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5.2.1: Solving Percent Problems

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Learning Objectives

• 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.

Identify the percent, amount, and base in this problem.

30 is 20% of what number?

Percent: The percent is the number with the % symbol: 20%.

Base : The base is the whole amount, which in this case is unknown.

Amount: The amount based on the percent is 30.

Percent=20%

Base=unknown

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?

Identify the percent, base, and amount in this problem:

What percent of 30 is 3?

The percent is unknown, because the problem states " What percent?" The base is the whole in the situation, so the base is 30. The amount is the portion of the whole, which is 3 in this case.

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.

\[\ \text { Percent } {\color{red}\cdot}\text { Base }{\color{blue}=}\text { Amount } \nonumber \]

In the examples below, the unknown is represented by the letter \(\ n\). The unknown can be represented by any letter or a box \(\ \square\) or even a question mark.

Write an equation that represents the following problem.

\(\ 20 \% \cdot n=30\)

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 \(\ 20 \% \cdot n=30\), you can divide 30 by 20% to find the unknown: \(\ n=30 \div 20 \%\).

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

\(\ n=30 \div 20 \%=30 \div 0.20=150\)

What percent of 72 is 9?

\(\ 12.5 \% \text { of } 72 \text { is } 9\).

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.

\(\ 10 \% \text { of } 72=0.1 \cdot 72=7.2\)

\(\ 20 \% \text { of } 72=0.2 \cdot 72=14.4\)

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

What is 110% of 24?

\(\ 26.4 \text { is } 110 \% \text { of } 24\).

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.

18 is what percent of 48?

• \(\ 0.375 \%\)
• \(\ 8.64 \%\)
• \(\ 37.5 \%\)
• \(\ 864 \%\)

Incorrect. You may have calculated properly, but you forgot to move the decimal point when you rewrote your answer as a percent. The equation for this problem is \(\ n \cdot 48=18\). The corresponding division is \(\ 18 \div 48\), so \(\ n=0.375\). Rewriting this decimal as a percent gives the correct answer, \(\ 37.5 \%\).

Incorrect. You may have used \(\ 18\) or \(\ 48\) as the percent, rather than the amount or base. The equation for this problem is \(\ n \cdot 48=18\). The corresponding division is \(\ 18 \div 48\), so \(\ n=0.375\). Rewriting this decimal as a percent gives the correct answer, \(\ 37.5 \%\).

Correct. The equation for this problem is \(\ n \cdot 48=18\). The corresponding division is \(\ 18 \div 48\), so \(\ n=0.375\). Rewriting this decimal as a percent gives \(\ 37.5 \%\).

Incorrect. You probably used 18 or 48 as the percent, rather than the amount or base, and also forgot to rewrite the percent as a decimal before multiplying. The equation for this problem is \(\ n \cdot 48=18\). The corresponding division is \(\ 18 \div 48\), so \(\ n=0.375\). Rewriting this decimal as a percent gives the correct answer, \(\ 37.5 \%\).

Using Proportions to Solve Percent Problems

Percent problems can also be solved by writing a proportion. A proportion is an equation that sets two ratios or fractions equal to each other. With percent problems, one of the ratios is the percent, written as \(\ \frac{n}{100}\). The other ratio is the amount to the base.

\(\ \text { Percent }=\frac{\text { amount }}{\text { base }}\)

Write a proportion to find the answer to the following question.

30 is 20% of 150.

18 is 125% of what number?

• \(\ 0.144\)
• \(\ 694 \frac{4}{9}\) (or about \(\ 694.4\))

Incorrect. You probably didn’t write a proportion and just divided 18 by 125. Or, you incorrectly set up one fraction as \(\ \frac{18}{125}\) and set this equal to the base, \(\ n\). The percent in this case is 125%, so one fraction in the proportion should be \(\ \frac{125}{100}\). The base is unknown and the amount is 18, so the other fraction is \(\ \frac{18}{n}\). Solving the proportion \(\ \frac{125}{100}=\frac{18}{n}\) gives \(\ n=14.4\).

Correct. The percent in this case is 125%, so one fraction in the proportion should be \(\ \frac{125}{100}\). The base is unknown and the amount is 18, so the other fraction is \(\ \frac{18}{n}\). Solving the proportion \(\ \frac{125}{100}=\frac{18}{n}\) gives \(\ n=14.4\).

Incorrect. You probably put the amount (18) over 100 in the proportion, rather than the percent (125). Perhaps you thought 18 was the percent and 125 was the base. The correct percent fraction for the proportion is \(\ \frac{125}{100}\). The base is unknown and the amount is 18, so the other fraction is \(\ \frac{18}{n}\). Solving the proportion \(\ \frac{125}{100}=\frac{18}{n}\) gives \(\ n=14.4\).

Incorrect. You probably confused the amount (18) with the percent (125) when you set up the proportion. The correct percent fraction for the proportion is \(\ \frac{125}{100}\). The base is unknown and the amount is 18, so the other fraction is \(\ \frac{18}{n}\). Solving the proportion \(\ \frac{125}{100}=\frac{18}{n}\) gives \(\ n=14.4\).

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.

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 of the $220 original price .

The coupon will take $33 off the original price.

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

\(\ \begin{array}{l} 10 \% \text { of } 220=0.1 \cdot 220=22 \\ 20 \% \text { of } 220=0.2 \cdot 220=44 \end{array}\)

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.

Evelyn bought some books at the local bookstore. Her total bill was $31.50, which included 5% tax. How much did the books cost before tax?

