PSAT/NMSQT Prep 2020 - Princeton Review 2020

Ratios, proportions, and percents
Data analysis

Learning Objectives

After completing this chapter, you will be able to:

· Set up and solve a proportion for a missing value

· Use proportions to perform unit conversions

· Calculate percents and percent change

80/600 SmartPoints^®

How Much Do You Know?

Directions

Try the questions that follow. Show your work so that you can compare your solutions to the ones found in the Check Your Work section immediately after this question set. The “Category” heading in the explanation for each question gives the title of the lesson that covers how to solve it. If you answered the question(s) for a given lesson correctly, and if your scratchwork looks like ours, you may be able to move quickly through that lesson. If you answered incorrectly or used a different approach, you may want to take your time on that lesson.

1. Seven out of every 250 students at a certain university who take a test are expected to score at least 90 percent. If the university gives this test to 12,000 students, how many would be expected to score at least 90 percent?

1. 176

2. 224

3. 300

4. 336

2. A homeowner wants to buy 81 square feet of grass for his yard, but the vendor he uses sells grass only by the square yard. How many square yards of grass does the homeowner need? (1 yard = 3 feet)

1. 9

2. 27

3. 243

4. 729

3. Ethanol can be mixed with gasoline to reduce automobile emissions. Much automotive gasoline is 15% ethanol by volume. An oil company tries decreasing the ethanol content to 6% to lower the cost. If a car with a 14-gallon tank is filled with the 15% blend and a second car with a 10-gallon tank is filled with the 6% blend, how many times more ethanol is in the first car than in the second car?

1. 1.5

2. 2.5

3. 3.5

4. 4.0

4. Kelania owns a bakery, and she adjusts the number of pounds of flour she orders each week based on the number she used the previous week. After the first week of the month, she decreased the number of pounds of flour she ordered by 25%, then increased the number by 10% the following week, then increased the number by an additional 50% in the last week of the month. What is the approximate total percent increase in the number of pounds of flour Kelania ordered from the start of the month until the end of the month?

1. 20%

2. 24%

3. 30%

4. 35%

5. The cost of tuition at a private nonprofit four-year college in 1988 was approximately $15,800. In 2013, the cost of tuition at the same type of college was approximately $30,100. If tuition experiences the same total percent increase over the next 25 years, approximately how much will tuition at a private nonprofit four-year college cost?

1. $44,400

2. $45,800

3. $57,300

4. $66,200

6.

Check Your Work

1. D

Difficulty: Easy

Category: Ratios and Proportions

Getting to the Answer: Assign a variable, say n, to the number of students expected to score at least 90 percent when 12,000 students take the test. Then, set up a proportion and solve for n:

The correct answer is (D).

2. A

Difficulty: Easy

Category: Unit Conversion

Getting to the Answer: Map out your route from starting units to end units, being mindful of the fact that the question deals with units of area (square units). The starting quantity is in square feet and the desired quantity is in square yards. The only conversion factor you need is 3 feet (ft) = 1 yard (yd). Setting up your route to square yards, you get:

This matches (A).

3. C

Difficulty: Medium

Category: Percents

Getting to the Answer: Starting with the 14-gallon tank, plug the known values into the three-part formula: 15% × 14 = ? → 0.15 × 14 = 2.1 gallons of ethanol. Repeat for the smaller tank: 6% × 10 = ? → 0.06 × 10 = 0.6 gallons of ethanol. The question asks how many times more ethanol is in the larger tank, so divide the quantities to get . This matches (C).

4. B

Difficulty: Medium

Category: Percent Change

Getting to the Answer: Remember to avoid merely adding the percentages together. Find each change individually. You’re not given a definite number of pounds of flour in the question, so assume Kelania starts with 100. To save a step with each change, calculate the total amount instead of the weekly increase or decrease. The first change is —25%; so the number of pounds of flour ordered the next week is 75% × 100 = 0.75 × 100 = 75. The second change is +10%, which corresponds to 110% × 75 = 1.1 × 75 = 82.5 pounds. The final change is +50%, which means there are now 150% × 82.5 = 1.5 × 82.5 = 123.75 pounds. The percent change is . Now, round to the nearest percent and you get 24%, which is (B).

5. C

Difficulty: Medium

Category: Percent Change

Getting to the Answer: Find the percent increase using the formula: . Then, apply the same percent increase to the amount for 2013. The amount of increase is 30,100 − 15,800 = 14,300, so the percent increase is over 25 years. If the total percent increase over the next 25 years is the same, the average cost of tuition will be 30,100 × 1.905 = 57,340.50, or about $57,300, which is (C).

Ratios and Proportions

Learning Objective

After this lesson, you will be able to:

· Set up and solve a proportion for a missing value

To answer a question like this:

1. A property is projected to be built with dimensions of 1,245 feet long by 274 feet wide. The contractor wishes to build an exact replica scale model of the property that is 6 feet long. Approximately how many inches wide will the scale model’s width be? (1 foot = 12 inches)

1. 12

2. 16

3. 25

4. 107

You need to know this:

A ratio is a comparison of one quantity to another. When writing ratios, you can compare one part of a group to another part of that group or you can compare a part of the group to the whole group. Suppose you have a bowl of apples and oranges: you can write ratios that compare apples to oranges (part to part), apples to total fruit (part to whole), and oranges to total fruit (part to whole).

