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

Ratios and proportions
Ratios, proportions, and percents
Data analysis

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:

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)

A. 12

B. 16

C. 25

D. 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).

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: inches. 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. 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?

A. 6,000

B.9,000

C.12,000

D. 18,000

2. 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 percent. 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?

A. 18.75

B.25

C.56.25

D. 100

3. 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?

A. 4

B.6

C.7

D. 8

4. 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?

A. 5

B.19

C.50

D. 188

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.

5. 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.)