Rates, ratios, proportions, and percentages - Problem solving & Data analysis - Math

PSAT/NMSQT Prep 2019 - Princeton Review 2019

Rates, ratios, proportions, and percentages
Problem solving & Data analysis
Math

CHAPTER OBJECTIVES

By the end of this chapter, you will be able to:

1. Use the Kaplan Method for Multi-Part Math Questions to answer Problem Solving and Data Analysis questions effectively

2. Solve multi-part question sets involving rates, ratios, and proportions

3. Use appropriate formulas to find percentages and single or multiple percent changes

SMARTPOINTS

Point Value

SmartPoint Category

Point Builder

Kaplan Method for Multi-Part Math Questions

80 Points

Rates, Ratios, Proportions, and Percentages

Prepare

The PSAT contains multiple-choice and grid-in questions, as well as multi-part math question sets. These question sets have multiple parts that are based on the same scenario and may require more analysis and planning than a typical multiple-choice question. To help you answer these questions effectively, use the Kaplan Method for Multi-Part Math Questions.

KAPLAN METHOD FOR MULTI-PART MATH QUESTIONS

1. Step 1: Read the first question in the set, looking for clues

2. Step 2: Identify and organize the information you need

3. Step 3: Based on what you know, plan your steps to navigate the first question

4. Step 4: Solve, step-by-step, checking units as you go

5. Step 5: Did I answer the right question?

6. Step 6: Repeat for remaining questions, incorporating results from the previous question if possible

The next few pages will walk you through each step in more detail.

Step 1: Read the first question in the set, looking for clues

· Focus all your energy here instead of diluting it over the whole set of questions; solving a multi-part question in pieces is far simpler than trying to solve all the questions in the set at once. Further, you may be able to use the results from earlier parts to solve subsequent ones. Don’t even consider the later parts of the question set until you’ve solved the first part.

· Watch for hints about what information you’ll actually need to use to answer the questions. Underlining key quantities is often helpful to separate what’s important from extraneous information.

Step 2: Identify and organize the information you need

If you think this sounds like the Kaplan Method for Math, you’re absolutely correct. You’ll use some of those same skills. The difference: a multi-part math question is just more involved with multiple pieces.

· What information am I given? Jot down key notes, and group related quantities to develop your strategy.

· What am I solving for? This is your target. As you work your way through subsequent steps, keep your target at the front of your mind. This will help you avoid unnecessary work (and subsequent time loss). You’ll sometimes need to tackle these questions from both ends, so always keep your goal in mind.

Expert Tip

Many students freeze when they encounter a question with multiple steps and seemingly massive amounts of information. Don’t worry! Take each piece one at a time, and you won’t be intimidated.

Step 3: Based on what you know, plan your steps to navigate the first question

· What pieces am I missing? Many students become frustrated when faced with a roadblock such as missing information, but it’s an easy fix. Sometimes you’ll need to do an intermediate calculation to reveal the missing piece or pieces of the puzzle.

Step 4: Solve, step-by-step, checking units as you go

· Work quickly but carefully, just as you’ve done on other PSAT math questions.

Step 5: Did I answer the right question?

· As is the case with the Kaplan Method for Math, make sure your final answer is the requested answer.

· Review the first question in the set.

· Double-check your units and your work.

Step 6: Repeat for remaining questions, incorporating results from the previous question if possible

· Now take your results from the first question and think critically about whether they fit into the subsequent questions in the set. Previous results won’t always be applicable, but when they are, they often lead to huge time savings. But be careful—don’t round results from the first question in your calculations for the second question—only the final answer should be rounded.

When you’ve finished, congratulate yourself for persevering through such a challenging task. A multi-part math question is likely to be one of the toughest on the PSAT. If you can ace these questions, you’ll be poised for a great score on Test Day. Don’t worry if the Kaplan Method seems complicated; we’ll walk through an example shortly.

Expert Tip

Because these question sets take substantially more time, consider saving multi-part math questions for last.

RATES, MEASUREMENT, AND UNIT CONVERSIONS

By now, you’ve become adept at using algebra to answer many PSAT math questions, which is great, because you’ll need those algebra skills to answer questions involving rates. You’re likely already familiar with many different rates—kilometers per hour, meters per second, and even miles per gallon are all considered rates.

A fundamental equation related to rates is “Distance = rate × time” (a.k.a. the DIRT equation—Distance Is Rate × Time). If you have two of the three components of the equation, you can easily find the third. An upcoming multi-part math example demonstrates this nicely.

You’ll notice units of measurement are important for rate questions (and others that require a unit conversion) and, therefore, also an opportunity to fall for trap answers if you’re not careful. How can you avoid this? Use the factor-label method (also known as dimensional analysis). The factor-label method is a simple yet powerful way to ensure you’re doing your calculations correctly and getting an answer with the requested units.

For example, suppose you’re asked to find the number of cups there are in two gallons. First, identify your starting quantity’s units (gallons), and then identify the end quantity’s units (cups). The next step is to piece together a path of relationships that will convert gallons into cups, canceling out units as you go. Keep in mind that you will often have multiple stepping stones between your starting and ending quantities, so don’t panic if you can’t get directly from gallons to cups.

The test makers won’t expect you to know English measurements by heart. Instead, they’ll provide conversion factors when needed. For example, a gallon is the same as 4 quarts, every quart contains 2 pints, and a pint equals 2 cups. And there you have it! Your map from gallons to cups is complete. The last step is to put it together as a giant multiplication question. Each relationship, called a conversion factor, is written as a fraction. The basic rules of fraction multiplication apply, so you can cancel a unit that appears in both the numerator and denominator.

