problem solving base rate and percentage

Solving Percent Problems

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7th grade foundations (Eureka Math/EngageNY)

Course: 7th grade foundations (eureka math/engageny)   >   unit 4.

• The meaning of percent
• Converting percents to decimals & fractions example
• Converting between percents, fractions, & decimals
• Finding a percent
• Percent of a whole number

Identifying percent amount and base

• Finding percents

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Video transcript

Lesson Percentage word problems (Type 3 problems, Finding the Base)

Percent , Rate , Base

Understanding percent , rate , and base is essential in various mathematical and real-life contexts. In this study guide, we will cover the basics of percent , rate , and base , and provide examples to help you grasp these concepts.

Percent means "per hundred" and is denoted by the symbol "%". It is used to express a number as a fraction of 100. For example, 25% is equivalent to the fraction 25/100 or the decimal 0.25.

A rate is a special ratio in which the two terms are in different units . For example, miles per hour (mph) is a rate . It compares the distance traveled to the time taken. Rates are often expressed using the word "per" or the symbol "/", such as 60 miles per hour or 60 mph.

The base is the original value in a percent problem. It is the whole or the original amount before a percentage is calculated. For example, if you're calculating 20% of 80, then 80 is the base .

Key Formulas

The following formulas are essential when dealing with percent , rate , and base :

Percent = (Part / Whole) * 100

Rate = (Part / Base )

Base = (Part / Rate )

Let's work through a few examples to illustrate these concepts:

Example 1: Calculating Percent

If you scored 35 out of 50 on a test, what is your score as a percentage?

Percent = (35 / 50) * 100 = 70%

Example 2: Calculating Rate

If a car travels 300 miles in 5 hours , what is its speed in miles per hour ?

Rate = 300 miles / 5 hours = 60 mph

Example 3: Finding the Base

If 15 is 20% of a number, what is the original number?

Base = 15 / 0.20 = 75

When studying percent , rate , and base , it's helpful to practice converting between fractions , decimals , and percentages . Additionally, working through real-life problems involving discounts, taxes, and tips can improve your understanding of these concepts.

Remember to use the key formulas and units to guide your problem-solving process. Understanding the relationship between percent , rate , and base will also make it easier to solve problems in various scenarios.

By mastering percent , rate , and base , you'll develop a valuable skill set for handling a wide range of mathematical and practical situations.

Good luck with your studies!

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Solving problems with percentages

• Price difference I
• Price difference II
• How many students?

To solve problems with percent we use the percent proportion shown in "Proportions and percent".

$$\frac{a}{b}=\frac{x}{100}$$

$$\frac{a}{{\color{red} {b}}}\cdot {\color{red} {b}}=\frac{x}{100}\cdot b$$

$$a=\frac{x}{100}\cdot b$$

x/100 is called the rate.

$$a=r\cdot b\Rightarrow Percent=Rate\cdot Base$$

Where the base is the original value and the percentage is the new value.

47% of the students in a class of 34 students has glasses or contacts. How many students in the class have either glasses or contacts?

$$a=r\cdot b$$

$$47\%=0.47a$$

$$=0.47\cdot 34$$

$$a=15.98\approx 16$$

16 of the students wear either glasses or contacts.

We often get reports about how much something has increased or decreased as a percent of change. The percent of change tells us how much something has changed in comparison to the original number. There are two different methods that we can use to find the percent of change.

The Mathplanet school has increased its student body from 150 students to 240 from last year. How big is the increase in percent?

We begin by subtracting the smaller number (the old value) from the greater number (the new value) to find the amount of change.

$$240-150=90$$

Then we find out how many percent this change corresponds to when compared to the original number of students

$$90=r\cdot 150$$

$$\frac{90}{150}=r$$

$$0.6=r= 60\%$$

We begin by finding the ratio between the old value (the original value) and the new value

$$percent\:of\:change=\frac{new\:value}{old\:value}=\frac{240}{150}=1.6$$

As you might remember 100% = 1. Since we have a percent of change that is bigger than 1 we know that we have an increase. To find out how big of an increase we've got we subtract 1 from 1.6.

$$1.6-1=0.6$$

$$0.6=60\%$$

As you can see both methods gave us the same answer which is that the student body has increased by 60%

Video lessons

A skirt cost $35 regulary in a shop. At a sale the price of the skirtreduces with 30%. How much will the skirt cost after the discount?

Solve "54 is 25% of what number?"

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Solving Percent Problems

Learning Objective(s)

·          Identify the amount, the base, and the percent in a percent problem.

·          Find the unknown in a percent problem.

Introduction

Percents are a ratio of a number and 100. So they are easier to compare than fractions, as they always have the same denominator, 100. A store may have a 10% off sale. The amount saved is always the same portion or fraction of the price, but a higher price means more money is taken off. Interest rates on a saving account work in the same way. The more money you put in your account, the more money you get in interest. It’s helpful to understand how these percents are calculated.

Parts of a Percent Problem

Jeff has a coupon at the Guitar Store for 15% off any purchase of $100 or more. He wants to buy a used guitar that has a price tag of $220 on it. Jeff wonders how much money the coupon will take off the original $220 price.

Problems involving percents have any three quantities to work with: the percent , the amount , and the base .

The percent has the percent symbol (%) or the word “percent.” In the problem above, 15% is the percent off the purchase price.

The base is the whole amount. In the problem above, the whole price of the guitar is $220, which is the base.

The amount is the number that relates to the percent. It is always part of the whole. In the problem above, the amount is unknown. Since the percent is the percent off , the amount will be the amount off of the price .

You will return to this problem a bit later. The following examples show how to identify the three parts, the percent, the base, and the amount.

The previous problem states that 30 is a portion of another number. That means 30 is the amount. Note that this problem could be rewritten: 20% of what number is 30?

Solving with Equations

Percent problems can be solved by writing equations. An equation uses an equal sign (= ) to show that two mathematical expressions have the same value.

Percents are fractions, and just like fractions, when finding a percent (or fraction, or portion) of another amount, you multiply.

The percent of the base is the amount.

Percent of the Base is the Amount.

Percent · Base = Amount

Once you have an equation, you can solve it and find the unknown value. To do this, think about the relationship between multiplication and division. Look at the pairs of multiplication and division facts below, and look for a pattern in each row.

Multiplication and division are inverse operations. What one does to a number, the other “undoes.”

