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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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Percentage Calculator

Calculator Use

Find a percentage or work out the percentage given numbers and percent values. Use percent formulas to figure out percentages and unknowns in equations. Add or subtract a percentage from a number or solve the equations.

How to Calculate Percentages

There are many formulas for percentage problems. You can think of the most basic as X/Y = P x 100. The formulas below are all mathematical variations of this formula.

Let's explore the three basic percentage problems. X and Y are numbers and P is the percentage:

• Find P percent of X
• Find what percent of X is Y
• Find X if P percent of it is Y

Read on to learn more about how to figure percentages.

1. How to calculate percentage of a number. Use the percentage formula: P% * X = Y

Example: What is 10% of 150?

• Convert the problem to an equation using the percentage formula: P% * X = Y
• P is 10%, X is 150, so the equation is 10% * 150 = Y
• Convert 10% to a decimal by removing the percent sign and dividing by 100: 10/100 = 0.10
• Substitute 0.10 for 10% in the equation: 10% * 150 = Y becomes 0.10 * 150 = Y
• Do the math: 0.10 * 150 = 15
• So 10% of 150 is 15
• Double check your answer with the original question: What is 10% of 150? Multiply 0.10 * 150 = 15

2. How to find what percent of X is Y. Use the percentage formula: Y/X = P%

Example: What percent of 60 is 12?

• Convert the problem to an equation using the percentage formula: Y/X = P%
• X is 60, Y is 12, so the equation is 12/60 = P%
• Do the math: 12/60 = 0.20
• Important! The result will always be in decimal form, not percentage form. You need to multiply the result by 100 to get the percentage.
• Converting 0.20 to a percent: 0.20 * 100 = 20%
• So 20% of 60 is 12.
• Double check your answer with the original question: What percent of 60 is 12? 12/60 = 0.20, and multiplying by 100 to get percentage, 0.20 * 100 = 20%

3. How to find X if P percent of it is Y. Use the percentage formula Y/P% = X

Example: 25 is 20% of what number?

• Convert the problem to an equation using the percentage formula: Y/P% = X
• Y is 25, P% is 20, so the equation is 25/20% = X
• Convert the percentage to a decimal by dividing by 100.
• Converting 20% to a decimal: 20/100 = 0.20
• Substitute 0.20 for 20% in the equation: 25/0.20 = X
• Do the math: 25/0.20 = X
• So 25 is 20% of 125
• Double check your answer with the original question: 25 is 20% of what number? 25/0.20 = 125

Remember: How to convert a percentage to a decimal

• Remove the percentage sign and divide by 100
• 15.6% = 15.6/100 = 0.156

Remember: How to convert a decimal to a percentage

• Multiply by 100 and add a percentage sign
• 0.876 = 0.876 * 100 = 87.6%

Percentage Problems

There are nine variations on the three basic problems involving percentages. See if you can match your problem to one of the samples below. The problem formats match the input fields in the calculator above. Formulas and examples are included.

What is P percent of X?

• Written as an equation: Y = P% * X
• The 'what' is Y that we want to solve for
• Remember to first convert percentage to decimal, dividing by 100
• Solution: Solve for Y using the percentage formula Y = P% * X

Example: What is 10% of 25?

• Written using the percentage formula: Y = 10% * 25
• First convert percentage to a decimal 10/100 = 0.1
• Y = 0.1 * 25 = 2.5
• So 10% of 25 is 2.5

Y is what percent of X?

• Written as an equation: Y = P% ? X
• The 'what' is P% that we want to solve for
• Divide both sides by X to get P% on one side of the equation
• Y ÷ X = (P% ? X) ÷ X becomes Y ÷ X = P%, which is the same as P% = Y ÷ X
• Solution: Solve for P% using the percentage formula P% = Y ÷ X

Example: 12 is what percent of 40?

• Written using the formula: P% = 12 ÷ 40
• P% = 12 ÷ 40 = 0.3
• Convert the decimal to percent
• P% = 0.3 × 100 = 30%
• So 12 is 30% of 40

Y is P percent of what?