Note

The PSAT will not require you to memorize conversions for conventional units. If the test asks you to convert miles into inches, for example, you will be provided with enough conversion factors to answer the question.

Follow along as we convert from gallons to quarts to pints to cups using the factor-label method:

The DIRT equation is actually a variation of this process. Suppose you travel at 60 mph for 5 hours. You would calculate the distance traveled using the equation miles. The units for hours cancel out, leaving only miles, which is precisely what you’re looking for, a distance. This built-in check is a great way to ensure your path to the answer is correct. If your units are off, check your steps for mistakes along the way. The PSAT will never ask you for a quantity such as miles4 or gallons3, so if you end up with funky units like that, you’ve made an error somewhere in your work.

Note

When using the factor-label method, don’t be afraid to flip fractions and rates to make the units cancel out as needed.

The following question demonstrates the factor-label method in a test-like question:

1. A homeowner wants to buy 81 square feet of grass for his yard, but the vendor he uses only sells grass 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

Work through the Kaplan Method for Math step-by-step to solve this question. The following table shows Kaplan’s strategic thinking on the left, along with suggested math scratchwork on the right.

Strategic Thinking

Math Scratchwork

Step 1: Read the question, identifying and organizing important information as you go


You’re asked how many square yards of grass the homeowner needs. You know he needs 81 square feet of grass.

81 ft2 grass needed

Step 2: Choose the best strategy to answer the question


How do you convert from square feet to square yards?

Use factor-label method

What are the starting and ending quantity units? Which conversion factors are needed?


You’re starting with square feet and need to convert to square yards. You know that 1 yd = 3 ft, but be careful: 1 yd2 is not the same as 3 ft2. Consider each feet-to-yards conversion separately.

starting qty: 81 ft2

end qty: ? yd2

You’ll need to multiply by your conversion factor twice. Remember your rules for exponents: to cancel out ft2, you’ll need to divide by ft2.

Step 3: Check that you answered the right question


You’ve correctly converted from square feet to square yards to get the correct answer, (A).

9 yd2

Note

The conversion from feet to yards is not the same as the conversion from square feet to square yards (or cubic feet to cubic yards). Trap answers will often use incorrect conversion factors. Be particularly careful when dealing with area or volume conversions that have multiple dimensions.

Next, you’ll walk through a test-like multi-part question that involves rates. Follow along with the Kaplan Method and think about how knowledge of rates and conversion factors is used to get to the answer.

Remember, even though these questions have multiple parts, you’ll rely on the same math skills you’d use in a simple multiple-choice question to solve each part. If you find that there are missing pieces or missing quantities, use techniques such as the factor-label method to bridge the gap. Also keep in mind that you may be able to use the answer from one part as a shortcut to answering the next part. If you do, don’t round until the final answer, especially on grid-in questions.

1. Questions 2 and 3 refer to the following information.

2. Three business professionals are traveling to New York City for a conference. Mr. Black is taking a train from Philadelphia that leaves at 7:00 A.M. Eastern Time (ET), Ms. Weiss will begin driving in from Newark at 6:15 A.M. ET, and Dr. Grey plans to catch a plane from Chicago that departs at 8:30 A.M. Central Time (one hour behind ET).

2. With traffic, Ms. Weiss averages a speed of 25 mph for the length of her 20-mile commute. Mr. Black’s train will get to NYC, a 100-mile journey, at 8:15 A.M. ET. How much longer, in minutes, will Mr. Black travel than Ms. Weiss?

3. Dr. Grey’s flight is delayed 45 minutes. If the plane flies at 332 mph for the 830-mile flight, how many hours after Ms. Weiss arrives will Dr. Grey arrive? Round your answer to the nearest tenth of an hour.

Work through the Kaplan Method for Multi-Part Math Questions step-by-step to solve this set of questions. The following table shows Kaplan’s strategic thinking on the left, along with suggested math scratchwork on the right.

Strategic Thinking

Math Scratchwork

Step 1: Read the first question in the set, looking for clues

You know the travel start times of the three colleagues, as well as Ms. Weiss’s speed and commute length and Mr. Black’s commute length and arrival time.

Departure times:

B: 7:00 a.m. ET

W: 6:15 a.m. ET

G: 8:30 a.m. CT

W: 25 mph, 20 mi traveled

B: 100 mi traveled, arrives at 8:15 ET

Step 2: Identify and organize the information you need

What do you need to find for this question? Of the information given, what will help answer the question? What can you dismiss?

You’re asked to find the time between Ms. Weiss and Mr. Black’s arrivals. Because the first question asks only about Mr. Black and Ms. Weiss, you can disregard any information about Dr. Grey for now. To answer this question, you need Mr. Black’s and Ms. Weiss’s travel times.

What about Mr. Black’s speed?

This question asks for the difference in arrival times. Because you’re given Mr. Black’s travel time, finding his speed is not necessary.

B travel time: ?

W travel time: ?

disregard G for now

Step 3: Based on what you know, plan your steps to navigate the first question

How do you find Mr. Black’s travel time? What about Ms. Weiss’s?

Finding Mr. Black’s travel time is straightforward; just take the difference between his departure and arrival times.

You can use the DIRT equation to find Ms. Weiss’s travel time.