When you have an equation such as 20% · n = 30, you can divide 30 by 20% to find the unknown: n =  30 ÷ 20%.

You can solve this by writing the percent as a decimal or fraction and then dividing.

n = 30 ÷ 20% =  30 ÷ 0.20 = 150

You can estimate to see if the answer is reasonable. Use 10% and 20%, numbers close to 12.5%, to see if they get you close to the answer.

10% of 72 = 0.1 · 72 = 7.2

20% of 72 = 0.2 · 72 = 14.4

Notice that 9 is between 7.2 and 14.4, so 12.5% is reasonable since it is between 10% and 20%.

This problem is a little easier to estimate. 100% of 24 is 24. And 110% is a little bit more than 24. So, 26.4 is a reasonable answer.

Using Proportions to Solve Percent Problems

Let’s go back to the problem that was posed at the beginning. You can now solve this problem as shown in the following example.

You can estimate to see if the answer is reasonable. Since 15% is half way between 10% and 20%, find these numbers.

10% of 220 = 0.1 · 220 = 22

20% of 220 = 0.2 · 220 = 44

The answer, 33, is between 22 and 44. So $33 seems reasonable.

There are many other situations that involve percents. Below are just a few.

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Definition:

The percent, base, and rate are connected with one another in terms of computation. To find the percentage , multiply the base by the rate. Remember that the rate must be changed from a percent to a decimal before multiplying can be done. Rate times base equals percentage.

PERCENTAGE (P=BxR) –  The result obtained when a number is multiplied by a percent.

BASE (B=P/R) –  The whole in a problem. The amount you are taking a percent of.

RATE (R=P/B) –  The ratio of amount to the base. It is written as a percent.

Applying Percentage, Base, and Rate Worksheets

This is a fantastic bundle which includes everything you need to know about Applying Percentage, Base, and Rate across 15+ in-depth pages. These are ready-to-use Common core aligned Grade 6 Math worksheets. Each ready to use worksheet collection includes 10 activities and an answer guide. Not teaching common core standards ? Don’t worry! All our worksheets are completely editable so can be tailored for your curriculum and target audience.

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Grade 6 Mathematics Module: Finding the Percentage, Base and Rate in a Given Problem

This Self-Learning Module (SLM) is prepared so that you, our dear learners, can continue your studies and learn while at home. Activities, questions, directions, exercises, and discussions are carefully stated for you to understand each lesson.

Each SLM is composed of different parts. Each part shall guide you step-by-step as you discover and understand the lesson prepared for you.

Pre-tests are provided to measure your prior knowledge on lessons in each SLM. This will tell you if you need to proceed on completing this module or if you need to ask your facilitator or your teacher’s assistance for better understanding of the lesson. At the end of each module, you need to answer the post-test to self-check your learning. Answer keys are provided for each activity and test. We trust that you will be honest in using these.

Please use this module with care. Do not put unnecessary marks on any part of this SLM. Use a separate sheet of paper in answering the exercises and tests. And read the instructions carefully before performing each task.

If you have any questions in using this SLM or any difficulty in answering the tasks in this module, do not hesitate to consult your teacher or facilitator.

This module was designed and written with you in mind. It is here to help you master the lessons on Finding the Percentage, Base and Rate or Percent in given problems. The scope of this module permits it to be used in many different learning situations. The language used recognizes your diverse vocabulary level. The lessons are arranged to follow the standard sequence of the course. But the order in which you read them can be changed to correspond with the textbook you are now using.

The module is divided into three lessons, namely:

• Lesson 1 – Finding the Percentage in a Given Problem
• Lesson 2 – Finding the Rate in a Given Problem
• Lesson 3 – Finding the Base in a Given Problem

After going through this module, you are expected to:

1. identify the percentage, rate and base in a given problem;

2. find the base, percentage or rate or percent in a given problem; and

3. solve routine and non-routine problems involving the percentage, rate and base using appropriate strategies and tools.

Grade 6 Mathematics Quarter 2 Self-Learning Module: Finding the Percentage, Base and Rate in a Given Problem

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Determining Percentage, Base, & Rate

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Solving Percent Problems

Percent is a great mathematical tool to express quantities and is used extensively in different things – from interest rates, discounts, and taxes to surveys, censuses, etc.

This article is your guide to percent and solving percent problems frequently appearing in major national examinations.

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Table of Contents

What does percent mean.

The word “percent” originated from the Latin phrase per centum, meaning “by hundred.” When we say “percent,” we refer to “parts per 100”. This means that a percent is a fraction with 100 as the denominator. The symbol % is used to indicate a percent.

For example, 3% means three parts per 100 or 3⁄100; 45% means 45⁄100; and 92% means 92⁄100.

Illustrating Percent

Suppose a vendor has 100 biscuits. If 10% of those biscuits are ube-flavored, 10⁄100 or 10 out of 100 biscuits are ube-flavored.

On the other hand, suppose there are 100 students in a school auditorium. If 42% of those students are honor students, 42⁄100 students, or 42 out of 100 students, are honor students.

Expressing Percent as Fraction and Decimal

Since percent means a fraction with 100 as the denominator, we can express a percent as a fraction or a decimal number .

Drop the percent sign and put 100 as the denominator to transform a percent into a fraction. For instance, 25% is simply 25⁄100.

Note that when 25⁄100 is reduced to its lowest terms, you will obtain ¼. This means that 25% is also equivalent to ¼. 

Furthermore, note that when you transform ¼ into its decimal form using the steps we have discussed in the previous reviewer , you will obtain 0.25. Hence, 25% is also equal to 0.25. 

There is an easier way to transform percent into decimals . Drop the percent sign and move the decimal point two places to the left of the given number.

For example, 54% is equivalent to 0.54

Example: Transform 3% to decimal form.

Suppose that your mom prepared ten pieces of your favorite cookies. You are excited to taste those cookies, but you realize that your brother ate 20% of the cookies that your mom prepared. What exactly is the number of cookies eaten by your brother?

To determine the answer to your question above, you must determine 20% of 10. This case involves the application of percentages.

The percentage is the result when you multiply a number by a percent. Returning to your problem about the number of cookies your brother ate, 20% of 10 can be determined if you multiply ten by 20%. The result after you multiply the numbers is called the percentage.

How To Find the Percentage

Follow these steps if you want to find the percentage:

Step 1: Convert the given percent (the one with the % sign) into decimals .

