Ratios, Proportions, and Percents for CBEST Exam Study Guide (page 4)
Ratios and proportions, along with their cousins, percents, are sure to appear on the CBEST. A good understanding of these topics can help you pick up valuable points on the math section of the test.
The Three-Step Ratio
The three-step ratio asks for the ratio of one quantity to another.
Sample Three-Step Ratio Question
Use the three steps to help you work out the following problem.
- Which of the following expresses the ratio of 2 yards to 6 inches?
- One yard is 36 inches, so 2 yards is 72 inches. Thus, the ratio becomes 72:6. (The quantities can also be put in yards.)
- This ratio can be expressed as or 72:6. In this problem, 72:6 is the form that is used in the answers.
- Since the answer is not there, reduce. 72 inches:6 inches = 12:1. The answer is choice e. Notice that choice c, 1:12, is backward, and therefore incorrect.
Three Success Steps for Three-Step Ratios
- Put the quantities in the same units of measurement (inches, yards, seconds, etc.).
- Put the quantities in order and in the form given by the answer choices.
- If the answer you come up with isn't a choice, reduce.
Try the three steps on the following problems.
- Find the ratio of 3 cups to 16 ounces.
- Find the ratio of 6 feet to 20 yards.
- Find the ratio of 2 pounds to 4 ounces.
- In a certain class, the ratio of children who preferred magenta to chartreuse was 3:4.What was the ratio of those who preferred magenta to the total students in the class? Hint: Add 3 and 4 to get the total.
- You can eliminate two of the choices for question 2 immediately. You know that 3:16 or 16:3 can't be right because the units haven't been converted yet.
- In question 3, choices b and c are the same ratios. There can't be two right answers, so they can be eliminated. Six feet is two yards. Reducing 2:20 makes 1:10.
Four Success Steps for Four-Step Ratios
- Label the categories of quantities in the problem to illustrate exactly what you're working with.
- Set up the complete set in ratio form.
- Set up the incomplete set in ratio form.
- Cross multiply to get the missing figure.
- In a certain factory, employees were either foremen or assembly workers. The ratio of foremen to assembly workers was 1 to 7.What is the ratio of the assembly workers to the total number of employees?
The Four-Step Ratio
The four-step ratio solution is used when there are two groups of numbers: the ratio set, or the small numbers; and the actual, real-life set, or the larger numbers. One of the sets will have both numbers given, and you will be asked to find one of the numbers from the other set.
Sample Four-Step Ratio Question
- The ratio of home games won to total games played was 13 to 20. If home teams won 78 games, how many games were played?
This problem can be solved in four steps.
- Notice the two categories: home team wins and total games played. Place one category over the other in writing.
- In the previous problem, the small ratio set is complete (13 to 20), and you're being asked to find the larger, real-life set. Work with the complete set first. Decide which numbers from the complete set go with each written category. Be careful; if you set up the ratio wrong, you will most probably get an answer that is one of the answer choices, but it will be the wrong answer.
- Determine whether the remaining number in the problem best fits home wins or total games. "If home teams won 78 games" indicates that the 78 goes in the home-team row. The number of total games played isn't given, so that spot is filled with an x.
- Now cross multiply. Multiply the two numbers on opposite corners: 20 × 78. Then divide by the number that is left (13).
Note: This step is frequently omitted by test takers in order to save time, but the omission of this step causes most of the mistakes made on ratio problems.
Notice which category is mentioned first: "The number of HOME games won to TOTAL games played …" Then check to see what number is first: "… was 13 to 20." Thirteen is first, so 13 goes with home games; 20 goes with the total games.
After you cross multiply and wind up with one fraction, you can divide a top number and the denominator by the same factor to avoid long computations. In the previous example, 13 ÷ 13 = 1 and 78 ÷ 13 = 6.
The problem would then be much simpler: 20 × 6 = 120.
Try the four steps on the following problems.
- On a blueprint, inch equals 2 feet. If a hall is supposed to be 56 feet wide, how many inches wide will the hall be on the blueprint?
- In a certain recipe, 2 cups of flour are needed to serve 5 people. If 20 guests are coming, how much flour will be needed?
- A certain district needs 2 buses for every 75 students who live out of town. If there are 225 students who live out of town, how many buses are needed?
There are five basic types of percent problems on the CBEST. As is true with most other types of problems on the CBEST, percent problems most often appear in word-problem format. Percents can be done by using ratios or by algebra. Since ratios have just been covered, this section will explain the ratio method.
Percents can be fairly simple if you memorize these few relationships:
Sample Finding Part of a Whole Question
- There are 500 flights out of Los Angeles every hour. Five percent are international flights. How many international flights leave Los Angeles every hour?