B: start @ 7:00 a.m., arrive @ 8:15 a.m.



W: d = rt

Step 4: Solve, step-by-step, checking units as you go

What is Mr. Black’s travel time? What’s Ms. Weiss’s?

You know it took Mr. Black 75 minutes.

For Ms. Weiss, plug her given speed (rate) and distance traveled into the DIRT equation, then solve for t.

How much longer will Mr. Black spend traveling?

Find the difference in travel times, making sure your answer is in the units requested (minutes).



B: 7:00 a.m. — 8:15 a.m. =

1 h 15 min = 75 min

W: 20 mi = 25 mi/h x t

t = h = 0.8 h = 48 min

75 min — 48 min = 27 min

Step 5: Did I answer the right question?

Mr. Black will travel 27 minutes longer than Ms. Weiss.





27 min

Note

You might be given extra information on questions like these. If you don’t need it to get to the answer, then don’t worry about it.

Now on to Step 6: Repeat for remaining questions in the set. Kaplan’s strategic thinking is on the left, along with suggested math scratchwork on the right.

Strategic Thinking

Math Scratchwork

Step 1: Read the second question in the set, looking for clues

From the introduction, you know Dr. Grey’s flight was supposed to depart at 8:30 A.M. CT. This question tells you there is a 45-minute delay, as well as Dr. Grey’s speed and distance traveled. You found Ms. Weiss’s travel time in the previous question.



8:30 a.m. CT planned start, delayed 45 min

plane speed: 332 mi/h

distance covered: 830 mi

W travel: 48 min

Step 2: Identify and organize the information you need

You need to determine how many hours separate the arrival times of Ms. Weiss and Dr. Grey.

W arrival: ?

G arrival: ?

Step 3: Based on what you know, plan your steps to navigate the second question

What calculations will yield Dr. Grey’s travel time?

You’re given Dr. Grey’s speed and distance, so you can use the DIRT equation to find his time in transit. Then add this time to his start time, taking the delay and time zone change into account.

G travel time: ?

d = rt

+ 45 min (delay) + 1 h (time zone)

Step 4: Solve, step-by-step, checking units as you go

How long is Dr. Grey’s flight?

Plug the rate and distance into the DIRT equation and solve for time.

What effect do the delay and time zone change have on his arrival time?

Add 45 minutes to the initial start time to account for the delay, and then add the 2 hours and 30 minutes flight time. Add another hour for the time zone change.

At what time did Ms. Weiss arrive in New York? How many hours before Dr. Grey’s arrival is this?

Add 48 minutes to Ms. Weiss’s start time to yield her arrival time. Lastly, find the difference between the two colleagues’ arrival times.

d = rt

830 mi = 332 mi/h x t

t = 2.5 h (2 h 30 min)

G: 8:30 a.m. → 9:15 a.m.(delay)

+ 2 h 30 min flt = 11:45 A.M.

CT→ 12:45 p.m. ET

W: 6:15 a.m. ET + 48 min = 7:03 a.m. ET

7:03 a.m. ET vs. 12:45 p.m. ET: diff = 5 h 42 min

Step 5: Did I answer the right question?

Adjust your answer so it’s in the requested format.

5 h 42 min = 5 h

= 5.7 h

As you just saw, using the Kaplan Method for Multi-Part Math Questions makes an intimidating question far more straightforward. You’ll have a chance to try it yourself later in this chapter.

RATIOS AND PROPORTIONS

Ratios and proportions are quite common in everyday life. Whether it’s making a double batch of meatballs or calculating the odds of winning the lottery, you’ll find that ratios and proportions are invaluable in myriad situations.

A ratio is a comparison of one quantity to another. When writing ratios, you can compare 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).

You can also combine ratios. If you have two ratios, a:b and b:c, you can derive a:c by finding a common multiple of the b terms. Take a look at the following table to see this in action:

a

:

b

:

c

3

:

4





3

:

5

9

:

12





12

:

20

9


:


20

What’s a common multiple of the b terms? The number 12 is a good choice because it’s the least common multiple of 3 and 4, thus reducing the need to simplify later. Where do you go from there? Multiply each ratio by the factor (use 3 for a:b and 4 for b:c) that will get you to b = 12.

The ratio a:c equals 9:20. Notice we didn’t merely say a:c is 3:5; this would be incorrect on Test Day (and likely a wrong-answer trap!).

A proportion is simply two ratios set equal to each other. Proportions are an efficient way to solve certain questions, but you must exercise caution when setting them up. Watching the units of each piece of the proportion will help you with this. Sometimes the PSAT will 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 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).

Alternatively, pick numerical values for a, b, c, and d; then simplify and confirm the two sides of the equation are equal. For example, take the two equivalent fractions and (a = 2, b = 3, c = 6, d = 9).

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

Let’s take a look at a test-like question that involves ratios:

4. A researcher is optimizing solvent conditions for a chemical reaction. The conventional protocols use either 7 parts dioxane (an organic solvent) and 3 parts water or 5 parts water and 2 parts methanol. The researcher wants to see what happens when she uses dioxane and methanol without deviating from the given protocols. What ratio of methanol to dioxane should she use?

1. 35:6

2. 7:2

3. 2:7

4. 6:35

Work through the Kaplan Method for Math step-by-step to solve this question. The following table shows Kaplan’s strategic thinking on the left, along with suggested math scratchwork on the right.