Again, to convert percent into its decimal form, we drop the percent sign and then move the decimal point two places to the left. Thus, 20% = 0.20

Step 2 : Multiply the decimal you have obtained from Step 1 to the given number. The result is the percentage.

To multiply 0.20 by 10, we ignored the decimal point for a while and multiplied the given decimals like whole numbers. We have obtained 0200. Since 0.20 has two decimal places while 10 has none, the final answer should have two decimal places. We count two digits from the right of 0200 and put the decimal point there. Hence, the answer is 02.00, which is equivalent to 2.

Hence, 20% of 10 is 2. This means that out of 10 cookies your mother prepared, 2 of those were eaten by your brother.

Let us have another example.

Example: What is 50% of 120?

Step 1 : Convert the given percent (the one with the % sign) into decimals.

We drop the % sign of 50% and move the decimal point two places to the left.

Thus, 50% = 0.50

Step 2: Multiply the decimal you have obtained from Step 1 to the given number. The result is the percentage.

To multiply 0.50 by 120, we ignored the decimal point for a while and multiplied the given decimals like whole numbers. Through this process, we have obtained 06000. Since 0.50 has two decimal places while 120 has none, the final answer should have two decimal places. We count two digits from the right of 06000 and put the decimal point there. Hence, the answer is 060.00, which is equivalent to 60.

Hence, 50% of 120 is 60.

Simple Tricks in Computing Percentages

We always want to make our computations in mathematics faster and more accurate. For this reason, I will share two tricks you can use when computing percentages.

Trick #1: You can compute some percentages using only mental computation.

If you want to determine the 25%, 50%, 75%, or 100% of a number, you can do so without the help of pen and paper.

• 25% is equivalent to 25⁄100 or ¼. Hence, to find the 25% of a number, divide the given number by 4. Example: 25% of 40 is just 40 ÷ 4 = 10.
• 50% is equivalent to 50⁄100 or ½. Thus, to find the 50% of a number, divide the given number by 2. This means 50% of a number is just half the given number. Example: 50% of 40 is just 40 ÷ 2 = 20.
• 75% is equivalent to 75⁄100 or ¾. Thus, to find the 75% of a number, multiply the given number by three and then divide the result by 4. Example: 75% of 40 is just 40 x 3 = 120 ÷ 4 = 30.
• 100% is equivalent to 100⁄100 or 1. Thus, 100% of a number is the number itself . Example: 100% of 40 is just 40 itself.

Trick #2: X% of a number Y is equal to Y% of number X

This trick means we can transfer the % sign to the other number, and the result will be the same.

Example : What is 40% of 25?

Using trick #2, we can transfer the % sign from 40% to 25. Thus, we have 25%. This means 40% of 25 is the same as 25% of 40.

Thus, applying our first trick on finding the 25% of a number, 40 ÷ 4 = 10; hence, 40% of 25 is 10.

Example : What is 92% of 50? 

92% of 50 is the same as 50% of 92. Hence, we can just divide 92 by 2 to obtain the answer, 92 ÷ 2 = 46

Therefore, 92% of 50 is 46.

Base and Rate

The base is the amount you are taking a percent of. Meanwhile, the rate is the percent you are calculating.

For example, if there are 50 students in a classroom and 20% of those students are honor students, it follows that ten students are honor students. 50 is the base since it is the amount we take a percent of. Meanwhile, 20% is the rate since we calculate the percentage. Lastly, 10 is the percentage.

The product of the base and the rate is the percentage .

Percentage = Base × Rate

Example: Determine the percentage, base, and rate if 20% of 90 is 18.

Since 90 x 20% = 90 x 0.20 = 18, 90 is the base, 20% is the rate, and 18 is the percentage.

Calculating Percentage, Base, and Rate

Formula to find the percentage.

The formula to find the percentage, as we have stated, is: 

We can manipulate the mathematical equation above to obtain the formulas for computing the base and the rate:

Formula to Find the Base

Base = Percentage ÷ Rate

Formula to Find the Rate

Rate = Percentage ÷ Base

Example 1: If 10% of a number is 90, what is the number?

We can interpret this question as 10% of ______ = 90. Since “of” is a signal word for multiplication, it also implies 10% x ______ = 90

This means that 10% is the rate while 90 is the percentage. The unknown number is the base. Thus, we need to compute the base.

Using the formula to find the base:

  Base = Percentage ÷ Rate

Base  = 90 ÷ 10%

Convert the given percent into decimal:

Base  = 90 ÷ 0.10

Now that you have already transformed the rate into decimal form, you may divide 90 by 0.10 to obtain the answer.

To perform division with decimal numbers , we need to transform the divisor (0.10) into a whole number by moving two decimal places to the right. Thus, the new divisor is 10. We also move two decimal places for the dividend (90). Thus, the new dividend is 9000.

We now perform long division with our new dividend and divisor:

To find the base, we compute 90 ÷ 0.10 = 900

Hence, the base is 900.

Example 2:  What percent of 720 is 90?

We can translate the question above in this form: _____% of 720 is 90 or _____% x 720 = 90. Therefore, 720 is the base, while 90 is the percentage. The missing number is the rate.

We will now use the formula for finding the rate.

Again, based on the given problem, the percentage is 90 while the base is 720

          Rate = 90 ÷ 720

Notice that the dividend (the first number) is smaller than the divisor (the second number). In this case, you may apply the same steps in transforming fractions into decimal form because  90 ÷ 720 is a proper fraction (i.e., 90⁄720).

Let us divide 90 by 720 using the steps in transforming fractions into decimal form .

We add some zeros and decimal points to proceed with the division process.

We can now divide 900 by 720.

Note that every time the remainder becomes smaller than the divisor, we add zeros to 900 and the remainder to continue the division process.

The quotient we obtained is 0.125. Thus, 0.125 is our rate.

However, the rate must always be expressed with a percent sign. To do this, we multiply 0.125 by 100 or move two decimal places to the right of it and put a percent sign. Thus, 0.125 is equal to 12.5%.

Therefore, the rate is 12.5%

The Percentage, Base, and Rate Triangle

What if you forgot the formula to determine the percentage, base, or rate in a particular problem? Don’t worry because there is a fun way to derive these formulas. 

Shown below is the Percentage, Base, and Rate Triangle . It is a triangle divided into three portions where P (for percentage) is written on the upper portion, and B (for base) and R (for rate) are written on the lower portions. There are also division signs in the triangle’s outer left and outer right parts and a multiplication sign below it.