The books cost $30 before tax.

Susana worked 20 hours at her job last week. This week, she worked 35 hours. In terms of a percent, how much more did she work this week than last week?

Since 35 is 175% of 20, Susana worked 75% more this week than she did last week. (You can think of this as, “Susana worked 100% of the hours she worked last week, as well as 75% more.”)

Percent problems have three parts: the percent, the base (or whole), and the amount. Any of those parts may be the unknown value to be found. To solve percent problems, you can use the equation, \(\ \text { Percent } \cdot \text { Base }=\text { Amount }\), and solve for the unknown numbers. Or, you can set up the proportion, \(\ \text { Percent }=\frac{\text { amount }}{\text { base }}\), where the percent is a ratio of a number to 100. You can then use cross multiplication to solve the proportion.

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How to Solve Percent Problems? (+FREE Worksheet!)

Learn how to calculate and solve percent problems using the percent formula.

Related Topics

• How to Find Percent of Increase and Decrease
• How to Find Discount, Tax, and Tip
• How to Do Percentage Calculations
• How to Solve Simple Interest Problems

Step by step guide to solve percent problems

• In each percent problem, we are looking for the base, or part or the percent.
• Use the following equations to find each missing section. Base \(= \color{black}{Part} \ ÷ \ \color{blue}{Percent}\) \(\color{ black }{Part} = \color{blue}{Percent} \ ×\) Base \(\color{blue}{Percent} = \color{ black }{Part} \ ÷\) Base

Percent Problems – Example 1:

\(2.5\) is what percent of \(20\)?

In this problem, we are looking for the percent. Use the following equation: \(\color{blue}{Percent} = \color{ black }{Part} \ ÷\) Base \(→\) Percent \(=2.5 \ ÷ \ 20=0.125=12.5\%\)

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Percent problems – example 2:.

\(40\) is \(10\%\) of what number?

Use the following formula: Base \(= \color{ black }{Part} \ ÷ \ \color{blue}{Percent}\) \(→\) Base \(=40 \ ÷ \ 0.10=400\) \(40\) is \(10\%\) of \(400\).

Percent Problems – Example 3:

\(1.2\) is what percent of \(24\)?

In this problem, we are looking for the percent. Use the following equation: \(\color{blue}{Percent} = \color{ black }{Part} \ ÷\) Base \(→\) Percent \(=1.2÷24=0.05=5\%\)

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Percent problems – example 4:.

\(20\) is \(5\%\) of what number?

Use the following formula: Base \(= \color{black}{Part} \ ÷ \ \color{blue}{Percent}\) \(→\) Base \(=20÷0.05=400\) \( 20\) is \(5\%\) of \(400\).

Exercises for Calculating Percent Problems

Solve each problem..

• \(51\) is \(340\%\) of what?
• \(93\%\) of what number is \(97\)?
• \(27\%\) of \(142\) is what number?
• What percent of \(125\) is \(29.3\)?
• \(60\) is what percent of \(126\)?
• \(67\) is \(67\%\) of what?

Download Percent Problems Worksheet

• \(\color{blue}{15}\)
• \(\color{blue}{104.3}\)
• \(\color{blue}{38.34}\)
• \(\color{blue}{23.44\%}\)
• \(\color{blue}{47.6\%}\)
• \(\color{blue}{100}\)

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by: Effortless Math Team about 4 years ago (category: Articles , Free Math Worksheets )

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Calcworkshop

How to Solve Percent Problems? 13 Amazing Examples!

// Last Updated: January 20, 2020 - Watch Video //

Ratios and proportions help us in solving problems with percents .

Jenn, Founder Calcworkshop ® , 15+ Years Experience (Licensed & Certified Teacher)

Because a percent is the ratio (fraction) of one number to another.

In other words, its a number divided by 100.

What is most important to note is that our overall goal is to translate each problem into an algebraic expression by looking for keywords such as “of” and “is,” and then solve for the unknown variable.

First, we will look at how a percent is created by understanding it is simply a proportion:

Percents as a Proportion

And we will quickly see that most problems will follow a very simple Percents Formula, as Khan Academy so accurately states.

Together will look at how to change percents to fractions, percents to decimals, and decimals to percents for various problems.

Additionally, we will look at real-world examples, such as finding the percent of commission, the price of an item with a percent discount, change in population, as well as mixture problems.

Solving Percents (How-To) – Video

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Enter the value(s) for the required question and click the adjacent Go button.

PERCENTAGES

This section will explain how to apply algebra to percentage problems.

In algebra problems, percentages are usually written as decimals.

Example 1. Ethan got 80% of the questions correct on a test, and there were 55 questions. How many did he get right?

The number of questions correct is indicated by:

Ethan got 44 questions correct.

Explanation: % means "per one hundred". So 80% means 80/100 = 0.80.

Example 2. A math teacher, Dr. Pi, computes a student’s grade for the course as follows:

a. Compute Darrel's grade for the course if he has a 91 on the homework, 84 for his test average, and a 98 on the final exam.

Darrel’s grade for the course is an 89.6, or a B+.

b. Suppose Selena has an 89 homework average and a 97 test average. What does Selena have to get on the final exam to get a 90 for the course?

The difference between Part a and Part b is that in Part b we don’t know Selena’s grade on the final exam.

So instead of multiplying 30% times a number, multiply 30% times E. E is the variable that represents what Selena has to get on the final exam to get a 90 for the course.

Because Selena studied all semester, she only has to get a 79 on the final to get a 90 for the course.