Strategic Thinking

Math Scratchwork

Step 1: Read the question, identifying and organizing important information as you go

You need the ratio of methanol to dioxane. You’re given two ratios: dioxane to water and water to methanol.

D:W = 7:3

W:M = 5:2

Step 2: Choose the best strategy to answer the question

How can you directly compare methanol to dioxane? What’s a common multiple of the two water components?

The two given ratios both contain water, but the water components are not identical. However, they share a common multiple: 15. Multiply each ratio by the factor that will make the water part equal 15.

What does the combined ratio look like?

Merging the two ratios lets you compare dioxane to methanol directly.

D:W = 7:3

W:M = 5:2

common multiple: 5 x 3 = 15

(7:3) x 5 = 35:15

(5:2) x 3 = 15:6

D:W:M = 35:15:6

D:M = 35:6

Step 3: Check that you answered the right question

The question asks for methanol to dioxane, so flip your ratio, and you’re done. Choice (D) is correct. Watch out for trap answer A. You aren’t looking for dioxane to methanol.

M:D = 6:35

Note

Beware of trap answers that contain incorrect ratios. Always confirm that you’ve found the ratio requested.

PERCENTAGES

Percentages aren’t just for test grades; you’ll find them frequently throughout life—discount pricing in stores, income tax brackets, and stock price trackers all use percents in some form. It’s critical that you know how to use them correctly, especially on Test Day.

Suppose you have a bag containing 10 blue marbles and 15 pink marbles, and you’re asked what percent of the marbles are pink. You can determine this easily by using the formula . Plug 15 in for the part and 10 + 15 (= 25) for the whole to get pink marbles.

Another easy way to solve many percent questions is to use the following statement: (blank) percent of (blank) is (blank). Translating from English into math, you obtain (blank)% × (blank) = (blank). As you saw with the DIRT equation in the rates section, knowledge of any two quantities will unlock the third.

Note

The percent formula requires the percent component to be in decimal form. Remember to move the decimal point appropriately before using this formula.

You might also be asked to determine the percent change in a given situation. Fortunately, you can find this easily using a variant of the percent formula:

Sometimes more than one change will occur. Be especially 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 calculate accordingly. We’ll demonstrate this in an upcoming question.

The following is a test-like question involving percentages:

5. Ethanol is almost always mixed with gasoline to reduce automobile emissions. Most tanks of gasoline are 15% ethanol by volume. An oil company tries decreasing the ethanol content to 6% to lower the cost of gas. 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

Work through the Kaplan Method for Math step-by-step to solve this question. The following table shows Kaplan’s strategic thinking on the left, along with suggested math scratchwork on the right.

Strategic Thinking

Math Scratchwork

Step 1: Read the question, identifying and organizing important information as you go

You need to find how many times more ethanol is in the 15% tank. The question supplies information about a 14-gallon tank and a 10-gallon tank, each containing a different ethanol/gasoline blend.

14 gal tank: 15% ethanol

10 gal tank: 6% ethanol

Step 2: Choose the best strategy to answer the question

What formula can you use to get to the answer?

How much ethanol is in the 14-gallon tank? How much ethanol is in the 10-gallon tank?

Plug the appropriate values into the percent formula. Remember to move the decimal points of your percents to get their decimal forms.

How many times more ethanol is in the larger tank?

Set up an equation to show 0.6 times a number equals 2.1. Solving for Z gives 3.5; that is, the larger tank contains 3.5 times more ethanol than the smaller one.


0.15 x 14 gal = 2.1 gal

0.06 x 10 gal = 0.6 gal



0.6 gal x Z = 2.1 gal

Step 3: Check that you answered the right question

The question asks for how many times more ethanol is in the 14-gallon tank, which is what you found: 3.5 matches (C).


Note

Resist the urge to merely take the difference between the two ethanol quantities. Make sure you’re answering the question posed.

Here’s an example of a multi-part question that tests your percentage expertise.

1. Questions 6 and 7 refer to the following information.

2. A bank normally offers a compound annual interest rate of 0.25% on any savings account with a minimum balance of $5,000. The bank is currently offering college students a higher rate, 0.42%, with a $1,000 minimum balance. Assume the average balances are kept constant at the required minima (e.g., all interest is withdrawn) for the following.

6. How much more interest does the regular account earn after three years than the student account?

7. What is the minimum balance a student would need to maintain to earn the same amount of interest as would be earned by saving money in the regular account? Round your answer to the nearest dollar.

Work through the Kaplan Method for Multi-Part Math Questions step-by-step to solve this question. The following table shows Kaplan’s strategic thinking on the left, along with suggested math scratchwork on the right.

Strategic Thinking

Math Scratchwork

Step 1: Read the first question in the set, looking for clues

The intro provides information on two account types.

regular acct: 0.25%, $5,000 min

student acct: 0.42%, $1,000 min

Step 2: Identify and organize the information you need

You need to find how much more interest the $5,000 account will have after three years.

difference in interest: ?

Step 3: Based on what you know, plan your steps to navigate the first question

What pieces needed to find the answer are missing? How do you find the difference in interest?

You’ll need the amount of interest that each account accrues after three years. Use the three-part percent formula to find annual interest, then find the interest after three years, then take the difference.

reg. int. = ?

stu. int. = ?

reg. int. x 3 = ?

stu. int. x 3 = ?

reg. — stu. = ?

(blank)% of (blank) is (blank)

Step 4: Solve, step-by-step, checking units as you go

How much interest does each account earn after one year? After three years?