- You are being asked to find a part of the 500 flights. The 500 flights is the whole. The percent is 5. You need to find the part. 5% is fairly small, and considering that 20% of 500 is 100, you know your answer will be less than 100.
- The second sentence has an implied pronoun. The sentence can be rephrased "Five percent of them are international flights." Them refers to the number 500.
- The question is How many… Use the other sentences to reconstruct the question so it includes all the necessary information. The problem is asking "5% of 500 (them) are how many (international flights)?" The question is now conveniently set up.
- Are is the verb; 500 and 5% are on the left side of the verb and how many is on the right side. How many is all by itself, so it goes on top of the ratio in the form of a variable; 500 is next to the of, so it goes on the bottom. At this point, check to see that the part is over the whole.
- The 5 goes over 100.
- The two are equal to each other.
- Twenty-five international flights leave every hour.
Eight Success Steps for Solving Percent Problems
Feel free to skip steps whenever you don't need them.
- Notice the numbers. Usually you are given two numbers and are asked to find a third. Are you given the whole, the part, or both ? Is the percent given ? Is the percent large or small? Is it more or less than half? Sometimes you can estimate the answer enough to eliminate some alien answers.
- If there are pronouns in the problem, write the number to which they refer above the pronoun.
- Find the question and underline the question word. Question words can include how much is, what is, find, etc. In longer word problems, you may have to translate the problem into a simple question you can use to find the answer.
- Notice the verb in the question. The quantity that is by itself on one side of the verb is considered the is.
Place this number over the number next to the of . If a question word is next to an is or of, put a variable in place of the number in that spot. If there is no is or no of, check to see whether one is implied. See whether you can rephrase the question, keeping the same meaning, but putting in the missing two-letter word. If all else fails, check to make sure the part is over the whole.
- Place the percent over 100. If there is no percent, put a variable over 100 .
- Set the two fractions equal to each other.
- Solve as you would a ratio.
- Be sure to answer the question that was asked.
Sample Finding Part of a Whole Question
- In a certain laboratory, 60%, or 12, of the mice worked a maze in less than one minute. How many mice were there in the laboratory?
Once again, follow the eight Success Steps to solving this problem.
- Twelve is part of the total number of mice in the laboratory. Sixty is the percent, which is more than half. Twelve must be more than half of the whole.
- There are no pronouns.
- The problem is asking, "60% of what number (total mice) is 12?"
- Is is the verb. The 12 is all by itself on the right of the verb. What number is next to the of. The 12 goes on top, the variable on the bottom.
- The 60 goes over 100.
- The two fractions are equal to each other.
- There were 20 mice in the laboratory.
Sample Percent Question
- Courtney sold a car for a friend for $6,000. Her friend gave her a $120 gift for helping with the sale. What percent of the sale was the gift?
- The number 6,000 is the whole and 120 the part.
- There are no pronouns, but there are words that stand for numbers. In the question at the end, the sale is 6,000 and the gift is 120.
- The question is written out clearly: "What percent of 6,000 (sale) was 120 (gift)?"
- Was is the verb. The number 120 is by itself on one side. It is the part, so it goes on top; 6,000 is near the of and is the whole, so it goes on the bottom.
- There is no percent, so x goes over 100.
- The two equal each other.
- The gift was 2% of the sale.
Sample Percent Change Question
A change problem is a little bit different than a basic percent problem. To solve it, just remember that change goes over old:
- The Handy Brush company made $500 million in sales this year. Last year, the company made $400 million. What was the percent increase in sales this year?
First of all, what was the change in sales? Yes, 100 million. You got that by subtracting the two numbers. Which number is the oldest? Last year is older than this year, so 400 is the oldest. Therefore, 100 goes over 400.
The percent is the unknown figure, so a variable is placed over 100 and the two are made equal to each other. Cross multiply and solve for x.
The answer is 25%. Note that if you had put 100 over 500, your answer would have come out differently.
Three Success Steps for Algebra Problems
In order to make a problem less confusing, try the WHO method:
- What numbers are on the same side as the variable? There are two sides to the equal sign, the right side and the left side.
- How are the numbers and the variable connected?
- The Opposite is what? For example, the opposite of subtraction is addition.
Sample Interest Question
- How much interest will Jill earn if she deposits $5,000 at 3% interest for six months?
Interest is a percent problem with time added. The formula for interest is I = PRT. I is the interest. P is the principal, R is the rate or percent, and T is the time in years. To find the interest, you simply multiply everything together. Be sure to put the time in years. You may change the percent to a decimal, or place it over 100.
$5,000 (principal) × 0.03 (percent) × (year) = $75.
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