How To Use the Percentage, Base, and Rate Triangle

Suppose you are looking for the base. You have to cover the B in the triangle and look at the remaining letters and the operation between them. Notice that if you cover B, the remaining letters are P and R, with a division sign between them. This means that to find the base, you must divide P by R.

Next topic:  Ratio and Proportion

Previous topic : Fundamental Operations on Fractions and Decimals

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Stormy Daniels Takes the Stand

The porn star testified for eight hours at donald trump’s hush-money trial. this is how it went..

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It’s 6:41 AM. I’m feeling a little stressed because I’m running late. It’s the fourth week of Donald J. Trump’s criminal trial. It’s a white collar trial. Most of the witnesses we’ve heard from have been, I think, typical white collar witnesses in terms of their professions.

We’ve got a former publisher, a lawyer, accountants. The witness today, a little less typical, Stormy Daniels, porn star in a New York criminal courtroom in front of a jury more accustomed to the types of witnesses they’ve already seen. There’s a lot that could go wrong.

From “The New York Times,” I’m Michael Barbaro. This is “The Daily.”

Today, what happened when Stormy Daniels took the stand for eight hours in the first criminal trial of Donald J. Trump. As before, my colleague Jonah Bromwich was inside the courtroom.

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It’s Friday, May 10th.

So it’s now day 14 of this trial. And I think it’s worth having you briefly, and in broad strokes, catch listeners up on the biggest developments that have occurred since you were last on, which was the day that opening arguments were made by both the defense and the prosecution. So just give us that brief recap.

Sure. It’s all been the prosecution’s case so far. And prosecutors have a saying, which is that the evidence is coming in great. And I think for this prosecution, which is trying to show that Trump falsified business records to cover up a sex scandal, to ease his way into the White House in 2016, the evidence has been coming in pretty well. It’s come in well through David Pecker, former publisher of The National Enquirer, who testified that he entered into a secret plot with Trump and Michael Cohen, his fixer at the time, to suppress negative stories about Trump, the candidate.

It came in pretty well through Keith Davidson, who was a lawyer to Stormy Daniels in 2016 and negotiated the hush money payment. And we’ve seen all these little bits and pieces of evidence that tell the story that prosecutors want to tell. And the case makes sense so far. We can’t tell what the jury is thinking, as we always say.

But we can tell that there’s a narrative that’s coherent and that matches up with the prosecution’s opening statement. Then we come to Tuesday. And that day really marks the first time that the prosecution’s strategy seems a little bit risky because that’s the day that Stormy Daniels gets called to the witness stand.

OK, well, just explain why the prosecution putting Stormy Daniels on the stand would be so risky. And I guess it makes sense to answer that in the context of why the prosecution is calling her as a witness at all.

Well, you can see why it makes sense to have her. The hush money payment was to her. The cover-up of the hush money payment, in some ways, concerns her. And so she’s this character who’s very much at the center of this story. But according to prosecutors, she’s not at the center of the crime. The prosecution is telling a story, and they hope a compelling one. And arguably, that story starts with Stormy Daniels. It starts in 2006, when Stormy Daniels says that she and Trump had sex, which is something that Trump has always denied.

So if prosecutors were to not call Stormy Daniels to the stand, you would have this big hole in the case. It would be like, effect, effect, effect. But where is the cause? Where is the person who set off this chain reaction? But Stormy Daniels is a porn star. She’s there to testify about sex. Sex and pornography are things that the jurors were not asked about during jury selection. And those are subjects that bring up all kinds of different complex reactions in people.

And so, when the prosecutors bring Stormy Daniels to the courtroom, it’s very difficult to know how the jurors will take it, particularly given that she’s about to describe a sexual episode that she says she had with the former president. Will the jurors think that makes sense, as they sit here and try to decide a falsifying business records case, or will they ask themselves, why are we hearing this?

So the reason why this is the first time that the prosecution’s strategy is, for journalists like you, a little bit confusing, is because it’s the first time that the prosecution seems to be taking a genuine risk in what they’re putting before these jurors. Everything else has been kind of cut and dry and a little bit more mechanical. This is just a wild card.

This is like live ammunition, to some extent. Everything else is settled and controlled. And they know what’s going to happen. With Stormy Daniels, that’s not the case.

OK, so walk us through the testimony. When the prosecution brings her to the stand, what actually happens?

It starts, as every witness does, with what’s called direct examination, which is a fancy word for saying prosecutors question Stormy Daniels. And they have her tell her story. First, they have her tell the jury about her education and where she grew up and her professional experience. And because of Stormy Daniels’s biography, that quickly goes into stripping, and then goes into making adult films.

And I thought the prosecutor who questioned her, Susan Hoffinger, had this nice touch in talking about that, because not only did she ask Daniels about acting in adult films. But she asked her about writing and directing them, too, emphasizing the more professional aspects of that work and giving a little more credit to the witness, as if to say, well, you may think this or you may think that. But this is a person with dignity who took what she did seriously. Got it.

What’s your first impression of Daniels as a witness?

It’s very clear that she’s nervous. She’s speaking fast. She’s laughing to herself and making small jokes. But the tension in the room is so serious from the beginning, from the moment she enters, that those jokes aren’t landing. So it just feels, like, really heavy and still and almost oppressive in there. So Daniels talking quickly, seeming nervous, giving more answers than are being asked of her by the prosecution, even before we get to the sexual encounter that she’s about to describe, all of that presents a really discomfiting impression, I would say.

And how does this move towards the encounter that Daniels ultimately has?

It starts at a golf tournament in 2006, in Lake Tahoe, Nevada. Daniels meets Trump there. There are other celebrities there, too. They chatted very briefly. And then she received a dinner invitation from him. She thought it over, she says. And she goes to have dinner with Trump, not at a restaurant, by the way. But she’s invited to join him in the hotel suite.

So she gets to the hotel suite. And his bodyguard is there. And the hotel door is cracked open. And the bodyguard greets her and says she looks nice, this and that. And she goes in. And there’s Donald Trump, just as expected. But what’s not expected, she says, is that he’s not wearing what you would wear to a dinner with a stranger, but instead, she says, silk or satin pajamas. She asked him to change, she says. And he obliges.