Keep in mind that ratios convey relative amounts, not necessarily actual amounts, and that they are typically expressed in lowest terms. For example, if there are 10 apples and 6 oranges in a bowl, the ratio of apples to oranges would likely be expressed as on the PSAT rather than as . However, if you know the ratio of apples to oranges and either the actual number of apples or the total number of pieces of fruit, you can find the actual number of oranges by setting up a proportion (see below).

HINT: Note that the PSAT may occasionally use the word “proportion” to mean “ratio.”

A proportion is simply two ratios set equal to each other, for example, . Proportions are an efficient way to solve certain problems, but you must exercise caution when setting them up. Noting the units of each piece of the proportion will help you put each piece of the proportion in the right place. Sometimes, the PSAT may ask you to determine whether certain proportions are equivalent—check this by cross-multiplying. You’ll get results that are much easier to compare.

Each derived ratio shown above except the last one is simply a manipulation of the first, so all except the last are correct. You can verify this via cross-multiplication (ad = bc in each case except the last).

Alternatively, you can pick equivalent fractions and (a = 2, b = 3, c = 6, d = 9). Cross-multiplication gives 2 × 9 = 3 × 6, which is a true statement. Dividing 2 and 3 by 6 and 9 gives , which is also true, and so on. However, attempting to equate and will not work.

If you know any three numerical values in a proportion, you can solve for the fourth. For example, say a fruit stand sells 3 peaches for every 5 apricots and you are supposed to calculate the number of peaches sold on a day when 20 apricots were sold. You would use the given information to set up a proportion and solve for the unknown:

You can now solve for the number of peaches sold, p, by cross-multiplying:

Alternatively, you could use the common multiplier to solve for p: the numerator and denominator in the original ratio must be multiplied by the same value to arrive at their respective terms in the new ratio. To get from 5 to 20 in the denominator, you multiply by 4, so you also have to multiply the 3 in the numerator by 4 to arrive at the actual number of peaches sold: 4(3) = 12.

You need to do this:

· Set up a proportion and solve for the unknown, either by cross-multiplying or by using the common multiplier.

Explanation:

The ratio of the length of the real property to that of the scale model is . You know the actual width (274 feet), so set up a proportion and solve for the scale model’s width:

The question asks for the answer in inches, not feet, so multiply by 12 inches per foot: . Hence, (B) is correct.

Try on Your Own

Directions

Take as much time as you need on these questions. Work carefully and methodically. There will be an opportunity for timed practice at the end of the chapter.

HINT: You can save time by making the numbers in Q1 more manageable before you attempt to solve. Try making the number 4,000 easier to work with. (But don’t forget to simplify both numerator and denominator!)

1.

2. For every 4,000 snowblowers produced by a snowblower factory, exactly 8 are defective. At this rate, how many snowblowers were produced during a period in which exactly 18 snowblowers were defective?

1. 6,000

2. 9,000

3. 12,000

4. 18,000

3. An engineer is monitoring construction of a 75-foot-long escalator. The difference in height between the two floors being connected was originally supposed to be 40 feet, but due to a calculation error, this figure must be reduced by 25%. The angle between the escalator and the floor must not change in order to comply with the building code. What is the change in length in feet between the original escalator measurement and its corrected value?

1. 18.75

2. 25

3. 56.25

4. 100

4. The number of cars that can safely pass through a stoplight is directly proportional to the length of time in seconds that the light is green. If 9 cars can safely pass through a light that stays green for 36 seconds, how many cars can safely pass through a light that stays green for 24 seconds?

1. 4

2. 6

3. 7

4. 8

5. If the total weight of 31 identical medieval coins is approximately 16 ounces, which of the following is closest to the weight, in ounces, of 97 of these coins?

1. 5

2. 19

3. 50

4. 188

6. HINT: For Q5, assign a variable as the common multiplier in the proportion of the pyramid’s length:width:height, then express the volume in terms of that common multiplier.

7. For a school project, a student wants to build a replica of the Great Pyramid of Giza out of modeling clay. The real Great Pyramid has a square base with side length 750 feet and a height of 500 feet. If the student has 162 cubic inches of clay for her model, what height will her pyramid be in inches?

(The formula for the volume of a pyramid is and is provided in your test booklet.)

8.

Unit Conversion

Learning Objective

After this lesson, you will be able to:

· Set up equivalent ratios to make units cancel

To answer a question like this:

1. City A and city B are 2,000 miles apart, while city C is double the distance from city A than city A is from city B. What is the approximate distance, in inches, between city A and city C? (1 mile = 5,280 feet and 1 foot = 12 inches)

1. 4.3 million

2. 52 million

3. 127 million

4. 253 million

You need to know this:

You can set up a proportion to perform unit conversions. This is especially useful when there are multiple conversions or when the units are unfamiliar.