Example 3. Sink Hardware store is having a 15% off sale. The sale price of a toilet is $97; find the retail price of the toilet.

a. Complete the table to find an equation relating the sale price to the retail price (the price before the sale).

Vocabulary: Retail price is the original price to the consumer or the price before the sale. Discount is how much the consumer saves, usually a percentage of the retail price. Sale Price is the retail price minus the discount.

b. Simplify the equation.

Explanation: The coefficient of R is one, so the arithmetic for combining like terms is 1 - 0.15 = .85. In other words, the sale price is 85% of the retail price.

c. Solve the equation when the sale price is $97.

The retail price for the toilet was $114.12. (Note: the answer was rounded to the nearest cent.)

The following diagram is meant as a visualization of problem 3.

The large rectangle represents the retail price. The retail price has two components, the sale price and the discount. So Retail Price = Sale Price + Discount If Discount is subtracted from both sides of the equation, a formula for Sale Price is found. Sale Price = Retail Price - Discount

Percentages play an integral role in our everyday lives, including computing discounts, calculating mortgages, savings, investments, and estimating final grades. When working with percentages, remember to write them as decimals, to create tables to derive equations, and to follow the proper procedures to solve equations.

Study Tip: Remember to use descriptive letters to describe the variables.

CHAPTER 1 REVIEW

This unit introduces algebra by examining similar models. You should be able to read a problem and create a table to find an equation that relates two variables. If you are given information about one of the variables, you should be able to use algebra to find the other variable.

Signed Numbers:

Informal Rules:

Adding or subtracting like signs: Add the two numbers and use the common sign.

Adding or subtracting unlike signs: Subtract the two numbers and use the sign of the larger, (more precisely, the sign of the number whose absolute value is largest.)

Multiplying or dividing like signs: The product or quotient of two numbers with like signs is always positive.

Multiplying or dividing unlike signs: The product or quotient of two numbers with unlike signs is always negative.

Order of operations: P lease E xcuse M y D ear A unt S ally 1. Inside P arentheses, (). 2. E xponents. 3. M ultiplication and D ivision (left to right) 4. A ddition and S ubtraction (left to right)

Study Tip: All of these informal rules should be written on note cards.

Introduction to Variables:

Generate a table to find an equation that relates two variables.

Example 6. A car company charges $14.95 plus 35 cents per mile.

Simplifying Algebraic Equations:

Combine like terms:

Solving Equations:

1. Simplify both sides of the equation. 2. Write the equation as a variable term equal to a constant. 3. Divide both sides by the coefficient or multiply by the reciprocal. 4. Three possible outcomes to solving an equation. a. One solution ( a conditional equation ) b. No solution ( a contradiction ) c. Every number is a solution (an identity )

Applications of Linear Equations:

This section summarizes the major skills taught in this chapter.

Example 9. A cell phone company charges $12.50 plus 15 cents per minute after the first six minutes.

a. Create a table to find the equation that relates cost and minutes.

c. If the call costs $23.50, how long were you on the phone?

If the call costs $23.50, then you were on the phone for approximately 79 minutes.

Literal Equations:

A literal equation involves solving an equation for one of two variables.

Percentages:

Write percentages as decimals.

Example 11. An English teacher computes his grades as follows:

Sue has an 87 on the short essays and a 72 on the research paper. If she wants an 80 for the course, what grade does Sue have to get on the final?

Sue has to get a 78.36 in the final exam to get an 80 for the course.

Study Tips:

1. Make sure you have done all of the homework exercises. 2. Practice the review test on the following pages by placing yourself under realistic exam conditions. 3. Find a quiet place and use a timer to simulate the test period. 4. Write your answers in your homework notebook. Make copies of the exam so you may then re-take it for extra practice. 5. Check your answers. 6. There is an additional exam available on the Beginning Algebra web page. 7. DO NOT wait until the night before the exam to study.

Math Topics

More solvers.

• Add Fractions
• Simplify Fractions

Percents (%)

When we say "percent" we are really saying "per 100".

One percent ( 1% ) means 1 per 100.

Try it Yourself:

Using Percent

Use the slider and try some different numbers (What is 40% of 80? What is 10% of 200? What is 90% of 10?)

Because "Percent" means "per 100" think:

"this should be divided by 100"

So 75% really means 75 100

And 100% is 100 100 , or exactly 1 (100% of any number is just the number, unchanged)

And 200% is 200 100 , or exactly 2 (200% of any number is twice the number)

A Percent can also be expressed as a Decimal or a Fraction

Read more about this at Decimals, Fractions and Percentages .

Some Worked Examples

Example: calculate 25% of 80.

25% = 25 100

And   25 100 × 80 = 20

So 25% of 80 is 20

Example: 15% of 200 apples are bad. How many apples are bad?

15% = 15 100

30 apples are bad

Example: if only 10 of the 200 apples are bad, what percent is that?

As a fraction, 10 200 = 0.05

As a percentage it is: 10 200 x 100 = 5%

5% of those apples are bad

Example: A Skateboard is reduced 25% in price. The old price was $120. Find the new price.

First, find 25% of $120:

And   25 100  × $120 = $30

25% of $120 is $30

So the reduction is $30

Take the reduction from the original price

$120 − $30 = $90

The Price of the Skateboard in the sale is $90

Calculation Trick

This little rule can make some calculations easier:

x% of y = y% of x

Example: 8% of 50

8% of 50 is the same as 50% of 8

And 50% of 8 is 4

So 8% of 50 is also 4

Percent vs Percentage

My Dictionary says "Percentage" is the "result obtained by multiplying a quantity by a percent". So 10 percent of 50 apples is 5 apples: the 5 apples is the percentage .