• The 'what' is X that we want to solve for
• Divide both sides by P% to get X on one side of the equation
• Y ÷ P% = (P% × X) ÷ P% becomes Y ÷ P% = X, which is the same as X = Y ÷ P%
• Solution: Solve for X using the percentage formula X = Y ÷ P%

Example: 9 is 60% of what?

• Writen using the formula: X = 9 ÷ 60%
• Convert percent to decimal
• 60% ÷ 100 = 0.6
• X = 9 ÷ 0.6
• So 9 is 60% of 15

What percent of X is Y?

• Written as an equation: P% * X = Y
• (P% * X) ÷ X = Y ÷ X becomes P% = Y ÷ X

Example: What percent of 27 is 6?

• Written using the formula: P% = 6 ÷ 27
• 6 ÷ 27 = 0.2222
• Convert decimal to percent
• P% = 0.2222 × 100
• P% = 22.22%
• So 22.22% of 27 is 6

P percent of what is Y?

• Written as an equation: P% × X = Y
• (P% × X) ÷ P% = Y ÷ P% becomes X = Y ÷ P%

Example: 20% of what is 7?

• Written using the formula: X = 7 ÷ 20%
• Convert the percent to a decimal
• 20% ÷ 100 = 0.2
• X = 7 ÷ 0.2
• So 20% of 35 is 7.

P percent of X is what?

Example: 5% of 29 is what.

• Written using the formula: 5% * 29 = Y
• 5% ÷ 100 = 0.05
• Y = 0.05 * 29
• So 5% of 29 is 1.45

Y of what is P percent?

• Written as an equation: Y / X = P%
• Multiply both sides by X to get X out of the denominator
• (Y / X) * X = P% * X becomes Y = P% * X
• Divide both sides by P% so that X is on one side of the equation
• Y ÷ P% = (P% * X) ÷ P% becomes Y ÷ P% = X

Example: 4 of what is 12%?

• Written using the formula: X = 4 ÷ 12%
• Solve for X: X = Y ÷ P%
• 12% ÷ 100 = 0.12
• X = 4 ÷ 0.12
• X = 33.3333
• 4 of 33.3333 is 12%

What of X is P percent?

• Multiply both sides by X to get Y on one side of the equation
• (Y ÷ X) * X = P% * X becomes Y = P% * X

Example: What of 25 is 11%?

• Written using the formula: Y = 11% * 25
• 11% ÷ 100 = 0.11
• Y = 0.11 * 25
• So 2.75 of 25 is 11%

Y of X is what percent?

• Solution: Solve for P% using the percentage formula P% = Y / X

Example: 9 of 13 is what percent?

• Written using the formula: P% = Y / X
• 9 ÷ 13 = P%
• 9 ÷ 13 = 0.6923
• Convert decimal to percent by multiplying by 100
• 0.6923 * 100 = 69.23%
• 9 ÷ 13 = 69.23%
• So 9 of 13 is 69.23%

Related Calculators

Find the change in percentage as an increase or decrease using the Percentage Change Calculator .

Solve decimal to percentage conversions with our Decimal to Percent Calculator .

Convert from percentage to decimals with the Percent to Decimal Calculator .

If you need to convert between fractions and percents see our Fraction to Percent Calculator , or our Percent to Fraction Calculator .

Weisstein, Eric W. " Percent ." From MathWorld -- A Wolfram Web Resource.

Cite this content, page or calculator as:

Furey, Edward " Percentage Calculator " at https://www.calculatorsoup.com/calculators/math/percentage.php from CalculatorSoup, https://www.calculatorsoup.com - Online Calculators

Last updated: October 20, 2023

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Percentage calculator

This online calculator solves the four basic types of percent problems and percentage increase/decrease problem. The calculator will generate a step-by-step explanation for each type of percentage problem.

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About this calculator

This calculator solves four types of percentage problems.

Type 1: What is 15% of 20?