For example, though these units of measurement are no longer commonly used, there are 8 furlongs in a mile and 3 miles in a league. Say you’re asked to convert 4 leagues to furlongs. A convenient way to do this is to set up a proportion so that equivalent units cancel:

Notice that all the units cancel out except the furlongs, which is the one you want.

You need to do this:

Set up a proportion to make equivalent units cancel. (Keep track of the units by writing them down next to the numbers in the proportion.) You should be left with the units you’re converting into.

Explanation:

City C is twice as far from city A as city A is from city B, so city C is 2(2,000) = 4,000 miles away from city A. Set up a proportion to convert to inches(in):

Therefore, (D) is correct.

Try on Your Own

Directions

Take as much time as you need on these questions. Work carefully and methodically. There will be an opportunity for timed practice at the end of the chapter.

1. Jack is taking a road trip. If he travels 180 miles while his car uses gasoline at a rate of 40 miles per gallon and then travels another 105 miles while his car uses gasoline at a rate of 35 miles per gallon, how many gallons of fuel has his car consumed?

1. 1.5

2. 3.0

3. 4.5

4. 7.5

2. HINT: Begin Q7 by figuring out the cost per ounce of each can of pineapple.

3. If an 8-ounce can of pineapple sells for $0.72 and a 20-ounce can costs $1.10, how many more cents does the 8-ounce can cost per ounce than the 20-ounce can? (100 cents = 1 dollar)

4. An artist is creating a rectangular tile mosaic. Her desired pattern uses 5 green tiles and 3 blue tiles per square foot of mosaic. If the artist’s entire mosaic is 12 feet by 18 feet, how many more green tiles than blue tiles will she need?

5. HINT: Be careful to use the correct units in Q9! What units are used in the question and what units appear in the answer choices?

6. The average college student reads at a rate of about 5 words per second. If the pages of Jorge’s textbook contain an average of 500 words per page, how long will it take him to read a 45-page chapter?

1. 50 minutes

2. 1 hour, 15 minutes

3. 1 hour, 25 minutes

4. 1 hour, 40 minutes

7. Each MRI scan given at a hospital produces about 3.6 gigabits of data. Every night, for 8 hours, the hospital backs up the files of the scans. The hospital computers can upload the MRI scans at a rate of 2 megabits per second. What is the maximum number of MRI scans that the hospital can upload each night? (1 gigabit = 1,024 megabits)

1. 15

2. 16

3. 56

4. 202

8.

Percents

Learning Objective

After this lesson, you will be able to:

· Calculate percents

To answer a question like this:

1. Teachers surveyed their students at two different schools about their favorite classes to find out how many students favored math class. At the first school, they asked 512 students and of those, 12.5% responded favorably. At the second school, 24.8% of 625 students responded favorably. What percent of all the students surveyed responded favorably?

1. 15.4%

2. 19.3%

3. 25.4%

4. 31.9%

You need to know this:

To calculate percents, use this basic equation:

Alternatively, use this statement: [blank] percent of [blank] is [blank]. Translating from English into math, you get [blank]% × [blank] = [blank].

You need to do this:

· Plug in the values for any two parts of the formula and solve for the third.

· In some calculations, you may find it convenient to express percents as decimals. To do this, use the formula above but stop before you multiply by 100% at the end.

Explanation:

Use a variation of the three-part percent formula to answer this question: , where the percent is expressed as a decimal.

First, find the number of students at each school who responded favorably using the formula. For the first school: 512 × 0.125 = 64. For the second school: 625 × 0.248 = 155. Next, find the total number of students who were surveyed at both schools, which was 512 + 625 = 1,137, and the total number who responded favorably, 64 + 155 = 219. Finally, find the percent of people who responded favorably by using the formula one more time:

Of all the students surveyed, about 19.3% responded favorably, making (B) the correct answer.

Try on Your Own

Directions

Take as much time as you need on these questions. Work carefully and methodically. There will be an opportunity for timed practice at the end of the chapter.

1. A company sells dolls for $20 each. It decides to offer a discount of 20% for a month to see how many new customers it can attract. How much will each doll sell for during the month of the discount?

1. $12

2. $14

3. $16

4. $18

2. HINT: For Q12, begin by figuring out what percent of the budget actually goes to lunch.

3. A high school’s Environment Club receives a certain amount of money from the school to host an all-day event. The club budgets 40% of the money for a guest speaker, 25% for educational materials, 20% to rent a hotel conference room, and the remainder for lunch. If the club plans to spend $225 on lunch for the participants, how much does it plan to spend on the guest speaker?

1. $375

2. $450

3. $525

4. $600

4. A bag of marbles contains 60 marbles that are either red, blue, or yellow. If there are 12 blue marbles, what percent of the bag is made of red and yellow marbles?

1. 50%

2. 60%

3. 70%

4. 80%

5. Questions 14 and 15 refer to the following information.

6. The following table shows the chemical makeup of one mole (a unit of measure commonly used in chemistry) of acetone and the approximate mass of a mole of each component element.

7. HINT: For Q14, use the percent formula. Which is the part and which is the whole?

8. Oxygen makes up what percent of the mass of one mole of acetone? Round your answer to the nearest whole percent.

9. If a chemist starts with 1,800 grams of acetone and uses up 930 grams, approximately how many moles of carbon are left? Round your answer to the nearest whole mole.