Plug in appropriate values. Remember to adjust the decimal point on the percents appropriately. Triple the interest amounts to get the total accrued interest after three years.

What’s the difference in interest earned?

Subtract.

0.0025 x $5,000 = $12.50

0.0042 x $1,000 = $4.20

$12.50 x 3 = $37.50

$4.20 x 3 = $12.60

$37.50 — $12.60 = $24.90

Step 5: Did I answer the right question?

You’ve found how much more interest the regular account makes after three years, so you’re done with the first question.

24.9

Note

Disregard the 0 in the hundredths place when gridding in your answer.

The first part of the question set is finished! Now on to Step 6: Repeat for the other question in the set. Kaplan’s strategic thinking is on the left, along with suggested math scratchwork on the right.

Strategic Thinking

Math Scratchwork

Step 1: Read the second question in the set, looking for clues

No new information here, but some pieces from the first part of the question set might be useful.


Step 2: Identify and organize the information you need

What does the second question ask you to find? Is there any information from the first question that will help?

The second question asks for the student account balance that will yield the same interest as a regular account at the minimum balance. You know the interest rates for a regular account and a student account at their respective minimum balances. You also know the interest earned annually (from the first question).

reg. acct: 0.25%, $12.50/yr

stu. acct: 0.42%,

$4.20/yr

Step 3: Based on what you know, plan your steps to navigate the second question

How can you determine when the two accounts will earn the same interest? Will algebra work here?

To answer the second question, you’ll need to find when annual student interest equals annual regular interest. Set up equations with interest earned as a function of the account balance, one for each account. You already know what the regular account makes in interest annually, so it’s just a matter of finding when the student equation equals that value.

Int. = rate × balance

Step 4: Solve, step-by-step, checking units as you go

When does the student account earn $12.50 in interest per year?

Plug 12.5 in for your dependent variable, then solve for x.



reg.: y = 0.0025x

12.5 = 0.0025 x 5,000

stu.: y = 0.0042x

12.5 = 0.0042x

x = 2,976.19

Step 5: Did I answer the right question?

The second question asks for the balance a student account needs to make the same interest as a regular account. Round to the nearest dollar, and you’re done!

2,976

Practice

Now you’ll have a chance to try a few test-like questions in a scaffolded way. We’ve provided some guidance, but you’ll need to fill in the missing parts of explanations or the step-by-step math to get to the correct answer. Don’t worry—after going through the worked examples at the beginning of this section, these questions should be completely doable.

8. Fuel efficiency is a measure of how many miles a car can go using a specified amount of fuel. It can change depending on the speed driven and how often the driver brakes and then accelerates, as well as other factors. Jack is taking a road trip. If he travels 180 miles at 40 miles per gallon and then another 105 miles at 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

The following table can help you structure your thinking as you go about solving this question. Kaplan’s strategic thinking is provided, as are bits of structured scratchwork.

Strategic Thinking

Math Scratchwork

Step 1: Read the question, identifying and organizing important information as you go

You need to find the amount of fuel Jack’s car consumed. The question provides information about two legs of Jack’s trip.

___ mi @ ___ mpg

___mi @ ___ mpg

Step 2: Choose the best strategy to answer the question

What hint does the unit “miles per gallon” give you? Is there a standard equation you can use?

The unit mpg is a rate, so straightforward math using the DIRT equation is appropriate.

180 miles is clearly a distance, and 40 miles per gallon is your rate. Therefore, you need to find the time. Don’t forget your units. Repeat for the second leg.

The ultimate goal is to find the total fuel consumption. Add the two gallon figures together.


Leg 1: _____ = _____ x t

t = _____

Leg 2: _____ = _____ x t

t = _____



_____ + _____ = _____

Step 3: Check that you answered the right question

If your answer is (D), you’re correct!



fuel consumed: _____ gal

Expert Tip

Sometimes distance or time units won’t look like those you’re used to (e.g., miles, minutes, etc.). Don’t let this deter you. If you have a rate, you can use the DIRT equation.

Here’s another test-like example to try:

9. Financial advisers are often hired to manage people’s retirement accounts by shifting how money is invested in stocks and bonds. During a particularly volatile stock market period, a financial adviser makes a number of changes to a client’s stock allocation. She first decreases this allocation by 25%, then increases it by 10%, then increases it by an additional 50%. What is the approximate net percent increase in this client’s stock allocation?

1. 20%

2. 24%

3. 30%

4. 35%

Note

It’s tempting to just add the percentages together, but this will lead you to a trap (incorrect) answer. Take the time to work through the question properly to get the points.

The following table can help you structure your thinking as you go about answering this question. Kaplan’s strategic thinking is provided, as are bits of structured scratchwork. If you’re not sure how to approach a question like this, start at the top and work your way down.

Strategic Thinking

Math Scratchwork

Step 1: Read the question, identifying and organizing important information as you go

You need to find the net percent change in this client’s stock allocation. The question provides a series of percent changes that this allocation undergoes.

start → _____ → _____ → _____ → end

Step 2: Choose the best strategy to answer the question

What kind of question is this? What’s a good starting point for calculations?

This is a percent change question. You aren’t given a concrete starting point, so pick a starting number that’s easy to use with percents, and get ready for a series of three-part percent formula calculations.

How many shares are left after each change?

Plug in the starting share count and the new percent, then solve for the new share count. Using the percent left in your calculation instead of the percent change will save you a step. When you compute the second and third changes, remember to use the final share count from the previous calculation, not the original number of shares.

