PSAT/NMSQT Prep 2022 - Eggert M.D., Strelka A. 2022

Reflect
Ratios, proportions, and percents
Data analysis

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.

6. 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).

7. 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.

8. 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.

9. 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).

10. 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.

11. C

Difficulty: Easy

Getting to the Answer: For discount questions, multiply by 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.

12. 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.

13. 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.

14. 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 1 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.

15. 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 1 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.

16. 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).

17. 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.

18. 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.

19. 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.

20. 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.

21. 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.

22. 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 . This is the number of businesses that had violations; 12% of 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.

23. 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.

24. 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:

Car: 0.09 × 4 quarts = 0.36 quarts of zinc

Truck: 0.04 × 6 quarts = 0.24 quarts of zinc

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).

25. 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).

26. 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).

27. 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.

28. 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.

29. 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.

30. 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.

31. 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.