100 (sorry, Steve). The 5% raise takes him to \$105. The 20% raise is 20% of the new, \$105 salary, not 20% of the old, \$100 salary. 20% × \$105 = \$21. So his new salary is \$105 + \$21 = \$126. That’s the same as if he had gotten a raise of 26% of his original \$100 salary. Choose (B).

4.DThe number of primates is ×2,400=600. The reduction in primates is ×600=150 primates, leaving 600 − 150 = 450 primates. Don’t forget, however, that this also reduces the total number of animals at the zoo to 2,400 − 150 = 2,250. The fraction of animals at the zoo that are primates is (part over whole), which reduces to . From the conversion chart, you know that’s the same as 20%. Choose (D).

5.BJust write each of these fractions as a decimal. You probably know some of them from the conversion chart, but you can always use the division method if you forget. The conversions are: = 0.125, = 0.2, = , = and = 0.75. 0.2 is the answer because it has only one digit to the right of the decimal point. Choose (B).

6.CYou want to find ×100. With Statement (1), you know the total number of books, but not the number of poetry books. The number of novels is irrelevant because you don’t know whether there are types of books other than poetry books and novels. Narrow the choices to (B), (C), and (E). With Statement (2), you know the number of poetry books, but not the total. Eliminate (B). With both statements, you know both the total and the number of poetry books, allowing you to find the percentage. Choose (C).

7.ADo two subtractions first (using the Bowtie method) and then do the division. In the numerator, you’ll need to convert the mixed fractions. 3 becomes = . 2 becomes = . Applying the Bowtie, you get = = for the numerator. Applying the Bowtie to the denominator, you get = = −. To divide fractions, you flip the divisor (in this case, the denominator) and multiply. This gives you × = − 14. Choose (A).

8.DThe 150 bricks left are of his initial inventory of bricks, so he had 4 × 150 = 600 bricks. He sold 600 − 150 = 450 bricks at \$0.25 each. His sales revenue was \$0.25 × 450 = \$112.50. Choose (D).

9.BTo get the least possible number of union members, you want as many pianists as possible and as few violinists as possible, because is less than . However, you have to have some violinists because the problem states that there are some of each. It’s impossible to have fractional violinists, so the number of violinists is a multiple of 3. It can’t be 3 violinists and 21 pianists because that would give you ×21=10 union pianists. Try 6 violinists and 18 pianists. That gives you ×18=9 union pianists. Combined with the ×6=4 union violinists, that means 9 + 4 = 13 union members total. Note that Joe Bloggs will probably pick the smallest answer when the question asks for the “least possible number.” Choose (B).

10.CYou need to convert the fractions to decimals, either with the conversion chart or by doing the division. = , = 0.2857..., = 0.375. If you want to check the other two answers (depending on where you are in the section), they are: = and = . Another way of solving is to turn the decimal into a fraction and reduce: 0.375 = = . Choose (C).

11.CUse the Bowtie to find the total area of the shaded regions: + = = = . That leaves of the area for the unshaded region. Choose (C).

12.CPaul has 80 marbles, so Max has 30% × 80 = 24 marbles more for a total of 80 + 24 = 104 marbles. Justin has 50% × 104 = 52 fewer marbles than Max for a total of 104 − 52 = 52 marbles. Choose (C).

13.BUse the percent change formula: ×100. With Statement (1), you know the final population, but not the original population. Narrow the choices to (B), (C), and (E). With Statement (2), you know that the population doubled, which is the same as increasing by 100%. Choose (B). You don’t need to be able to find the actual numbers as long as you can answer the question, so (C) is a trap.

14.AYou’ll need to work this problem from the innermost fraction outward. It helps if you rewrite the problem as you work each part. 1 + = = for the bottom. For the next step, 2÷ = 2× = . Next, 2+ = = . Last, 1÷ = 1× = . Choose (A).

15.EUse translation to set up the equation: 12 = ××n. That simplifies to 12 = n. Multiply both sides by to get n = 150. Choose (E).

16.CUse the percent change formula. The difference in fuel efficiency is 45 − 30 = 15. The original number was 30, so the percent change is = = 50%. The key is dividing by the right number. Choose (C).

17.EThe percent change formula is ×100. Statement (1) initially looks like it might be sufficient. However, unless you also know how the overall student populations in the two years compare to each other, you can’t determine whether 45% of the later group is more or less than 30% of the earlier group or by how much. Narrow the choices to (B), (C), and (E). With Statement (2) alone, you know the amount of the change (+750), but not the original number of engineering majors. Eliminate (B). Even with both statements together, you can’t find the original number of engineering majors. You could set up the equation 0.45y − 0.3x = 750, but you can’t solve it because there are two variables. Choose (E).

18.DTo get the numerator of the fraction, just multiply 4 × 6 = 24; then add in the two decimal places to get 0.24. To divide by 0.12, move the decimal point two places to the right in each decimal to make it 24 ÷ 12 = 2. Choose (D).

19.BRestate the information as “\$30,000 is 20% of the sale price.” Then you can translate that sentence to the equation 30,000 = ×n. Solving the equation, you find that n = \$150,000. However, you need to know how much she owes. That’s \$150,000 – \$30,000 = \$120,000. Choose (B).

20.CFirst, put the fractions in order. You can quickly compare fractions to by seeing if the numerator is greater or less than one half the denominator. So is more than , and the other fractions are less. You need the Bowtie to put the other three fractions in order. Comparing to , you get 2 × 8 = 16 versus 5 × 3 = 15, so is greater. Comparing to , you get 3 × 11 = 33 versus 8 × 4 = 32, so is greater. So the final order is , , , , and . The second smallest fraction is . Choose (C).

21.BIt’s helpful to make up a number for the original sale price, say \$100,000. That means the price in 1991 was \$80,000 and the price in 1992 was \$60,000. Use the percent change formula. The difference is \$80,000 – \$60,000 = \$20,000. The original number is \$80,000 because you’re measuring only the 1991 to 1992 time period. So the percent change is = = 25%. Choose (B).

22.CThe easiest way to solve this problem is to convert each fraction to a decimal and then add them. = 0.5, = 0.003, and = 0.0009. This is just like the way you convert decimals to fractions, except reversed. Adding the numbers, you get 1 + 0.5 + 0.003 + 0.0009 = 1.5039. Choose (C).

23.EThe number of people wearing hats is ×400=100. The number of people wearing fedoras is ×100=20. So the number of people not wearing fedoras is 400 − 20 = 380. Choose (E).

24.BTo find the fraction, you want . With Statement (1), you know the number of cookies with nuts, but not the total. Narrow the choices to (B), (C), and (E). With Statement (2), you can find that 100% − 90% = 10% of the cookies contain nuts. A percent is just another way of showing a fraction, so you can answer the question even though you can’t find the actual numbers. Choose (B).

25.CTry plugging in a number (pretty self-explanatory, but we have more information about this in our “Solutions Beyond Algebra” chapter) to solve this problem. Say that the number of hours currently worked by the workforce is 50, and that each hour the workforce is able to produce 2 units of production. This means the workforce is currently producing 100 total units in 50 hours. The manager wants to decrease the work hours by 20%. To determine this reduction, calculate 20% of 50, which is 10 hours. So, if the number of hours is reduced by 20%, the workforce will work only 40 hours. Since the manager wants the production to be the same, the work force must now produce 100 units in 40 hours. So, the work force must produce 2.5 units per hour, which is 0.5 more units an hour than before. Because 0.5 is one fourth of 2, the original production rate, the workforce must produce 25% more units. Choose (C).

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