Am I finished?

You found the final stock share count, now you need to find the percent change.

assume ___ shares @ start

after 25%:

_____% x _____ = _____

shares left: _____

after + 10%:

_____% x _____ = _____

shares left: _____

after + 50%:

_____% x _____ = _____

shares left: _____


Step 3: Check that you answered the right question

If you chose (B), you’re correct.



net change: _____%

Now try your hand at a multi-part question.

1. Questions 10 and 11 refer to the following information.

2. An artist is creating a rectangular tile mosaic. Her desired pattern uses 5 green tiles and 3 blue tiles per square foot of mosaic.

10.If the artist’s entire mosaic is 12 feet by 18 feet, how many more green tiles than blue will she need?

11.The artist discovers her tile vendor has a very limited supply of green tiles, so she alters her design so that it requires 2 green tiles and 3 blue tiles per square foot of mosaic. She also decides to add red tiles to her mosaic. How many red tiles will the artist need to make the ratio of red tiles to green tiles 3:4 ?

The following table can help you structure your thinking as you go about answering this question. Kaplan’s strategic thinking is provided, as are bits of structured scratchwork. If you’re not sure how to approach a question like this, start at the top and work your way down.

Strategic Thinking

Math Scratchwork

Step 1: Read the first question in the set, looking for clues

Included are the size of the mosaic and the ratio of green to blue tiles.

mosaic dimensions: ___ x ___

green:blue → ___:___

Step 2: Identify and organize the information you need

You must find how many more green tiles than blue tiles the artist will use.



diff. between green & blue: ?

Step 3: Based on what you know, plan your steps to navigate the first question

What’s missing? What calculations will get you the tile counts? How are the mosaic dimensions important?

You’re missing the number of green tiles and blue tiles, as well as the number of times the pattern appears in the mosaic. You know the pattern appears once per square foot of mosaic, so finding the mosaic area will tell you how many times the pattern repeats. Multiply this number by the tile ratio to find how many of each color the artist needs, then take the difference.













# green tiles: ?

# blue tiles: ?

# pattern appearances: ?

A = ?

A x green:blue = # green: # blue

Step 4: Solve, step-by-step, checking units as you go

What’s the area of the mosaic?

Use the rectangle area formula to find the area of the mosaic (and the number of times the pattern repeats).

How many green tiles and blue tiles will the artist require? How many more green will she need?

Multiply the area by each number in the tile ratio, and then take the difference.



A
= ____ x ____ = ____

green: ____ x ____= ____

blue: ____ x ____= ____


Step 5: Did I answer the right question?

If you came up with 432 more green tiles, great job! You’re correct.



difference: ____

Fantastic! Now repeat for the other question in the set. Once again, Kaplan’s strategic thinking is provided in the table that follows, as are bits of structured scratchwork. If you’re not sure how to approach the second part, start at the top and work your way down.

Strategic Thinking

Math Scratchwork

Step 1: Read the second question in the set, looking for clues

You know the new ratio of green to blue tiles and what the ratio of red to green tiles should be, as well as the area of the mosaic (from the previous question).

green:blue → ____ : ____

red:green → ____: ____

area: ____

Step 2: Identify and organize the information you need

To answer this question, you need to find how many red tiles are required to make the ratio of red to green 3:4.



# red tiles: ?

Step 3: Based on what you know, plan your steps to navigate the second question

Anything missing? How does the new green:blue ratio affect the green tile count?

The ratios here are different, so you’ll need new tile counts for green and red (blue stays constant). First, find the adjusted number of green tiles needed. Then determine the number of red tiles needed to satisfy the given ratio using a proportion that compares red to green.

# green tiles: ?

# red tiles: ?

Step 4: Solve, step-by-step, checking units as you go

Given the new green:blue ratio, how many green tiles are now needed?

Multiply the number of pattern appearances from the first question by the green tile component of the adjusted ratio.

What red tile count will satisfy the desired red:green ratio? Could a proportion be useful?

Set up a proportion using the red:green ratio and the new green tile count to find the number of red tiles required. Be careful when setting it up.

# green: ____ x ____ = ____



Step 5: Did I answer the right question?

Did you get 324 red tiles? If so, congrats! You’re correct.

____ red tiles

Perform

Now that you’ve seen the variety of ways in which the PSAT can test you on ratios, rates, proportions, and percentages, try the following questions to check your understanding. Give yourself 5 minutes to answer the following four questions. Make sure you use the Kaplan Method for Math as often as you can (as well as the Kaplan Method for Multi-Part Math Questions when necessary). Remember, you want to emphasize speed and efficiency in addition to simply getting the correct answer.

12.An engineer is monitoring construction of a 75-foot 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

13.Grocery stores often sell larger quantities of certain foods at reduced prices so that customers ultimately get more food for less money. Suppose 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?

image

14.

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

2. An electronics store is having a Black Friday Blowout sale. All items are 40% off, and the first 50 customers will receive an additional 25% off the reduced prices.

14.Two people purchase a home theater system that normally costs $2,200. The first person is one of the first 50 customers. The second person arrives much later in the day. How much more, in dollars, does the first customer save than the second customer?

image

15.In addition to price reductions, the store is offering to pay part of the sales tax on customers’ purchases: instead of paying the regular 6.35% sales tax, customers pay only half this rate. Assuming the same conditions from the previous question and that you are among the first 50 in the store, what is the total difference in price between the reduced price (with reduced tax) and the full price with standard tax? Round your answer to the nearest dollar.

image

On your own

You may use your calculator for all questions in this section.