He goes, and he puts on a dress shirt and dress pants. And they sit down at the hotel suite’s dining room table. And they have a kind of bizarre dinner. Trump is asking her very personal questions about pornography and safe sex. And she testifies that she teased him about vain and pompous he is. And then at some point, she goes to the bathroom. And she sees that he has got his toiletries in there, his Old Spice, his gold tweezers.

Very specific details.

Yeah, we’re getting a ton of detail in this scene. And the reason we’re getting those is because prosecutors are trying to elicit those details to establish that this is a credible person, that this thing did happen, despite what Donald Trump and his lawyers say. And the reason you can know it happened, prosecutors seem to be saying, is because, look at all these details she can still summon up.

She comes out of the bathroom. And she says that Donald Trump is on the hotel bed. And what stands out to me there is what she describes as a very intense physical reaction. She says that she blacked out. And she quickly clarifies, she doesn’t mean from drugs or alcohol. She means that, she says, that the intensity of this experience was such that, suddenly, she can’t remember every detail. The prosecution asks a question that cuts directly to the sex. Essentially, did you start having sex with him? And Daniels says that she did. And she continues to provide more details than even, I think, the prosecution wanted.

And I think we don’t want to go chapter and verse through this claimed sexual encounter. But I wonder what details stand out and which details feel important, given the prosecution’s strategy here.

All the details stand out because it’s a story about having had sex with a former president. And the more salacious and more private the details feel, the more you’re going to remember them. So we’ll remember that Stormy Daniels said what position they had sex in. We’ll remember that she said he didn’t use a condom. Whether that’s important to the prosecution’s case, now, that’s a much harder question to answer, as we’ve been saying.

But what I can tell you is, as she’s describing having had sex with Donald Trump, and Donald Trump is sitting right there, and Eric Trump, his son, is sitting behind him, seeming to turn a different color as he hears this embarrassment of his father being described to a courtroom full of reporters at this trial, it’s hard to even describe the energy in that room. It was like nothing I had ever experienced. And it was just Daniels’s testimony and, seemingly, the former President’s emotions. And you almost felt like you were trapped in there with both of them as this description was happening.

Well, I think it’s important to try to understand why the prosecution is getting these details, these salacious, carnal, pick your word, graphic details about sex with Donald Trump. What is the value, if other details are clearly making the point that she’s recollecting something?

Well, I think, at this point, we can only speculate. But one thing we can say is, this was uncomfortable. This felt bad. And remember, prosecutor’s story is not about the sex. It’s about trying to hide the sex. So if you’re trying to show a jury why it might be worthwhile to hide a story, it might be worth —

Providing lots of salacious details that a person would want to hide.

— exposing them to how bad that story feels and reminding them that if they had been voters and they had heard that story, and, in fact, they asked Daniels this very question, if you hadn’t accepted hush money, if you hadn’t signed that NDA, is this the story you would have told? And she said, yes. And so where I think they’re going with this, but we can’t really be sure yet, is that they’re going to tell the jurors, hey, that story, you can see why he wanted to cover that up, can’t you?

You mentioned the hush money payments. What testimony does Daniels offer about that? And how does it advance the prosecution’s case of business fraud related to the hush money payments?

So little evidence that it’s almost laughable. She says that she received the hush money. But we actually already heard another witness, her lawyer at the time, Keith Davidson, testify that he had received the hush money payment on her behalf. And she testified about feeling as if she had to sell this story because the election was fast approaching, almost as if her leverage was slipping away because she knew this would be bad for Trump.

That feels important. But just help me understand why it’s important.

Well, what the prosecution has been arguing is that Trump covered up this hush money payment in order to conceal a different crime. And that crime, they say, was to promote his election to the presidency by illegal means.

Right, we’ve talked about this in the past.

So when Daniels ties her side of the payment into the election, it just reminds the jurors maybe, oh, right, this is what they’re arguing.

So how does the prosecution end this very dramatic, and from everything you’re saying, very tense questioning of Stormy Daniels about this encounter?

Well, before they can even end, the defense lawyers go and they consult among themselves. And then, with the jury out of the room, one of them stands up. And he says that the defense is moving for a mistrial.

On what terms?

He says that the testimony offered by Daniels that morning is so prejudicial, so damning to Trump in the eyes of the jury, that the trial can no longer be fair. Like, how could these jurors have heard these details and still be fair when they render their verdict? And he says a memorable expression. He says, you can’t un-ring that bell, meaning they heard it. They can’t un-hear it. It’s over. Throw out this trial. It should be done.

Wow. And what is the response from the judge?

So the judge, Juan Merchan, he hears them out. And he really hears them out. But at the end of their arguments, he says, I do think she went a little too far. He says that. He said, there were things that were better left unsaid.

By Stormy Daniels?

By Stormy Daniels. And he acknowledges that she is a difficult witness. But, he says, the remedy for that is not a mistrial, is not stopping the whole thing right now. The remedy for that is cross-examination. If the defense feels that there are issues with her story, issues with her credibility, they can ask her whatever they want. They can try to win the jury back over. If they think this jury has been poisoned by this witness, well, this is their time to provide the antidote. The antidote is cross-examination. And soon enough, cross-examination starts. And it is exactly as intense and combative as we expected.

We’ll be right back.

So, Jonah, how would you characterize the defense’s overall strategy in this intense cross-examination of Stormy Daniels?

People know the word impeach from presidential impeachments. But it has a meaning in law, too. You impeach a witness, and, specifically, their credibility. And that’s what the defense is going for here. They are going to try to make Stormy Daniels look like a liar, a fraud, an extortionist, a money-grubbing opportunist who wanted to take advantage of Trump and sought to do so by any means necessary.

And what did that impeachment strategy look like in the courtroom?

The defense lawyer who questions Stormy Daniels is a woman named Susan Necheles. She’s defended Trump before. And she’s a bit of a cross-examination specialist. We even saw her during jury selection bring up these past details to confront jurors who had said nasty things about Trump on social media with. And she wants to do the same thing with Daniels. She wants to bring up old interviews and old tweets and things that Daniels has said in the past that don’t match what Daniels is saying from the stand.

What’s a specific example? And do they land?

Some of them land. And some of them don’t. One specific example is that Necheles confronts Daniels with this old tweet, where Daniels says that she’s going to dance down the street if Trump goes to jail. And what she’s trying to show there is that Daniels is out for revenge, that she hates Trump, and that she wants to see him go to jail. And that’s why she’s testifying against him.