10.

Percent Change

Learning Objective

After this lesson, you will be able to:

· Calculate percent change

To answer a question like this:

1. For three days in a row, Bonnie changes the amount of lemonade made for her lemonade stand. First, she increases the amount made by 25%. Then, she decreases it by 10%. Finally, she increases it by 35%. What is the net percent increase in the amount of lemonade Bonnie made on the third day compared to original amount before the three days of changes, to the nearest whole percent? (The percent sign is understood after your answer. For example, if the answer is 15.1%, grid in 15.)

You need to know this:

You can determine the percent change in a given situation by applying this formula:

Sometimes, more than one change will occur. Be careful here, as it can be tempting to take a “shortcut” by just adding two percent changes together (which will almost always lead to an incorrect answer). Instead, you’ll need to find the total amount of the increase or decrease and then apply the formula.

You need to do this:

· Calculate the actual increase or decrease.

· Divide by the original amount (not the new amount!).

· Multiply by 100%

Explanation:

There isn’t any starting number given, so pick 100 and then calculate the actual change. An increase of 25% brings the amount of lemonade made to 125. A 10% decrease from 125 brings the amount of lemonade made to 125 − 12.5 = 112.5. Lastly, an increase of 35% puts the final amount of lemonade made at 112.5 + 0.35(112.5) = 112.5 + 39.375 = 151.875. The actual increase, then, is 151.875 − 100 = 51.875.

Now, plug this increase into the percent change formula, using your starting value of 100 as the denominator:

Round up and grid in 52.

Try on Your Own

Directions

Take as much time as you need on these questions. Work carefully and methodically. There will be an opportunity for timed practice at the end of the chapter.

1. A used car dealership initially prices a car at $12,000. When the car fails to sell, the dealership reduces the price to $10,500. During a holiday sale, the dealership drops the price of the car an additional 5 percent below the reduced price. To the nearest tenth of a percent, what is the total percent discount from the car’s initial price to the holiday sale price?

1. 15.5%

2. 16.9%

3. 17.5%

4. 20%

2. HINT: The first thing you’ll want to do for Q17 is figure out how much sand and gravel was sold this year.

3. Last year, a sand and gravel company sold 280 tons of gravel and 220 tons of sand. This year, the company sold 20 percent more gravel by weight and 25 percent more sand by weight than it sold last year. By approximately what percent did the total weight of sand and gravel sold increase this year over last year?

1. 22

2. 45

3. 56

4. 111

4. HINT: Remember that when picking numbers for Q18, you don’t have to pick realistic values. Pick numbers that are easy to work with in the given situation. For percent questions, the number is usually 100.

5. Over the past decade, the population of a certain town increased by 20 percent. If the population of the town increases 15 percent over the next decade, by what percent will the population have increased over the entire two-decade period?

1. 33%

2. 35%

3. 38%

4. 43%

6. Malik purchased a mutual fund to help save for his daughter’s college costs. During the first year, the price of the fund increased by 15 percent. The following year, the price increased by an additional 12 percent. To the nearest percent, what is the percent increase in the price of the stock for the two years?

1. 24

2. 27

3. 28

4. 29

7. HINT: What information do you need to determine in order to answer Q20 correctly?

8. There are currently 6,210 fish in a lake. If the number of fish in the lake increased by 15 percent during the last year and 20 percent during the year before that, how many more fish are in the lake currently than in the lake two years ago?

9.

On Test Day

When a question features multiple percentages, you have to make a key strategic decision: can I do the arithmetic on the percentages themselves and get the answer right away or do I have to calculate each percentage individually and do the arithmetic on the actual values?

For example, suppose a car traveling 50 miles per hour increases its speed by 20 percent and then decreases its speed by 20 percent. Can you just say that its final speed is 50 miles per hour since +20% − 20% = 0? No, because after a 20% increase, the car’s speed becomes 120% of the original: 1.2(50) = 60. When the car “decreases its speed by 20 percent,” that 20 percent is calculated based on the new speed, 60, not the original speed, and 20 percent of 60 is greater than 20 percent of 50. Thus, the car’s final speed is lower than its starting speed: 50(1.2)(0.8) = 48 miles per hour.

By contrast, suppose you have to find how many more meat eaters than vegans live in a certain region where there are 13,450 residents, given that 62 percent of them eat meat and 8 percent of them are vegan. It may be tempting to find 62 percent of 13,450 (0.62 × 13,450 = 8,339), then find 8 percent of 13,450 (0.08 × 13,450 = 1,076), and finally subtract those two numbers to get the answer (8,339 − 1,076 = 7,263). This is a waste of time, though. Instead, you can quickly find the difference between the two percentages (62 − 8 = 54) and take 54 percent of the total to get the answer in one step: 13,450 × 0.54 = 7,263, the same answer.

If you can do arithmetic using the percentages but choose to do arithmetic on the raw numbers instead, you’ll waste time doing unnecessary work. But if you can’t do arithmetic on the percentages (as in the first example) but do anyway, then you’ll get an incorrect answer. So, being able to tell whether you can or can’t do the arithmetic on the percentages is a useful skill.