1. During fairly heavy traffic, the number of cars that can safely pass through a stoplight during a left turn signal is directly proportional to the length of time in seconds that the signal is green. If 9 cars can safely pass through a light that lasts 36 seconds, how many cars can safely pass through a light that lasts 24 seconds?

1. 4

2. 6

3. 7

4. 8

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. The average college student reads prose text written in English at a rate of about 5 words per second. If the pages of Jorge’s world history 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

4. According to data provided by the College Board, the cost of tuition and fees for a private nonprofit four-year college in 1988 was approximately $15,800 (in 2013 dollars). In 2013, the cost of tuition and fees at the same type of college was approximately $30,100. If the cost of education experiences the same total percent increase over the next 25 years, approximately how much will tuition and fees at a private nonprofit four-year college cost (in 2013 dollars)?

1. $44,400

2. $45,800

3. $57,300

4. $66,200

5. A radiology center administers magnetic resonance images (MRIs) to check for abnormalities in a patient’s body. One MRI scan typically produces about 3.6 gigabits of data. Every night, for 8 hours, the hospital backs up the files of the scans on a secure remote server. The hospital computers can upload the images at a rate of 2 megabits per second. What is the maximum number of MRI scans that the hospital can upload to the remote server each night? (1 gigabit = 1,024 megabits)

1. 15

2. 16

3. 56

4. 202

6. A chef is preparing ingredients to make a large quiche. The recipe calls for 5 cups of milk and 2 eggs. There is also a lower-fat option that calls for 4 cups of water and 3 eggs. The caterer wants to see what happens when he uses both milk and water. To keep the same ratio of liquid to eggs, what ratio of milk to water should he use?

1. 1:1

2. 5:4

3. 8:15

4. 15:8

7.

1. Questions 7 and 8 refer to the following information.

2. image

Verona owns a landscaping company, and she is mulching all of the flowerbeds at the houses in a planned retirement community. The figure shows the layout of the community and the times that Verona started mulching the flowerbeds at two of the houses. According to the architect’s plans for the community, the flowerbed for each house is approximately 0.006 acres in size.

7. How many minutes will it take Verona to mulch all of the flowerbeds in the community?

image

8. Verona uses a landscaper’s wagon that holds about 15 bags of mulch. Each bag of mulch covers 24 square feet of flowerbed. How many times, including the first time, will Verona need to fill the wagon, assuming she loads 15 bags of mulch each time? (1 acre = 43,560 square feet)

image

9. A cybercafé is a place that provides Internet access to the public, usually for a fee, along with snacks and drinks. Suppose a cybercafé charges a base rate of $25 to join for a year, an additional $0.30 per visit for the first 50 visits, and $0.10 for every visit after that. How much does the cybercafé charge for a year in which 72 visits are made?

1. $32.20

2. $36.60

3. $42.20

4. $46.60

10.A high school’s Environment Club receives a certain amount of money from the school to host an all-day Going Green Teach-In. The club budgets 40% 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

11.image

A certain necklace is made up of beads of the following colors: red, blue, orange, indigo, silver, and clear. The necklace contains 120 beads. According to the pie chart shown, how many beads of the necklace are not blue?

1. 20

2. 24

3. 80

4. 96

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

1.

2.

3.

4.

13.The Occupational Safety and Health Act (OSHA) was passed in 1970 to “assure safe and healthful working conditions.” Its provisions cover all non-farm employers. Some small businesses, however, are exempt from certain reporting and inspection requirements if they have fewer than 10 employees. A certain city has 2,625 businesses in its jurisdiction and has a ratio of 5:2 of businesses that are exempt from inspections to those that are not exempt. Of the businesses that were required to have inspections, 12% had safety violations and were required to address the deficiencies. How many covered businesses did not have to address any OSHA safety issues?

1. 90

2. 315

3. 660

4. 2,310

14.Carmen is a traffic engineer. The city she works for recently added a number of new intersections with stoplights and wants to redesign the traffic system. The distance between the new intersections increases as one moves down Main Street. Carmen has been given the task of determining the length of the green lights at the intersections along Main Street, according to the following guidelines:

o The length of each green light should be greater than or equal to 8 seconds but less than or equal to 30 seconds.

o Each green light should be at least 25% longer than the one at the intersection before it.

o The length of each green light must be a whole second.

Which list of light lengths meets the city guidelines and includes as many intersections as possible?

4. 8, 12, 16, 20, 25, 30

5. 8, 10, 13, 17, 22, 28

6. 8, 10, 12.5, 16, 20, 25

7. 8, 10, 12, 15, 18, 22, 27

15.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?

0.

1. 1

2.

3. 45

16.An Iinternet provider charges k dollars for the first hour of use in a month and m dollars per hour for every additional hour used that month. If Jared paid $65.50 for his Internet use in one month, which of the following expressions represents the number of hours he used the Internet that month?

0.

1.

2.

3.

17.Engine oil often contains additives that are designed to prevent certain common engine problems. One such additive is 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 pan than in the truck’s? (1 quart = 2 pints)

0. 0.34

1. 0.67

2. 1.5

3. 3

18.When a consignment store gets a used piece of furniture to sell, it researches the original price and then marks the used piece down 40%. Every 30 days after that, the price 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 October 5, what is the final selling price, not including tax?