And Daniels is very interesting during the cross-examination. It’s almost as if she’s a different person. She kind of squares her shoulders. And she sits up a little straighter. And she leans forward. Daniels is ready to fight. But it doesn’t quite land. The tweet actually says, I’ll dance down the street when he’s selected to go to jail.

And Daniels goes off on this digression about how she knows that people don’t get selected to go to jail. That’s not how it works. But she can’t really unseat this argument, that she’s a political enemy of Donald Trump. So that one kind of sticks, I would say. But there are other moves that Necheles tries to pull that don’t stick.

So unlike the prosecution, which typically used words like adult, adult film, Necheles seems to be taking every chance she can get to say porn, or pornography, or porn star, to make it sound base or dirty. And so when she starts to ask Daniels about actually being in pornography, writing, acting, and directing sex films, she tries to land a punch line, Necheles does. She says, so you have a lot of experience making phony stories about sex appear to be real, right?

As if to say, perhaps this story you have told about entering Trump’s suite in Lake Tahoe and having sex with him was made up.

Just another one of your fictional stories about sex. But Daniels comes back and says, the sex in the films, it’s very much real, just like what happened to me in that room. And so, when you have this kind of combat of a lawyer cross-examining very aggressively and the witness fighting back, you can feel the energy in the room shift as one lands a blow or the other does. But here, Daniels lands one back. And the other issue that I think Susan Necheles runs into is, she tries to draw out disparities from interviews that Daniels gave, particularly to N-TOUCH, very early on once the story was out.

It’s kind of like a tabloid magazine?

But some of the disparities don’t seem to be landing quite like Necheles would want. So she tries to do this complicated thing about where the bodyguard was in the room when Daniels walked into the room, as described in an interview in a magazine. But in that magazine interview, as it turns out, Daniels mentioned that Trump was wearing pajamas. And so, if I’m a juror, I don’t care where the bodyguard is. I’m thinking about, oh, yeah, I remember that Stormy Daniels said now in 2024 that Trump was wearing pajamas.

I’m curious if, as somebody in the room, you felt that the defense was effective in undermining Stormy Daniels’s credibility? Because what I took from the earlier part of our conversation was that Stormy Daniels is in this courtroom on behalf of the prosecution to tell a story that’s uncomfortable and has the kind of details that Donald Trump would be motivated to try to hide. And therefore, this defense strategy is to say, those details about what Trump might want to hide, you can’t trust them. So does this back and forth effectively hurt Stormy Daniels’s credibility, in your estimation?

I don’t think that Stormy Daniels came off as perfectly credible about everything she testified about. There are incidents that were unclear or confusing. There were things she talked about that I found hard to believe, when she, for instance, denied that she had attacked Trump in a tweet or talked about her motivations. But about what prosecutors need, that central story, the story of having had sex with him, we can’t know whether it happened.

But there weren’t that many disparities in these accounts over the years. In terms of things that would make me doubt the story that Daniels was telling, details that don’t add up, those weren’t present. And you don’t have to take my word for that, nor should you. But the judge is in the room. And he says something very, very similar.

What does he say? And why does he say it?

Well, he does it when the defense, again, at the end of the day on Thursday, calls for a mistrial.

With a similar argument as before?

Not only with a similar argument as before, but, like, almost the exact same argument. And I would say that I was astonished to see them do this. But I wasn’t because I’ve covered other trials where Trump is the client. And in those trials, the lawyers, again and again, called for a mistrial.

And what does Judge Marchan say in response to this second effort to seek a mistrial?

Let me say, to this one, he seems a little less patient. He says that after the first mistrial ruling, two days before, he went into his chambers. And he read every decision he had made about the case. He took this moment to reflect on the first decision. And he found that he had, in his own estimation, which is all he has, been fair and not allowed evidence that was prejudicial to Trump into this trial. It could continue. And so he said that again. And then he really almost turned on the defense. And he said that the things that the defense was objecting to were things that the defense had made happen.

He says that in their opening statement, the defense could have taken issue with many elements of the case, about whether there were falsified business records, about any of the other things that prosecutors are saying happened. But instead, he says, they focused their energy on denying that Trump ever had sex with Daniels.

And so that was essentially an invitation to the prosecution to call Stormy Daniels as a witness and have her say from the stand, yes, I had this sexual encounter. The upshot of it is that the judge not only takes the defense to task. But he also just says that he finds Stormy Daniels’s narrative credible. He doesn’t see it as having changed so much from year to year.

Interesting. So in thinking back to our original question here, Jonah, about the idea that putting Stormy Daniels on the stand was risky, I wonder if, by the end of this entire journey, you’re reevaluating that idea because it doesn’t sound like it ended up being super risky. It sounded like it ended up working reasonably well for the prosecution.

Well, let me just assert that it doesn’t really matter what I think. The jury is going to decide this. There’s 12 people. And we can’t know what they’re thinking. But my impression was that, while she was being questioned by the prosecution for the prosecution’s case, Stormy Daniels was a real liability. She was a difficult witness for them.

And the judge said as much. But when the defense cross-examined her, Stormy Daniels became a better witness, in part because their struggles to discredit her may have actually ended up making her story look more credible and stronger. And the reason that matters is because, remember, we said that prosecutors are trying to fill this hole in their case. Well, now, they have. The jury has met Stormy Daniels. They’ve heard her account. They’ve made of it what they will. And now, the sequence of events that prosecutors are trying to line up as they seek prison time for the former President really makes a lot of sense.

It starts with what Stormy Daniels says with sex in a hotel suite in 2006. It picks up years later, as Donald Trump is trying to win an election and, prosecutors say, suppressing negative stories, including Stormy Daniels’s very negative story. And the story that prosecutors are telling ends with Donald Trump orchestrating the falsification of business records to keep that story concealed.

Well, Jonah, thank you very much. We appreciate it.

Of course, thanks for having me.

The prosecution’s next major witness will be Michael Cohen, the former Trump fixer who arranged for the hush money payment to Stormy Daniels. Cohen is expected to take the stand on Monday.

Here’s what else you need to know today. On Thursday, Israeli Prime Minister Benjamin Netanyahu issued a defiant response to warnings from the United States that it would stop supplying weapons to Israel if Israel invades the Southern Gaza City of Rafah. So far, Israel has carried out a limited incursion into the city where a million civilians are sheltering, but has threatened a full invasion. In a statement, Netanyahu said, quote, “if we need to stand alone, we will stand alone.”