Luckily, the fundamental principle is simple: you can always do arithmetic on the percentages as long as the percentages are out of the same total. If the totals are different, then you must convert the percentages into actual values. Practice applying this principle on the following question.

1. There are 400 seniors and 420 juniors in a certain high school. Of the seniors, 65% are eligible for an advanced placement world history course. Among the juniors, 75% are not eligible to enroll in that course. How many more seniors than juniors could enroll in the course?

The answer and explanation can be found at the end of this chapter.

How Much Have You Learned?

Directions

For testlike practice, give yourself 15 minutes to complete this question set. Be sure to study the explanations, even for questions you got right. They can be found at the end of this chapter.

1. A certain city has 2,625 businesses and has a ratio of 5:2 of businesses that do not require safety inspections to those that do require safety inspections. Of the businesses that were required to have inspections, 12% had safety violations. How many businesses that required inspections did NOT have any safety violations?

1. 90

2. 315

3. 660

4. 2,310

2. An average consumer car can travel 120 miles per hour under controlled conditions. An average race car can travel 210 miles per hour. How many more miles can the race car travel in 30 seconds than the consumer car?

1.

2. 1

3.

4. 45

3. Engine oil often contains zinc, which reduces engine wear. Company A’s oil contains 4% zinc and company B’s oil contains 9%. Suppose a car uses 8 pints of company B’s oil and a truck uses 6 quarts of company A’s oil. How many times more zinc is in the car’s oil than in the truck’s? (1 quart = 2 pints)

1. 0.34

2. 0.67

3. 1.5

4. 3

4. When a consignment store gets a used piece of furniture to sell, it researches the original price and then reduces that price by 40% to determine the price of the used piece. Every 30 days after that, the price of the used piece is marked down an additional 20% until it is sold. The store gets a piece of used furniture on July 15. If the original price of the furniture was $1,050 and it is sold on September 5, what is the final selling price?

1. $258.05

2. $322.56

3. $504.00

4. $630.00

5. An amusement park is building a scale model of an airplane for a three-dimensional ride. The real airplane measures 220 feet from nose to tail. The amusement park plans to make the model airplane 36 feet long. If the wingspan of the real plane is 174 feet, approximately how many feet long should the wingspan on the ride’s model airplane be?

1. 17

2. 28

3. 35

4. 45

6. Luca is from a country that uses the metric system and is visiting his cousin Drew in the United States, which uses the standard, or English, system. He gives Drew a family recipe for bread. The recipe is for one loaf and calls for 180 milliliters of milk. Drew wants to make 5 loaves. If 1 U.S. cup equals 236.588 milliliters, approximately how many cups of milk will Drew need?

1.

2.

3.

4.

7. There are about 3 feet per meter and 1,000 meters per kilometer. Two cities are 1,800,000 feet apart; about how many kilometers apart are the two cities?

8. A professional speedway for Motorsport events wishes to build two new racetracks with the same proportions but different sizes, one for adult races and one for teen races. The racetracks will each have two straightaways, one shorter and one longer.

If the adult track has straightaways that are 100 meters long and 150 meters long, then what is the length of the longer teen track straightaway if the shorter teen track straightaway is 50 meters long?

1. 25

2. 50

3. 75

4. 100

9. Last year, a farmer had 350 acres planted in corn and 160 acres planted in soybeans. This year, the farmer reduced the acreage planted in corn by 20% and reduced the acreage planted in soybeans by 15%. By approximately what percent did the farmer reduce the total acreage planted in corn and soybeans?

1. 16.4%

2. 16.9%

3. 17.5%

4. 18.4%

10.A store starts a sale to attract more customers. Mandy notices in the newspaper that the store has comforters on sale for 40% less than the original price. If Mandy pays $89.10 after the discount and a 10% sales tax on the discounted price, what is the original price of the comforter?

1. $133

2. $135

3. $139

4. $141

11.

Reflect

Directions: Take a few minutes to recall what you’ve learned and what you’ve been practicing in this chapter. Consider the following questions, jot down your best answer for each one, and then compare your reflections to the expert responses on the following page. Use your level of confidence to determine what to do next.

What is a ratio and how is it different from a proportion?

If you’re given a ratio of one quantity to another, what can you say about the total number of quantities?

When doing unit conversions, how can you make sure you’re doing them correctly?

Suppose the value of something increases by 20 percent. How can you calculate the final value in the fewest number of steps? What if the value decreases by 20 percent?

What is the percent change formula and what is the biggest pitfall to avoid when using it?

EXPERT RESPONSES

What is a ratio and how is it different from a proportion?

A ratio is the relative comparison of one quantity to another. For example, if the ratio of dogs to cats in an animal shelter is 3 to 5, then there are 3 dogs for every 5 cats. A proportion is two ratios set equal to each other.

If you’re given a ratio of one quantity to another, what can you say about the total number of quantities?