But in practice people use both words the same way.

• Math Test Prep & More

Solving Percentage Problems

There are two easy ways of solving Percentage problems. We will look at translating the word problem into algebraic expressions in this video. This method requires you to just write the word problem mathematically and solve the resulting equation

We will learn how to find the percent of a number. We will also learn what percentage a number is of another number, and finally we will use percentage to find a number

Let’s look at these individually and we will use the idea of proportion to make these even easier.

Finding the percentage of a number

Find 60% of 200.

Here we know 60% is 60/100 .

“of” in mathematics means multiplication

So writing this algebraically we will have

  60/100  x 200

From here we just simplify to get 120

so 60% of 200 is 120. Watch video for details

Find 50% of 24.

This will be 50/100 times 24.

Again we know that "of" in mathematics means multiplication

We can solve this to get 12

Therefore 50% of 24 is 12. Watch video for details

What percentage of a number is another number?

Let's look at this question. What percent of 10 is 2?

The statement “What percent” is what we are looking for, so let’s represent it by P%.

We know that P% is P/100.

Also we know “of” means multiplication.

“Is” is the same as “is equal to”.

So we have equal to 2.

We will have P/100 x 10 = 2

From here we can just solve for P.

So our answer is 20%. Watch video for details

Please do not get confused when things are twisted. It is the same procedure.

8 is what percentage of 24?

So we know we can write it as

8 = P% x 24.

Again we change the P% to P/100.

"of" means multiplication

"is" means "equal to"

So now we solve for P.

After solving we will have 33.33%

Use percent to find a number

5% of what number is 20.

Here again we can just follow the mathematical meaning.

First we know 5% means 5/100.

“of” means multiplication.

“What number” is what we are looking for, so we will represent it by N.

We know “is” means equal to.

Then we have our 20.

So we have 5/100 x N = 20

We can now simply solve for N.

 N = 400. Watch video for details

This brings us to the end of this Lesson. Have a Great Day

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Solved Examples on Percentage

The solved examples on percentage will help us to understand how to solve step-by-step different types of percentage problems. Now we will apply the concept of percentage to solve various real-life examples on percentage.

Solved examples on percentage:

1.  In an election, candidate A got 75% of the total valid votes. If 15% of the total votes were declared invalid and the total numbers of votes is 560000, find the number of valid vote polled in favour of candidate.

Total number of invalid votes = 15 % of 560000

                                       = 15/100 × 560000

                                       = 8400000/100

                                       = 84000

Total number of valid votes 560000 – 84000 = 476000

Percentage of votes polled in favour of candidate A = 75 %

Therefore, the number of valid votes polled in favour of candidate A = 75 % of 476000

= 75/100 × 476000

= 35700000/100

2. A shopkeeper bought 600 oranges and 400 bananas. He found 15% of oranges and 8% of bananas were rotten. Find the percentage of fruits in good condition.

Total number of fruits shopkeeper bought = 600 + 400 = 1000

Number of rotten oranges = 15% of 600

                                    = 15/100 × 600

                                    = 9000/100

                                    = 90

Number of rotten bananas = 8% of 400

                                   = 8/100 × 400

                                   = 3200/100

                                   = 32

Therefore, total number of rotten fruits = 90 + 32 = 122

Therefore Number of fruits in good condition = 1000 - 122 = 878

Therefore Percentage of fruits in good condition = (878/1000 × 100)%

                                                                 = (87800/1000)%

                                                                 = 87.8%

3. Aaron had $ 2100 left after spending 30 % of the money he took for shopping. How much money did he take along with him?

Solution:            

Let the money he took for shopping be m.

Money he spent = 30 % of m

                      = 30/100 × m

                      = 3/10 m

Money left with him = m – 3/10 m = (10m – 3m)/10 = 7m/10

But money left with him = $ 2100

Therefore 7m/10 = $ 2100          

m = $ 2100× 10/7

m = $ 21000/7

Therefore, the money he took for shopping is $ 3000.

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Digital SAT Math

Course: digital sat math   >   unit 3, percentages | lesson.

• Percents — Basic example
• Percents — Harder example
• Percentages: foundations

What are percentages?

• Calculate percentages using part and whole values
• Switch between equivalent forms of percentages
• Calculate percent change

How do we calculate percentages?

Finding a percentage, calculating a percent value.

• The part is 21 ‍   .
• The whole is 24 ‍   .
• The whole is 8 ‍   .
• The percentage is 150 ‍   .
• The part is what we need to find.

Finding complementary percentages

• 56 % ‍   of the marbles are blue.
• So, 100 % − 56 % = 44 % ‍   the marbles are red.
• 45 ‍   is the part/whole/percentage .
• 117 ‍   is the part/whole/percentage .
• We need to solve for the part/whole/percentage .
• Your answer should be
• an integer, like 6 ‍  
• a simplified proper fraction, like 3 / 5 ‍  
• a simplified improper fraction, like 7 / 4 ‍  
• a mixed number, like 1   3 / 4 ‍  
• an exact decimal, like 0.75 ‍  
• a multiple of pi, like 12   pi ‍   or 2 / 3   pi ‍  

What forms can percentages take?

Converting percentages to decimals and fractions, switching between forms of percentages.

• The ratio 50 : 100 ‍   , which reduces to 1 : 2 ‍   .
• The fraction 50 100 ‍   , which reduces to 1 2 ‍   .
• The decimal value 0.5 ‍   .