Meanwhile, high level ceasefire negotiations between Israel and Hamas have been put on hold in part because of anger over Israel’s incursion into Rafah.

A reminder, tomorrow, we’ll be sharing the latest episode of our colleague’s new show, “The Interview” This week on “The Interview,” Lulu Garcia-Navarro talks with radio host Charlamagne Tha God about his frustrations with how Americans talk about politics.

If me as a Black man, if I criticize Democrats, then I’m supporting MAGA. But if I criticize, you know, Donald Trump and Republicans, then I’m a Democratic shill. Why can’t I just be a person who deals in nuance?

Today’s episode was produced by Olivia Natt and Michael Simon Johnson. It was edited by Lexie Diao, with help from Paige Cowett, contains original music by Will Reid and Marion Lozano, and was engineered by Alyssa Moxley. Our theme music is by Jim Brunberg and Ben Landsverk of Wonderly.

That’s it for “The Daily.” I’m Michael Barbaro. See you on Monday.

• May 13, 2024   •   27:46 How Biden Adopted Trump’s Trade War With China
• May 10, 2024   •   27:42 Stormy Daniels Takes the Stand
• May 9, 2024   •   34:42 One Strongman, One Billion Voters, and the Future of India
• May 8, 2024   •   28:28 A Plan to Remake the Middle East
• May 7, 2024   •   27:43 How Changing Ocean Temperatures Could Upend Life on Earth
• May 6, 2024   •   29:23 R.F.K. Jr.’s Battle to Get on the Ballot
• May 3, 2024   •   25:33 The Protesters and the President
• May 2, 2024   •   29:13 Biden Loosens Up on Weed
• May 1, 2024   •   35:16 The New Abortion Fight Before the Supreme Court
• April 30, 2024   •   27:40 The Secret Push That Could Ban TikTok
• April 29, 2024   •   47:53 Trump 2.0: What a Second Trump Presidency Would Bring
• April 26, 2024   •   21:50 Harvey Weinstein Conviction Thrown Out

Hosted by Michael Barbaro

Featuring Jonah E. Bromwich

Produced by Olivia Natt and Michael Simon Johnson

Edited by Lexie Diao

With Paige Cowett

Original music by Will Reid and Marion Lozano

Engineered by Alyssa Moxley

Listen and follow The Daily Apple Podcasts | Spotify | Amazon Music | YouTube

This episode contains descriptions of an alleged sexual liaison.

What happened when Stormy Daniels took the stand for eight hours in the first criminal trial of former President Donald J. Trump?

Jonah Bromwich, one of the lead reporters covering the trial for The Times, was in the room.

On today’s episode

Jonah E. Bromwich , who covers criminal justice in New York for The New York Times.

Background reading

In a second day of cross-examination, Stormy Daniels resisted the implication she had tried to shake down Donald J. Trump by selling her story of a sexual liaison.

Here are six takeaways from Ms. Daniels’s earlier testimony.

There are a lot of ways to listen to The Daily. Here’s how.

We aim to make transcripts available the next workday after an episode’s publication. You can find them at the top of the page.

The Daily is made by Rachel Quester, Lynsea Garrison, Clare Toeniskoetter, Paige Cowett, Michael Simon Johnson, Brad Fisher, Chris Wood, Jessica Cheung, Stella Tan, Alexandra Leigh Young, Lisa Chow, Eric Krupke, Marc Georges, Luke Vander Ploeg, M.J. Davis Lin, Dan Powell, Sydney Harper, Mike Benoist, Liz O. Baylen, Asthaa Chaturvedi, Rachelle Bonja, Diana Nguyen, Marion Lozano, Corey Schreppel, Rob Szypko, Elisheba Ittoop, Mooj Zadie, Patricia Willens, Rowan Niemisto, Jody Becker, Rikki Novetsky, John Ketchum, Nina Feldman, Will Reid, Carlos Prieto, Ben Calhoun, Susan Lee, Lexie Diao, Mary Wilson, Alex Stern, Dan Farrell, Sophia Lanman, Shannon Lin, Diane Wong, Devon Taylor, Alyssa Moxley, Summer Thomad, Olivia Natt, Daniel Ramirez and Brendan Klinkenberg.

Our theme music is by Jim Brunberg and Ben Landsverk of Wonderly. Special thanks to Sam Dolnick, Paula Szuchman, Lisa Tobin, Larissa Anderson, Julia Simon, Sofia Milan, Mahima Chablani, Elizabeth Davis-Moorer, Jeffrey Miranda, Renan Borelli, Maddy Masiello, Isabella Anderson and Nina Lassam.

Jonah E. Bromwich covers criminal justice in New York, with a focus on the Manhattan district attorney’s office and state criminal courts in Manhattan. More about Jonah E. Bromwich

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1.12: Percents Part 2 and Error Analysis

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You may use a calculator throughout this module.

Recall: The amount is the answer we get after finding the percent of the original number. The base is the original number, the number we find the percent of. We can call the percent the rate.

When we looked at percents in a previous module, we focused on finding the amount. In this module, we will learn how to find the percentage rate and the base.

\(\text{Amount}=\text{Rate}\cdot\text{Base}\)

\(A=R\cdot{B}\)

We can translate from words into algebra.

• “is” means equals
• “of” means multiply
• “what” means a variable

Solving Percent Problems: Finding the Rate

Suppose you earned \(56\) points on a \(60\)-point quiz. To figure out your grade as a percent, you need to answer the question “\(56\) is what percent of \(60\)?” We can translate this sentence into the equation \(56=R\cdot60\).

Exercises \(\PageIndex{1}\)

1. \(56\) is what percent of \(60\)?

2. What percent of \(120\) is \(45\)?

1. \(93\%\) or \(93.3\%\)

2. \(37.5\%\)

Be aware that this method gives us the answer in decimal form and we must move the decimal point to convert the answer to a percent.

Also, if the instructions don’t explicitly tell you how to round your answer, use your best judgment: to the nearest whole percent or nearest tenth of a percent, to two or three significant figures, etc.

Solving Percent Problems: Finding the Base

Suppose you earn \(2\%\) cash rewards for the amount you charge on your credit card. If you want to earn $ \(50\) in cash rewards, how much do you need to charge on your card? To figure this out, you need to answer the question “\(50\) is \(2\%\) of what number?” We can translate this into the equation \(50=0.02\cdot{B}\).