This can be solved in two steps:

Step 1: Write 15% as a decimal number: 15% = 0.15

Step 2: Multiply 0.15 and 20 to get the answer: 0.15 * 20 = 3

Type 2: The number 25 is what percentage of 80?

Step 1: Solve the equation: 80 * x = 25 . The solution is x = 0.3125

Step 2: Multiply 0.3125 by 100% to get the answer: 0.3125 * 100% = 31.25 %

Type 3: The number 32 is 8% of what number?

As usual, this problem requires to steps:

Step 1: Write 8% as a decimal number: 8% = 0.08

Step 2: Divide 32 and 0.08: 32 ÷ 0.08 = 400

Type 4: percentage increase: What is the percentage increase from 20 to 90?

In this type of problem we use formula

Using our numbers we have:

Type 4: percentage decrease: What is the percentage decrease from 160 to 40?

Now we will use formula

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Solving percent problems.

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Video transcript

Home / United States / Math Classes / 6th Grade Math / Solving Problems Based on Percentage

Solving Problems Based on Percentage

Percent is an alternate method of representing fractions and decimals. Here we will learn different methods of calculati ng the percent and the steps involved in each method. We will also look at some examples that will help you gain a better understanding of the concept. ...Read More Read Less

Table Of Contents

What is meant by percentage?

Solving problems based on percentages, finding the percentage of a number, finding the whole number from the percent, finding the whole using the ratio method, solved examples.

• Frequently Asked Questions

In mathematics, a percentage is a number or ratio that represents a fraction of 100. The symbol “ % ” is frequently used to represent it, and it has a few hundred years of history. While we are on the topic of percentages, one example will be, the decimal 0.35, or the fraction \(\frac{7}{20}\) , which is equivalent to 35 percent, or 35%.

By solving problems based on percentages, we can find the missing values and find the values of various unknowns in a given problem.

Find 40% of 200.

\(\frac{40}{100}\times 200\)              Write the percentage as a fraction

\(\frac{2}{5}\times 200=800\)            Simplify

First, write the percentage as a fraction or decimal. Then, divide the fraction or decimal by the part. This method applies to any situation in which a percentage and its value are given. 

If 2 percent equals 80, multiply 80 by 100 and divide it by 2 to get 4000.

Prove that 20% of 120 is 24. 

20% =\(\frac{20}{100}\)       Write the percent as a fraction or decimal.

Using multiplication equation:

\(\frac{20}{100}\times 120=24\)      Simplify

To prove the reverse of this solution we use the  division equation:

\(\frac{24}{\frac{20}{100}}\)      Simplify

\(\frac{2400}{20}=120\)    

A ratio table is the table that shows the comparison between two units and shows the relationship between them.

Example 1: What is 25% of 50?

We have 25% of 50.

So, 25% of 50 = \(\frac{1}{4}\times 50\) Write the percentage as a fraction or decimal.

                       = \(\frac{50}{4}\)     Simplify.

                       = 12.5  

Example 2: Using the ratio table, answer the following question:

What is 60% of 200?

We have 60% of 200.

Now, we have to use the ratio table to find the part. Let one row represent the part and the other row represent the whole row in the table and find the equivalent ratio of 200.

The first column represents the percentage = \(\frac{60}{100}\)

So, 60% of 200 is 120.

Example 3: Find the whole of the number.

                  50% of what number is 45.

We have: 50% of what number is 45?

Use division equation

\(\frac{45}{50%}\)    Write the percentage as a fraction or decimal

\(=\frac{45}{\frac{1}{2}}\)  Simplify

So, \(45\times 2=90\)

Hence, 50% of 90 is 45

Example 4: Find the whole of the number using the ratio table.

140% of what number is 84

We have to find 140% of what number is 84.

Use the ratio table to find the part. Let one be the part and the other be the whole row in the table. Now, find the equivalent ratio of 200.

So, 140% of 60 is 84.

Example 5: A rectangular hall’s width is 60 percent of its length.

What are the room’s dimensions?

Solution:  

Calculate the width of the room by taking 60% of 15 feet.

\(60%\times 15\) Write the percentage as a fraction or decimal.