0. $258.05

1. $322.56

2. $403.20

3. $630.00

19.In the United States, the maintenance and construction of airports, transit systems, and major roadways is largely funded through a federal tax on gasoline. Based on the 2011 statistics given below, what was the federal gasoline tax rate in cents per gallon?

o The average motor vehicle was driven approximately 11,340 miles per year.

o The national average fuel economy for noncommercial vehicles was 21.4 miles per gallon.

o The average American household owned 1.75 vehicles.

o The average household paid $170.63 annually in federal gasoline taxes.

4. 1.2

5. 5.43

6. 7.97

7. 18.4

20.An amusement park is building a scale model of an airplane for a 3-D ride. The real airplane measures 220 feet, 6 inches from nose to tail. The amusement park plans to make the ride 36 feet, 9 inches long. If the wingspan of the real plane is 176.5 feet, how long in inches should the wingspan on the ride be?

(1 foot = 12 inches)

0. 7 feet, 3 inches

1. 29 feet, 5 inches

2. 35 feet, 2 inches

3. 45 feet, 11 inches

21.

0. Questions 21 and 22 refer to the following information.

1. Three planes depart from three different airports at 8:00 A.M., all traveling to Fort Lauderdale International Airport (FLL). The first plane is a small passenger plane that must travel 110 miles to reach FLL. The second plane is a large passenger plane that must travel 825 miles. The third plane is a cargo plane that must travel 640 miles.

21.The small passenger plane traveled at an average speed of 200 miles per hour. The cargo plane arrived at FLL at 9:15 A.M. How many minutes before the cargo plane arrived did the small passenger plane arrive?

image

22.For the first of the distance of the trip, the large passenger plane flew through fairly heavy cloud cover at an average speed of 300 miles per hour. For the remaining portion of the trip, the sky was clear and the plane flew at an average speed of 500 miles per hour. Due to a backlog of planes at the Fort Lauderdale airport, it was forced to circle overhead in a holding pattern for 35 minutes after arriving at the airport. At what time did the large passenger plane land at FLL? Use only digits for your answer (for example, enter 11:15 as 1115).

image

22.

22. Questions 23 and 24 refer to the following information.

23. A hedge fund usually consists of a small group of investors that employs aggressive, high-risk methods of investing in hopes of earning large capital gains. They always require a large minimum investment. Hedge fund portfolio managers usually charge a percentage of the value of the portfolio as a management fee. A certain company charges an annual fee of 3.5% with a minimum initial investment of $100,000 for lower-risk portfolios and an annual fee of 1.25% with a minimum investment of $250,000 for higher-risk portfolios.

23.If Christi manages one of each type of portfolio that is opened with only the minimum initial investment, and neither portfolio gains or loses money over 4 years, what is the difference in total fees that Christi will collect between the two portfolios?

image

24.What percent gain would the higher-risk portfolio need to make in 1 year for the annual fee to be as much as that of the lower-risk portfolio? Enter your answer as a decimal number (for example, enter 25% as .25).

image

23.

24. Questions 25 and 26 refer to the following information.

25. An infomercial is advertising a new product that has just been made available in stores. The retail price of the item is $160. If you purchase the product through the infomercial, you receive 10% off the in-store retail price, and you do not have to pay sales tax. The company is also offering an additional 20% off the discounted infomercial price to the first 250 callers.

25.If the sales tax in your state is 5.5%, how much would you save by purchasing the item through the infomercial as one of the first 250 callers?

image

26.Malik is one of the first 250 callers and buys 5 of the item. He calls back later and is the 410th caller and buys 3 more. What is the average price that Malik paid per item?

image

24.

26. Questions 27 and 28 refer to the following information.

27. A chemical solvent is a substance that dissolves another. For example, acetone is a solvent and is the primary ingredient in fingernail polish remover. It is also one of the few solvents that will safely remove adhesives from skin. The following table shows the chemical makeup of 1 mole (a unit of measure commonly used in chemistry) of acetone:

Chemical Makeup of 1 Mole of Acetone

Element

Number of Moles

Mass per Mole (grams)

Oxygen

1

15.9994

Carbon

3

12.0107

Hydrogen

6

1.00794

27.Oxygen makes up what percent of the mass of 1 mole of acetone? Round your answer to the nearest whole percent.

image

28.If a chemist starts with 1,800 grams of acetone and uses 871.1871 grams, how many moles of carbon are left? Round your answer to the nearest whole mole.

image

25.

28. Questions 29 and 30 refer to the following information.

29. Great Britain has not adopted the euro as its form of currency and instead utilizes the British pound. Rosslyn is from Great Britain but is vacationing throughout Europe. When she arrives in France, she realizes she forgot to exchange her British pounds for euros. Her bank does not exist in France, so she must use a local bank, which charges a 5% fee to exchange money for noncustomers.

29.Rosslyn wants to get 1,800 euros. The bank representative tells her that the total amount she will need, including the exchange fee, is 1,512 pounds. What is the current exchange rate from euros to British pounds?

image

30.When she returns to Great Britain, Rosslyn converts the 65 euros she has left back to British pounds at her own bank, where there is no fee to exchange the money. She is informed that the euros-to-pounds exchange rate has increased slightly over the course of her vacation. If Rosslyn lost a total of 74 pounds, including the fee she paid to the bank in France, what was the new euros-to-pounds exchange rate?

image