3. $ \(50\) is \(2\%\) of what number?

4. \(5\%\) of what number is \(36\)?

3. $ \(2,500\)

Solving Percent Problems: Using Proportions

Recall that a percent is a ratio, a fraction out of \(100\). Instead of translating word for word as we have just been doing, we can set up a proportion with the percentage rate over \(100\). Because the base is the original amount, it corresponds to \(100\%\).

Let’s try Exercises 1 through 4 again, using proportions.

5. \(56\) is what percent of \(60\)?

6. What percent of \(120\) is \(45\)?

7. $ \(50\) is \(2\%\) of what number?

8. \(5\%\) of what number is \(36\)?

5. \(93\%\) or \(93.3\%\)

6. \(37.5\%\)

7. $ \(2,500\)

Now that we have looked at both methods, you are free to use whichever method you prefer: percent equations or proportions.

9. An \(18\%\) tip will be added to a dinner that cost $ \(107.50\). What is the amount of the tip?

10. The University of Oregon women’s basketball team made \(13\) of the \(29\) three-points shots they attempted during a game against UNC. What percent of their three-point shots did the team make?

11. \(45\%\) of the people surveyed answered “yes” to a poll question. If \(180\) people answered “yes”, how many people were surveyed altogether?

9. $ \(19.35\)

10. \(44.8\%\) or \(45\%\)

11. \(400\) people were surveyed

Solving Percent Problems: Percent Increase

When a quantity changes, it is often useful to know by what percent it changed. If the price of a candy bar is increased by \(50\) cents, you might be annoyed because it’s it’s a relatively large percentage of the original price. If the price of a car is increased by \(50\) cents, though, you wouldn’t care because it’s such a small percentage of the original price.

To find the percent of increase:

• Subtract the two numbers to find the amount of increase.
• Using this result as the amount and the original number as the base, find the unknown percent.

Notice that we always use the original number for the base, the number that occurred earlier in time. In the case of a percent increase, this is the smaller of the two numbers.

12. The price of a candy bar increased from $ \(0.89\) to $ \(1.39\). By what percent did the price increase?

13. The population of Portland in 2010 was \(583,793\). The estimated population in 2019 was \(654,741\). Find the percent of increase in the population. [1]

12. \(56.2\%\) increase

13. \(12.2\%\) increase

Solving Percent Problems: Percent Decrease

Finding the percent decrease in a number is very similar.

To find the percent of decrease:

• Subtract the two numbers to find the amount of decrease.

Again, we always use the original number for the base, the number that occurred earlier in time. For a percent decrease, this is the larger of the two numbers.

14. During a sale, the price of a candy bar was reduced from $ \(1.39\) to $ \(0.89\). By what percent did the price decrease?

15. The number of students enrolled at Clackamas Community College decreased from \(7,439\) in Summer 2019 to \(4,781\) in Summer 2020. Find the percent of decrease in enrollment.

14. \(36.0\%\) decrease

15. \(35.7\%\) decrease

Relative Error

In an earlier module, we said that a measurement will always include some error, no matter how carefully we measure. It can be helpful to consider the size of the error relative to the size of what is being measured. As we saw in the examples above, a difference of \(50\) cents is important when we’re pricing candy bars but insignificant when we’re pricing cars. In the same way, an error of an eighth of an inch could be a deal-breaker when you’re trying to fit a screen into a window frame, but an eighth of an inch is insignificant when you’re measuring the length of your garage.

The expected outcome is what the number would be in a perfect world. If a window screen is supposed to be exactly \(25\) inches wide, we call this the expected outcome, and we treat it as though it has infinitely many significant digits. In theory, the expected outcome is \(25.000000...\)

To find the absolute error , we subtract the measurement and the expected outcome. Because we always treat the expected outcome as though it has unlimited significant figures, the absolute error should have the same precision (place value) as the measurement , not the expected outcome .

To find the relative error , we divide the absolute error by the expected outcome. We usually express the relative error as a percent. In fact, the procedure for finding the relative error is identical to the procedures for finding a percent increase or percent decrease!

To find the relative error:

• Subtract the two numbers to find the absolute error.
• Using the absolute error as the amount and the expected outcome as the base, find the unknown percent.

Exercisew \(\PageIndex{1}\)

16. A window screen is measured to be \(25\dfrac{3}{16}\) inches wide instead of the advertised \(25\) inches. Determine the relative error, rounded to the nearest tenth of a percent.

17. The contents of a box of cereal are supposed to weigh \(10.8\) ounces, but they are measured at \(10.67\) ounces. Determine the relative error, rounded to the nearest tenth of a percent.

16. \(0.1875\div25\approx0.8\%\)

17. \(0.13\div10.8\approx1.2\%\)

The tolerance is the maximum amount that a measurement is allowed to differ from the expected outcome. For example, the U.S. Mint needs its coins to have a consistent size and weight so that they will work in vending machines. A dime (10 cents) weighs \(2.268\) grams, with a tolerance of \(\pm0.091\) grams. [2] This tells us that the minimum acceptable weight is \(2.268-0.091=2.177\) grams, and the maximum acceptable weight is \(2.268+0.091=2.359\) grams. A dime with a weight outside of the range \(2.177\leq\text{weight}\leq2.359\) would be unacceptable.

A U.S. nickel (5 cents) weighs \(5.000\) grams with a tolerance of \(\pm0.194\) grams.

18. Determine the lowest acceptable weight and highest acceptable weight of a nickel.

19. Determine the relative error of a nickel that weighs \(5.21\) grams.

A U.S. quarter (25 cents) weighs \(5.670\) grams with a tolerance of \(\pm0.227\) grams.

20. Determine the lowest acceptable weight and highest acceptable weight of a quarter.

21. Determine the relative error of a quarter that weighs \(5.43\) grams.

18. \(4.806\) g; \(5.194\) g

19. \(0.21\div5.000=4.2\%\)

20. \(5.443\) g; \(5.897\) g

21. \(0.24\div5.670\approx4.2\%\)

• www.census.gov/quickfacts/fact/table/portlandcityoregon,OR,US/PST045219 ↵
• https://www.usmint.gov/learn/coin-and-medal-programs/coin-specifications and https://www.thesprucecrafts.com/how-much-do-coins-weigh-4171330 ↵