Given a ratio, you know that the total must be a multiple of the sum of the ratio’s parts. For example, if the ratio of dogs to cats is 3 to 5, then the total number of dogs and cats must be a multiple of 3 + 5, or 8. This means that when the PSAT gives you one ratio, it’s actually giving you several. If you’re told that dogs:cats = 3:5, then you also know that dogs:total = 3:8 and cats:total = 5:8. You can use this “hidden” knowledge to your advantage.

When doing unit conversions, how can you make sure you’re doing them correctly?

To do unit conversions correctly, set up the conversion in whichever way makes units cancel. For example, to convert 3 feet into inches, you multiply 3 feet by 12 inches per foot, because it cancels out the feet unit. If instead you multiplied 3 feet by 1 foot per 12 inches, then the resulting units would be “feet squared per inch,” which makes no sense.

Suppose the value of something increases by 20 percent. How can you calculate the final value in the fewest number of steps? What if the value decreases by 20 percent?

The fastest way to increase a value by 20 percent is to multiply it by 1.2, which is 100% + 20% = 120%. Similarly, to decrease something by 20 percent, you multiply it by 0.8, which is 100% − 20% = 80%.

What is the percent change formula and what is the biggest pitfall to avoid when using it?

The percent change formula is as follows:

A common mistake is to put the new amount on the bottom of the fraction rather than the original amount.

NEXT STEPS

If you answered most questions correctly in the “How Much Have You Learned?” section, and if your responses to the Reflect questions were similar to those of the PSAT expert, then consider Ratios, Proportions, and Percents an area of strength and move on to the next chapter. Come back to this topic periodically to prevent yourself from getting rusty.

If you don’t yet feel confident, review those parts of this chapter that you have not yet mastered and try the questions you missed again. As always, be sure to review the explanations closely.

Answers and Explanations

1. B

Difficulty: Easy

Strategic Advice: When ratios involve large numbers, simplify if possible to make the calculations easier.

Getting to the Answer: Let b equal the number of snowblowers produced. Set up a proportion and solve for b. Be sure to match the units in the numerators and in the denominators on both sides:

This matches (B).

2. A

Difficulty: Medium

Getting to the Answer: Draw a diagram to make sense of the given situation. Your diagram should look similar to what’s shown:

The two triangles are similar, which means you can use a proportion to answer the question. First, find the correct height by taking 25% of 40, which is 10, and deducting that from 40 to get 30. Keeping the heights on the left of your proportion and the hypotenuses on the right, you have . Reduce the left side to get , then cross-multiply to eliminate the fractions: 225 = 4x. Solving for x yields 56.25. But don’t stop yet: the question asks for the difference in escalator length, not the new length. Subtract 56.25 from 75 to get 18.75, which matches (A).

3. B

Difficulty: Easy

Getting to the Answer: To answer a question that says “directly proportional,” set two ratios equal to each other and solve for the missing amount. Don’t forget—match the units in the numerators and in the denominators on both sides.

Let c equal the number of cars that can safely pass through a light that lasts 24 seconds. Set up the proportion and solve:

Therefore, (B) is correct.

4. C

Difficulty: Easy

Getting to the Answer: Set up a proportion and cross-multiply to solve:

Choice (C) is correct. Given that the choices are spaced far apart, you could have used estimating to answer this question. Since 97 is about 3 times 31, look for the choice that is about 3 times 16, which is 48. Only (C) is close.

5. 6

Difficulty: Hard

Getting to the Answer: The formula for the volume of a pyramid is . The pyramid has a square base, so the length and width are equal. The proportion of the length:width:height is 750:750:500, which reduces to 3:3:2. Expressed with a common multiplier, this is 3x:3x:2x.

Putting these proportions into the volume equation gives you:

The question states that the student has 162 cubic inches of modeling clay, so this value can be input for V to solve for x as follows:

The last piece to remember is that height in the equation was replaced with 2x, so the height of the student’s model will be 6 inches. Grid in 6.

1. D

Difficulty: Easy

Getting to the Answer: To determine the equations to find the gallons of fuel used during each leg of Jack’s trip, set up the units to cancel out. Plugging in values for the first leg, you get:

For the second leg:

Added together, there are 4.5 + 3 = 7.5 gallons of fuel used, which matches (D).

2. 3.5

Difficulty: Easy

Getting to the Answer: Whenever you’re asked for the cost of something per a set measurement, think unit rates. First, determine the price per ounce (i.e., the unit rate) for the smaller can:

Next, do the same for the larger can:

Take the difference to get $0.035, which is 3.5 cents. Pay careful attention to the way the question is worded—grid in 3.5, not .035, because the question asks how many more cents, not dollars.

3. 432

Difficulty: Medium

Getting to the Answer: Determine the number of times the pattern repeats and then find the corresponding number of green tiles. The question states that the ratio of green to blue tiles is 5:3 and that the pattern appears once per square foot. There are 12 × 18 = 216 square feet in the mosaic, meaning there are 5 × 216 = 1,080 green tiles and 3 × 216 = 648 blue tiles. Taking the difference gives 1,080 − 648 = 432 more green tiles than blue. Grid in 432.

