1.4: Percentages in Many Forms

INTRODUCTION

Recall from Preparation 1.3 the following:

• Educated guess: One type of estimation might be called an educated guess about something that has not been measured exactly. In Collaboration 1.2, you used estimations of the world population. This quantity cannot be measured exactly—it would be impossible to count how many people live on the earth at any given time. Scientists can use good data and mathematical techniques to estimate the population, but it will always be an estimate.
• Convenient estimation: Sometimes estimations are used when it is inconvenient or not worthwhile to make an exact count. Imagine that you need to know how much paint to buy to paint the baseboard trim in your house. (The baseboard trim is the piece of wood that follows along the bottom of the walls.) You need to know the length of the baseboard. You could measure the length of each wall to the nearest 1/8 inch and carefully subtract the width of halls and doors. It would be much quicker and just as effective to measure to the nearest foot or half foot. If you were cutting a piece of baseboard to go along the floor, however, you would want an exact measurement.
• Estimated calculation: This usually involves rounding numbers to make calculations simpler. This collaboration (1.3) focuses on estimating and calculating percentages. You will find in this course that percentages are used in many contexts. One of the most important skills you will develop is understanding and being comfortable working with percentages in a variety of situations.

SPECIFIC OBJECTIVES

By the end of this collaboration, you should understand that

• estimation is a valuable skill.
• standard benchmarks can be used in estimation.
• there are many strategies for estimating.
• percentages are an important quantitative concept.

By the end of this collaboration, you should be able to

• use a few standard benchmarks to estimate percentages.
• estimate the percent one number is of another.
• estimate the percent of a number, including situations involving percentages less than one.
• calculate the percent one number is of another.
• calculate the percent of a number, including situations involving percentages less than one.

PROBLEM SITUATION 1: ESTIMATIONS WITH PERCENTAGES

This collaboration will help you improve your estimation skills. Strong estimation skills allow you to make quick calculations when it is inconvenient or unnecessary to calculate exact results. You can also use estimation to check the results of a calculation. If the answer you calculated is not close to your estimate, you know that you need to check your work in your calculation.

In this course, you will make a lot of estimations and be asked to explain the strategies you used to generate your estimations. There is not one best strategy. It is important that you develop strategies that make sense to you. An estimation strategy is wrong only if it is mathematically incorrect (like saying that 25% is 1/2). In the following section, you will practice your use of estimation strategies to answer questions and calculate percentages.

PROBLEM SITUATION 2: FROM ESTIMATION TO EXACT CALCULATION

Being able to calculate with percentages is also very important. In the situation in Question 1, an estimate of the sale price would help you to decide whether to buy the coat. However, the store owner needs to make an exact calculation to know how much to charge. In Question 2, an estimate helps the officer get a sense of the situation, but if she is writing a report, she will want exact figures.

In the problems below, you will find exact calculations instead of estimates for the same situations you encountered in Questions 1–3. You may use a calculator. Show your work.

(4) You are shopping for a coat and find one that is on sale. The coat’s regular price is $87.99.

(a) What is the exact sale price based on each of the following discounts? Remember, we are trying to find the sale price here, so we need to figure out the discount first, then subtract to find the sale price. Do this on your own first before sharing in your group.

(i) 20% off

(ii) 25% off

(iii) 35% off

(iv) 70% off

(b) Explain to your group your sale price calculations for (i) - (iv) above. Enter the sale price for each of the discounts (i) - (iv). Round your calculations to 2 decimal places.

(5) For each precinct, what is the exact percentage of violent incidents that involve a weapon? Round your calculation to the nearest whole percent.

Precinct 1:

Precinct 2:

(6) You have a credit card that awards you a “cash back bonus.” This means that every time you use your credit card to make a purchase, you earn back a percentage of the money you spend. Your card gives you a bonus of 0.5%. Calculate your exact bonus if you charge $462 on your credit card. Round to the nearest cent.

FURTHER APPLICATIONS

(7) Estimate an answer to each of the following. Explain your estimation strategy. Record an estimate and strategy for (i) - (iv).

(i) 62% of 87

(ii) 22% of 203

(iii) 37 is what percent of 125

(iv) 2 is what percent of 310

MAKING CONNECTIONS

Record the important mathematical ideas from the discussion.