Translating percentage word problems

• "what" means x ‍  
• "is" means = ‍  
• "of" means multiplied by
• "percent" means divided by 100 ‍  

In what form should I enter my answer?

• a proper fraction, like 1 / 2 ‍   or 6 / 10 ‍  
• an improper fraction, like 10 / 7 ‍   or 14 / 8 ‍  

How do we calculate percent changes?

Percentage word problems, calculating percent change.

• Find the difference between the initial and final values.
• Divide the difference by the initial value.
• Convert the decimal to a percentage by multiplying the quotient by 100 ‍   .
• $ 40 ‍   is the final value .
• 20 ‍   is the % ‍   change.
• We're solving for the initial value ( x ‍   ).
• 63.89 ‍   is the initial/final/percent change .
• 68.86 ‍   is the initial/final/percent change .
• We need to solve for the initial/final/percent change .
• (Choice A)   362 ‍   A 362 ‍  
• (Choice B)   414 ‍   B 414 ‍  
• (Choice C)   468 ‍   C 468 ‍  
• (Choice D)   512 ‍   D 512 ‍  
• (Choice A)   3 ‍   A 3 ‍  
• (Choice B)   5 ‍   B 5 ‍  
• (Choice C)   8 ‍   C 8 ‍  
• (Choice D)   15 ‍   D 15 ‍  
• (Choice A)   3 % ‍   A 3 % ‍  
• (Choice B)   30 % ‍   B 30 % ‍  
• (Choice C)   43 % ‍   C 43 % ‍  
• (Choice D)   70 % ‍   D 70 % ‍  

Things to remember

• Convert the resulting decimal to a percentage.

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A Plan to Remake the Middle East

While talks for a cease-fire between israel and hamas continue, another set of negotiations is happening behind the scenes..

This transcript was created using speech recognition software. While it has been reviewed by human transcribers, it may contain errors. Please review the episode audio before quoting from this transcript and email [email protected] with any questions.

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

[MUSIC CONTINUES]

Today, if and when Israel and Hamas reach a deal for a ceasefire fire, the United States will immediately turn to a different set of negotiations over a grand diplomatic bargain that it believes could rebuild Gaza and remake the Middle East. My colleague Michael Crowley has been reporting on that plan and explains why those involved in it believe they have so little time left to get it done.

It’s Wednesday, May 8.

Michael, I want to start with what feels like a pretty dizzying set of developments in this conflict over the past few days. Just walk us through them?

Well, over the weekend, there was an intense round of negotiations in an effort, backed by the United States, to reach a ceasefire in the Gaza war.

The latest ceasefire proposal would reportedly see as many as 33 Israeli hostages released in exchange for potentially hundreds of Palestinian prisoners.

US officials were very eager to get this deal.

Pressure for a ceasefire has been building ahead of a threatened Israeli assault on Rafah.

Because Israel has been threatening a military offensive in the Southern Palestinian city of Rafah, where a huge number of people are crowded.

Fleeing the violence to the North. And now they’re packed into Rafah. Exposed and vulnerable, they need to be protected.

And the US says it would be a humanitarian catastrophe on top of the emergency that’s already underway.

Breaking news this hour — very important breaking news. An official Hamas source has told The BBC that it does accept a proposal for a ceasefire deal in Gaza.

And for a few hours on Monday, it looked like there might have been a major breakthrough when Hamas put out a statement saying that it had accepted a negotiating proposal.

Israeli Prime Minister Benjamin Netanyahu says the ceasefire proposal does not meet his country’s requirements. But Netanyahu says he will send a delegation of mediators to continue those talks. Now, the terms —

But those hopes were dashed pretty quickly when the Israelis took a look at what Hamas was saying and said that it was not a proposal that they had agreed to. It had been modified.

And overnight —

Israeli troops stormed into Rafah. Video showing tanks crashing over a sign at the entrance of the city.

— the Israelis launched a partial invasion of Rafah.

It says Hamas used the area to launch a deadly attack on Israeli troops over the weekend.

And they have now secured a border crossing at the Southern end of Gaza and are conducting targeted strikes. This is not yet the full scale invasion that President Biden has adamantly warned Israel against undertaking, but it is an escalation by Israel.

So while all that drama might suggest that these talks are in big trouble, these talks are very much still alive and ongoing and there is still a possibility of a ceasefire deal.

And the reason that’s so important is not just to stop the fighting in Gaza and relieve the suffering there, but a ceasefire also opens the door to a grand diplomatic bargain, one that involves Israel and its Arab neighbors and the Palestinians, and would have very far-reaching implications.

And what is that grand bargain. Describe what you’re talking about?

Well, it’s incredibly ambitious. It would reshape Israel’s relationship with its Arab neighbors, principally Saudi Arabia. But it’s important to understand that this is a vision that has actually been around since well before October 7. This was a diplomatic project that President Biden had been investing in and negotiating actually in a very real and tangible way long before the Hamas attacks and the Gaza war.

And President Biden was looking to build on something that President Trump had done, which was a series of agreements that the Trump administration struck in which Israel and some of its Arab neighbors agreed to have normal diplomatic relations for the first time.

Right, they’re called the Abraham Accords.

That’s right. And, you know, Biden doesn’t like a lot of things, most things that Trump did. But he actually likes this, because the idea is that they contribute to stability and economic integration in the Middle East, the US likes Israel having friends and likes having a tight-knit alliance against Iran.