4. B

Difficulty: Medium

Getting to the Answer: Pay careful attention to the units. As you read the question, decide how and when you will need to convert units. The answer choices are given in hours and minutes, but it’s easier to solve for m in minutes by setting up one large conversion:

Because 75 minutes is not an answer choice, convert it to hours and minutes: 75 minutes = 1 hour, 15 minutes, (B).

5. A

Difficulty: Hard

Getting to the Answer: Don’t let all the technical words in this question overwhelm you. Solve it step-by-step, examining the units as you go. First, use the factor-label method to determine the number of megabits the computer can upload in one night (8 hours):

Next, convert this amount to gigabits (because the information about the scans is given in gigabits, not megabits):

Finally, each scan produces about 3.6 gigabits of data, so divide this number by 3.6 to determine how many scans the computer can upload to the remote server: scans. You should round this number down to 15, (A), because the computer cannot complete the 16th scan in the time allowed.

1. C

Difficulty: Easy

Getting to the Answer: For discount problems, multiply the complement of the discount instead of subtracting the discount. The complement is 100% − discount (20%) = 80%. This will give the price of the doll after the discount. Use d as the variable for the price of the discounted doll:

(C) is correct.

2. D

Difficulty: Medium

Getting to the Answer: The total budget can be represented by 100%, so start there. The percent of the budget spent on lunch is 100% − 40% − 25% − 20% = 15%. You’re told that the club plans to spend $225 on lunch. Let x be the total amount of the budget in dollars. Then 15% of x is 225, so 0.15x = 225. Solving this equation for x yields x = 1,500. The total budget is $1,500. Of this amount, 40% was budgeted for a guest speaker, or 0.4 × $1,500 = $600. Choice (D) is correct.

3. D

Difficulty: Medium

Getting to the Answer: The question asks about the percent that is NOT blue marbles, so calculate the percent of blue marbles and subtract that from 100%.

Choice (D) is correct.

4.  28

Difficulty: Medium

Getting to the Answer: Use the formula:

To use the formula, find the part of the mass represented by oxygen: there is 1 mole of oxygen and it has a mass of 16 grams. Next, find the whole mass of one mole of acetone: 1 mole oxygen (16 g) + 3 moles carbon (3 × 12 = 36 g) + 6 moles hydrogen (6 × 1 = 6 g) = 16 + 36 + 6 = 58. Now, use the percent formula:

Before you grid in your answer, make sure you follow the directions—round to the nearest whole percent. Grid in 28.

5.  45

Difficulty: Hard

Strategic Advice: The question contains several steps. Think about the units given in the question and how you can use what you know to find what you need.

Getting to the Answer: Start with grams of acetone: the chemist starts with 1,800 and uses up 930, so there are 1,800 − 930 = 870 grams left. From the previous question, you know that one mole of acetone has a mass of 58 grams, so there are moles of acetone left. Don’t grid in this amount because you’re not finished yet! The question asks for the number of moles of carbon, not acetone. According to the table, each mole of acetone contains 3 moles of carbon, so there are 15 × 3 = 45 moles of carbon left. Grid in 45.

1. B

Difficulty: Medium

Getting to the Answer: To find the total percent discount, you’ll first need to determine the total discount in actual price. The first change is given: $12,000 − $10,500 = $1,500. The second change is a 5% discount, which can be calculated using the price after the first reduction: (.05)$10,500 = $525. Therefore, the total drop in price is $1,500 + $525 = $2,025. Now, apply the percent change formula:

Since the question asks for the percent change to the nearest tenth of a percent, round the answer to 16.9%. That’s choice (B).

2. A

Difficulty: Medium

Getting to the Answer: To find the total percent increase in weight of sand and gravel sold, first find the total actual change in weight. The total weight sold this year for gravel is 280 + (0.20 × 280) = 336. The total weight sold for sand is 220 + (0.25 × 220) = 275. Therefore, this year’s total weight is 336 + 275 = 611. Last year’s total weight was 280 + 220 = 500. Thus, the change from last year to this year is 611 — 500 = 111. Now, apply the percent change formula:

Choice (A) is correct.

3. C

Difficulty: Medium

Getting to the Answer: One key piece of information not given is the initial population of the town. Since this is a percent change question, pick 100 for the initial population. (Your number doesn’t have to be realistic, only easy to work with.) Over the last decade, the population increased by 20%, so that is an increase from 100 to 120. Over the next decade, if the population increases by 15%, the total population would become 120 + (0.15)120 = 120 + 18 = 138. Therefore, the total increase in actual population over the two decades is 138 — 100 = 38. Now, apply the percent change formula:

Choice (C) is correct.

4. D

Difficulty: Medium

Getting to the Answer: Since the price of the mutual fund is not given and this is a percentage change question, plug in $100 for the initial price and then calculate the changes in sequence. After the first year, the price would have been $100 × (1 + 0.15) = $115. Then, the price at the end of the second year would have been $115 × (1 + 0.12) = $128.80. This is a 28.8% increase from the original $100 price. The question asks for the answer to the nearest percent, which is 29%, so (D) is correct.