President Biden agrees with the Saudis and with the Israelis, that Iran is really the top threat to everybody here. So, how can you build on this? How can you expand it? Well, the next and biggest step would be normalizing relations between Israel and Saudi Arabia.

And the Saudis have made clear that they want to do this and that they’re ready to do this. They weren’t ready to do it in the Trump years. But Mohammed bin Salman, the Crown Prince of Saudi Arabia, has made clear he wants to do it now.

So this kind of triangular deal began to take shape before October 7, in which the US, Israel, and Saudi Arabia would enter this three way agreement in which everyone would get something that they wanted.

And just walk through what each side gets in this pre-October 7th version of these negotiations?

So for Israel, you get normalized ties with its most important Arab neighbor and really the country that sets the tone for the whole Muslim world, which is Saudi Arabia of course. It makes Israel feel safer and more secure. Again, it helps to build this alliance against Iran, which Israel considers its greatest threat, and it comes with benefits like economic ties and travel and tourism. And Prime Minister Benjamin Netanyahu has been very open, at least before October 7th, that this was his highest diplomatic and foreign policy priority.

For the Saudis, the rationale is similar when it comes to Israel. They think that it will bring stability. They like having a more explicitly close ally against Iran. There are economic and cultural benefits. Saudi Arabia is opening itself up in general, encouraging more tourism.

But I think that what’s most important to the Crown Prince, Mohammed bin Salman, is what he can get from the United States. And what he has been asking for are a couple of essential things. One is a security agreement whose details have always been a little bit vague, but I think essentially come down to reliable arms supplies from the United States that are not going to be cut off or paused on a whim, as he felt happened when President Biden stopped arms deliveries in 2021 because of how Saudi was conducting its war in Yemen. The Saudis were furious about that.

Saudi Arabia also wants to start a domestic nuclear power program. They are planning for a very long-term future, possibly a post-oil future. And they need help getting a nuclear program off the ground.

And they want that from the US?

And they want that from the US.

Now, those are big asks from the us. But from the perspective of President Biden, there are some really enticing things about this possible agreement. One is that it will hopefully produce more stability in the region. Again, the US likes having a tight-knit alliance against Iran.

The US also wants to have a strong relationship with Saudi Arabia. You know, despite the anger at Mohammed bin Salman over the murder of the Saudi dissident Jamal Khashoggi, the Biden administration recognizes that given the Saudis control over global oil production and their strategic importance in the Middle East, they need to have a good relationship with them. And the administration has been worried about the influence of China in the region and with the Saudis in particular.

So this is an opportunity for the US to draw the Saudis closer. Whatever our moral qualms might be about bin Salman and the Saudi government, this is an opportunity to bring the Saudis closer, which is something the Biden administration sees as a strategic benefit.

All three of these countries — big, disparate countries that normally don’t see eye-to-eye, this was a win-win-win on a military, economic, and strategic front.

That’s right. But there was one important actor in the region that did not see itself as winning, and that was the Palestinians.

[MUSIC PLAYING]

First, it’s important to understand that the Palestinians have always expected that the Arab countries in the Middle East would insist that Israel recognize a Palestinian state before those countries were willing to essentially make total peace and have normal relations with Israel.

So when the Abraham Accords happened in the Trump administration, the Palestinians felt like they’d been thrown under the bus because the Abraham Accords gave them virtually nothing. But the Palestinians did still hold out hope that Saudi Arabia would be their savior. And for years, Saudi Arabia has said that Israel must give the Palestinians a state if there’s going to be a normal relationship between Israel and Saudi Arabia.

Now the Palestinians see the Saudis in discussions with the US and Israel about a normalization agreement, and there appears to be very little on offer for the Palestinians. And they are feeling like they’re going to be left out in the cold here.

Right. And in the minds of the Palestinians, having already been essentially sold out by all their other Arab neighbors, the prospect that Saudi Arabia, of all countries, the most important Muslim Arab country in the region, would sell them out, had to be extremely painful.

It was a nightmare scenario for them. And in the minds of many analysts and US officials, this was a factor, one of many, in Hamas’s decision to stage the October 7th attacks.

Hamas, like other Palestinian leaders, was seeing the prospect that the Middle East was moving on and essentially, in their view, giving up on the Palestinian cause, and that Israel would be able to have friendly, normal relations with Arab countries around the region, and that it could continue with hardline policies toward the Palestinians and a refusal, as Prime Minister Benjamin Netanyahu has said publicly, to accept a Palestinian state.

Right. So Michael, once Hamas carries out the October 7th attacks in an effort to destroy a status quo that it thinks is leaving them less and less relevant, more and more hopeless, including potentially this prospect that Saudi Arabia is going to normalize relations with Israel, what happens to these pre-October 7th negotiations between the US, Saudi Arabia, and Israel?

Well, I think there was a snap assumption that these talks were dead and buried. That they couldn’t possibly survive a cataclysm like this.

But then something surprising happened. It became clear that all the parties were still determined to pull-off the normalization.

And most surprisingly of all, perhaps, was the continued eagerness of Saudi Arabia, which publicly was professing outrage over the Israeli response to the Hamas attacks, but privately was still very much engaged in these conversations and trying to move them forward.

And in fact, what has happened is that the scope of this effort has grown substantially. October 7th didn’t kill these talks. It actually made them bigger, more complicated, and some people would argue, more important than ever.

We’ll be right back.

Michael, walk us through what exactly happens to these three-way negotiations after October 7th that ends up making them, as you just said, more complicated and more important than ever?

Well, it’s more important than ever because of the incredible need in Gaza. And it’s going to take a deal like this and the approval of Saudi Arabia to unlock the kind of massive reconstruction project required to essentially rebuild Gaza from the rubble. Saudi Arabia and its Arab friends are also going to be instrumental in figuring out how Gaza is governed, and they might even provide troops to help secure it. None of those things are going to happen without a deal like this.

Fascinating.

But this is all much more complicated now because the price for a deal like this has gone up.

And by price, you mean?

What Israel would have to give up. [MUSIC PLAYING]

From Saudi Arabia’s perspective, you have an Arab population that is furious at Israel. It now feels like a really hard time to do a normalization deal with the Israelis. It was never going to be easy, but this is about as bad a time to do it as there has been in a generation at least. And I think that President Biden and the people around him understand that the status quo between Israel and the Palestinians is intolerable and it is going to lead to chaos and violence indefinitely.

So now you have two of the three parties to this agreement, the Saudis and the Americans, basically asking a new price after October 7th, and saying to the Israelis, if we’re going to do this deal, it has to not only do something for the Palestinians, it has to do something really big. You have to commit to the creation of a Palestinian state. Now, I’ll be specific and say that what you hear the Secretary of State, Antony Blinken, say is that the agreement has to include an irreversible time-bound path to a Palestinian state.

We don’t know exactly what that looks like, but it’s some kind of a firm commitment, the likes of which the world and certainly the Israelis have not made before.

Something that was very much not present in the pre-October 7th vision of this negotiation. So much so that, as we just talked about, the Palestinians were left feeling completely out in the cold and furious at it.

That’s right. There was no sign that people were thinking that ambitiously about the Palestinians in this deal before October 7th. And the Palestinians certainly felt like they weren’t going to get much out of it. And that has completely changed now.

So, Michael, once this big new dimension after October 7th, which is the insistence by Saudi Arabia and the US that there be a Palestinian state or a path to a Palestinian state, what is the reaction specifically from Israel, which is, of course, the third major party to this entire conversation?

Well, Israel, or at least its political leadership, hates it. You know, this is just an extremely tough sell in Israel. It would have been a tough sell before October 7th. It’s even harder now.

Prime Minister Benjamin Netanyahu is completely unrepentantly open in saying that there’s not going to be a Palestinian state on his watch. He won’t accept it. He says that it’s a strategic risk to his country. He says that it would, in effect, reward Hamas.

His argument is that terrorism has forced a conversation about statehood onto the table that wasn’t there before October 7th. Sure, it’s always in the background. It’s a perennial issue in global affairs, but it was not something certainly that the US and Israel’s Arab neighbors were actively pushing. Netanyahu also has — you know, he governs with the support of very right-wing members of a political coalition that he has cobbled together. And that coalition is quite likely to fall apart if he does embrace a Palestinian state or a path to a Palestinian state.

Now, he might be able to cobble together some sort of alternative, but it creates a political crisis for him.

And finally, you know, I think in any conversation about Israel, it’s worth bearing in mind something you hear from senior US officials these days, which is that although there is often finger pointing at Netanyahu and a desire to blame Netanyahu as this obstructionist who won’t agree to deals, what they say is Netanyahu is largely reflecting his population and the political establishment of his country, not just the right-wingers in his coalition who are clearly extremist.

But actually the prevailing views of the Israeli public. And the Israeli public and their political leaders across the spectrum right now with few exceptions, are not interested in talking about a Palestinian state when there are still dozens and dozens of Israeli hostages in tunnels beneath Gaza.

So it very much looks like this giant agreement that once seemed doable before October 7th might be more important to everyone involved than ever, given that it’s a plan for rebuilding Gaza and potentially preventing future October 7th’s from happening, but because of this higher price that Israel would have to pay, which is the acceptance of a Palestinian state, it seems from everything you’re saying, that this is more and more out of reach than ever before and hard to imagine happening in the immediate future. So if the people negotiating it are being honest, Michael, are they ready to acknowledge that it doesn’t look like this is going to happen?

Well, not quite yet. As time goes by, they certainly say it’s getting harder and harder, but they’re still trying, and they still think there’s a chance. But both the Saudis and the Biden administration understand that there’s very little time left to do this.

Well, what do you mean there’s very little time left? It would seem like time might benefit this negotiation in that it might give Israel distance from October 7th to think potentially differently about a Palestinian state?

Potentially. But Saudi Arabia wants to get this deal done in the Biden administration because Mohammed bin Salman has concluded this has to be done under a Democratic president.

Because Democrats in Congress are going to be very reluctant to approve a security agreement between the United States and Saudi Arabia.

It’s important to understand that if there is a security agreement, that’s something Congress is going to have to approve. And you’re just not going to get enough Democrats in Congress to support a deal with Saudi Arabia, who a lot of Democrats don’t like to begin with, because they see them as human rights abusers.

But if a Democratic president is asking them to do it, they’re much more likely to go along.

Right. So Saudi Arabia fears that if Biden loses and Trump is president, that those same Democrats would balk at this deal in a way that they wouldn’t if it were being negotiated under President Biden?

Exactly. Now, from President Biden’s perspective, politically, think about a president who’s running for re-election, who is presiding right now over chaos in the Middle East, who doesn’t seem to have good answers for the Israeli-Palestinian question, this is an opportunity for President Biden to deliver what could be at least what he would present as a diplomatic masterstroke that does multiple things at once, including creating a new pathway for Israel and the Palestinians to coexist, to break through the logjam, even as he is also improving Israel’s relations with Saudi Arabia.