5. 1710

Difficulty: Hard

Getting to the Answer: Work backward using the known information to find the actual number of fish in the lake for each of the past two years. You know the increase in the number of fish over the last year was 15%, so you can set up the following equation, where x represents the actual number of fish last year:

Now, do the same process for the previous year. You know that 5,400 represents the number of fish after an increase of 20%, so you can call the original number of fish at the beginning of the first year y and use the following equation to find y:

Now you have the starting number of fish from two years ago, so you can subtract this from the current number to find the actual change in the number of fish over the entire two years: 6,210 — 4,500 = 1,710.

Grid in 1710.

1. 155

Difficulty: Medium

Category: Percents

Strategic Advice: The total numbers of seniors and juniors are different, so you’ll have to apply the given percentages to the number of students in each class to determine the numbers of students eligible for the course in each class, then subtract the results to find the difference. Be careful: the percentage for juniors is stated as those who are NOT eligible.

Getting to the Answer: The number of seniors eligible is . The percentage of juniors who are not eligible is 75%, so the percentage who are eligible is 100% − 75% = 25%. Thus, the number of eligible juniors is . The difference is 260 − 105 = 155. Grid in 155.

1. C

Difficulty: Medium

Category: Percents

Getting to the Answer: Break the actual question into short steps. First, find the number of businesses that were required to have inspections. There are 2,625 businesses. The part to part ratio of businesses that require inspections to those that do not is 2:5, so the ratio of those business that require inspections to the total number of businesses is . So, the total number of businesses that need inspections is . Thus, the number of businesses that had violations, 12% of the those, is 0.12 × 750 = 90.

Finally, find the number of businesses that did NOT have violations by subtracting 90 from 750 to get 750 − 90 = 660 businesses that did not have any safety issues. Choice (C) is correct.

2. A

Difficulty: Medium

Category: Unit Conversion

Getting to the Answer: Let the units in this question guide you to the solution. The speeds of the cars are given in miles per hour, but the question asks about the number of miles each car can travel in 30 seconds, so convert miles per hour to miles per second and then multiply by 30 seconds. The difference in speeds between the two vehicles is 210 — 120 = 90 miles per hour. Now, set up a conversion:

The race car can travel miles farther in 30 seconds, so (A) is correct.

3. C

Difficulty: Hard

Category: Unit Conversion

Getting to the Answer: It’s easiest to compare two amounts when they are written in the same units, so start by converting the car’s pints to quarts and then go from there. The conversion from pints to quarts is straightforward:

Next, find the amount of zinc in each car using the percent formula: Percent × whole = part. Write the percents as decimals and multiply:

Finally, compare the amount in the car to the amount in the truck: . The car has 1.5 times as much zinc in its oil as the truck, which is (C).

4. C

Difficulty: Hard

Category: Percent Change

Getting to the Answer: Draw a chart or diagram with the various price reductions for each 30 days. Determine the percent change and new price for each date.

You can stop here because the item was sold on September 5, which is not 30 days after August 15. The final selling price is $504, (C).

5. B

Difficulty: Medium

Category: Ratios and Proportions

Getting to the Answer: Set up a proportion. Try writing the proportion in words first, then fill in the values:

The ride’s model airplane wingspan is approximately 28 feet, (B).

6. C

Difficulty: Easy

Category: Unit Conversion

Getting to the Answer: Break the question into short steps and solve each step, checking units as you go. First, find the total number of milliliters (mL) of milk Drew will need for 5 loaves:

Second, convert the total number of milliliters needed to cups:

Drew will need about 3.8 or cups of milk, (C). Note that estimating 900 divided by 236 as a little less than 4 is sufficient for answering this question without the use of a calculator, based on the answer choices.

7. 600

Difficulty: Easy

Category: Unit Conversion

Getting to the Answer: For unit conversions, start by deciding whether to multiply or divide. The question asks you to convert from feet to kilometers, so you should divide because a foot is a smaller measurement than a kilometer is. In other words, there will be fewer kilometers than feet. Convert feet into meters, and then meters into kilometers:

Grid in 600.

8. C

Difficulty: Easy

Category: Ratios and Proportions

Getting to the Answer: Set up a proportion and cross-multiply to solve for the unknown:

Thus, (C) is correct.

9. D

Difficulty: Medium

Category: Percent Change

Getting to the Answer: To find the total percent reduction in total acreage, first find the total actual reduction in acreage. For corn, that’s (0.20 × 350) = 70 acres; for soybeans, it’s 0.15(160) = 24 acres. Therefore, the total change in acreage planted in corn and soybeans is 70 + 24 = 94 acres. The total number of acres planted in corn and soybeans was 350 + 160 = 510 acres. Use this information to apply the percent change formula:

(D) is correct.

10.B

Difficulty: Hard

Category: Percent Change

Getting to the Answer: Set up an equation with the original price having the variable C. Instead of subtracting the 40% discount, use the complement and multiply by that: 100% − 40% = 60% = 0.6, so the discounted price is 0.60C. The sales tax is a percentage of this discounted price, so multiply by 100% + tax (10%) = 110% = 1.1, which gives 1.1(0.60C). This amount should equal what Mandy actually paid: 1.1(0.60C) = 89.10. Solve for C:

Thus, (B) is correct. You could also solve this question using backsolving.