So Biden and the Crown Prince hope that they can somehow persuade Bibi Netanyahu that in spite of all the reasons that he thinks this is a terrible idea, that this is a bet worth taking on Israel’s and the region’s long-term security and future?

That’s right. Now, no one has explained very clearly exactly how this is going to work, and it’s probably going to require artful diplomacy, possibly even a scenario where the Israelis would agree to something that maybe means one thing to them and means something else to other people. But Biden officials refuse to say that it’s hopeless and they refuse to essentially take Netanyahu’s preliminary no’s for an answer. And they still see some way that they can thread this incredibly narrow needle.

Michael, I’m curious about a constituency that we haven’t been talking about because they’re not at the table in these discussions that we are talking about here. And that would be Hamas. How does Hamas feel about the prospect of such a deal like this ever taking shape. Do they see it as any kind of a victory and vindication for what they did on October 7th?

So it’s hard to know exactly what Hamas’s leadership is thinking. I think they can feel two things. I think they can feel on the one hand, that they have established themselves as the champions of the Palestinian people who struck a blow against Israel and against a diplomatic process that was potentially going to leave the Palestinians out in the cold.

At the same time, Hamas has no interest in the kind of two-state solution that the US is trying to promote. They think Israel should be destroyed. They think the Palestinian state should cover the entire geography of what is now Israel, and they want to lead a state like that. And that’s not something that the US, Saudi Arabia, or anyone else is going to tolerate.

So what Hamas wants is to fight, to be the leader of the Palestinian people, and to destroy Israel. And they’re not interested in any sort of a peace process or statehood process.

It seems very clear from everything you’ve said here that neither Israel nor Hamas is ready to have the conversation about a grand bargain diplomatic program. And I wonder if that inevitably has any bearing on the ceasefire negotiations that are going on right now between the two of them that are supposed to bring this conflict to some sort of an end, even if it’s just temporary?

Because if, as you said, Michael, a ceasefire opens the door to this larger diplomatic solution, and these two players don’t necessarily want that larger diplomatic solution, doesn’t that inevitably impact their enthusiasm for even reaching a ceasefire?

Well, it certainly doesn’t help. You know, this is such a hellish problem. And of course, you first have the question of whether Israel and Hamas can make a deal on these immediate issues, including the hostages, Palestinian prisoners, and what the Israeli military is going to do, how long a ceasefire might last.

But on top of that, you have these much bigger diplomatic questions that are looming over them. And it’s not clear that either side is ready to turn and face those bigger questions.

So while for the Biden administration and for Saudi Arabia, this is a way out of this crisis, these larger diplomatic solutions, it’s not clear that it’s a conversation that the two parties that are actually at war here are prepared to start having.

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

On Tuesday afternoon, under intense pressure from the US, delegations from Israel and Hamas arrived in Cairo to resume negotiations over a potential ceasefire. But in a statement, Israel’s Prime Minister Benjamin Netanyahu made clear that even with the talks underway, his government would, quote, “continue to wage war against Hamas.”

Here’s what else you need to know today. In a dramatic day of testimony, Stormy Daniels offered explicit details about an alleged sexual encounter with Donald Trump that ultimately led to the hush money payment at the center of his trial. Daniels testified that Trump answered the door in pajamas, that he told her not to worry that he was married, and that he did not use a condom when they had sex.

That prompted lawyers for Trump to seek a mistrial based on what they called prejudicial testimony. But the judge in the case rejected that request. And,

We’ve seen a ferocious surge of anti-Semitism in America and around the world.

In a speech on Tuesday honoring victims of the Holocaust, President Biden condemned what he said was the alarming rise of anti-Semitism in the United States after the October 7th attacks on Israel. And he expressed worry that too many Americans were already forgetting the horrors of that attack.

The Jewish community, I want you to know I see your fear, your hurt, and your pain. Let me reassure you, as your president, you’re not alone. You belong. You always have and you always will.

Today’s episode was produced by Nina Feldman, Clare Toeniskoetter, and Rikki Novetsky. It was edited by Liz O. Baylen, contains original music by Marion Lozano, Elisheba Ittoop, and Dan Powell, 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 tomorrow.

• 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
• April 25, 2024   •   40:33 The Crackdown on Student Protesters

Hosted by Michael Barbaro

Featuring Michael Crowley

Produced by Nina Feldman ,  Clare Toeniskoetter and Rikki Novetsky

Edited by Liz O. Baylen

Original music by Marion Lozano ,  Elisheba Ittoop and Dan Powell

Engineered by Alyssa Moxley

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

If and when Israel and Hamas reach a deal for a cease-fire, the United States will immediately turn to a different set of negotiations over a grand diplomatic bargain that it believes could rebuild Gaza and remake the Middle East.

Michael Crowley, who covers the State Department and U.S. foreign policy for The Times, explains why those involved in this plan believe they have so little time left to get it done.

On today’s episode

Michael Crowley , a reporter covering the State Department and U.S. foreign policy for The New York Times.

Background reading :

Talks on a cease-fire in the Gaza war are once again at an uncertain stage .

Here’s how the push for a deal between Israel and Saudi Arabia looked before Oct. 7 .

From early in the war, President Biden has said that a lasting resolution requires a “real” Palestinian state .

Here’s what Israeli officials are discussing about postwar Gaza.

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.

Michael Crowley covers the State Department and U.S. foreign policy for The Times. He has reported from nearly three dozen countries and often travels with the secretary of state. More about Michael Crowley

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