= \(0.6\times 15\)      Simplify

We can al so understand it with the help of a diagram:

The width is 9 feet.

Area of the rectangle = \(\text{length}\times \text{width}\)

                                    = \(15\times 9\)

                                    = 135

Hence, the area of the given room is 135 \(feet^2\).

Example 6: You have won a camping trip at an auction at your school fair that cost $80. Your bid is 40% of your maximum bid for the price of the camping trip. How much more would you be willing to pay for the trip if you hadn’t already paid the full price?

You are given the winning camping bid that represents the maximum bid as well as the percentage of your maximum bid. You must calculate how much more you would have paid for the camping trip if you had known how much more you were willing to pay.

Your winning bid is the part, and your maximum bid is the whole.

Create a model based on the fact that 40% of the total is $80 to determine the highest bid. Then divide the winning bid by the maximum bid to find out how much more you were willing to pay.

The maximum bid is $200 and the winning bid is $80. So, you would be willing to bid $200 – $80 = $120 more for the tickets.

How do you calculate a percentage?

To calculate a percentage, divide the given value by the total value and multiply the result by 100. That is “(value/total value) x 100%”. This is the formula for calculating percentages.

In mathematics, a percentage is a number or ratio that represents a fraction of 100 in mathematics. Percentage is usually represented by the symbol “%”. It is also written simply as “percent” or “pct”. For example, the decimal 0.35, or the fraction \(\frac{35}{100}\), is equivalent to 0.35.

What is the purpose of percentages?

Percentages are used to figure out “how much” or “how many” of something is to be taken from a given value. Percentage makes it easier to calculate the exact amount or figure being discussed. In order to determine whether a percentage increase or decrease has occurred, a comparison of fractions is done. This aids in calculating percentages of profit and loss, for example in real life situations.

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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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Percentage Word Problems 5th Grade

Welcome to our Percentage Word Problems. In this area, we have a selection of percentage problem worksheets for 5th grade designed to help children learn to solve a range of percentage problems.

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Percentage Learning

Percentages are another area that children can find quite difficult. There are several key areas within percentages which need to be mastered in order.

Our selection of percentage worksheets will help you to find percentages of numbers and amounts, as well as working out percentage increases and decreases and converting percentages to fractions or decimals.

Key percentage facts:

• 50% = 0.5 = ½
• 25% = 0.25 = ¼
• 75% = 0.75 = ¾
• 10% = 0.1 = 1 ⁄ 10
• 1% = 0.01 = 1 ⁄ 100

Percentage Word Problems

How to work out percentages of a number.

This page will help you learn to find the percentage of a given number.

There is also a percentage calculator on the page to support you work through practice questions.

• Percentage Of Calculator

This is the calculator to use if you want to find a percentage of a number.

Simple choose your number and the percentage and the calculator will do the rest.

Here you will find a selection of worksheets on percentages designed to help your child practise how to apply their knowledge to solve a range of percentage problems..

The sheets are graded so that the easier ones are at the top.

The sheets have been split up into sections as follows:

• spot the percentage problems where the aim is to use the given facts to find the missing percentage;
• solving percentage of number problems, where the aim is to work out the percentage of a number.

Each of the sheets on this page has also been split into 3 different worksheets:

• Sheet A which is set at an easier level;
• Sheet B which is set at a medium level;
• Sheet C which is set at a more advanced level for high attainers.

Spot the Percentages Problems

• Spot the Percentage 1A
• PDF version
• Spot the Percentage 1B
• Spot the Percentage 1C
• Spot the Percentage 2A
• Spot the Percentage 2B
• Spot the Percentage 2C

Percentage of Number Word Problems

• Percentage of Number Problems 1A
• Percentage of Number Problems 1B
• Percentage of Number Problems 1C
• Percentage of Number Problems 2A
• Percentage of Number Problems 2B
• Percentage of Number Problems 2C
• Percentage of Number Problems 3A
• Percentage of Number Problems 3B
• Percentage of Number Problems 3C

More Recommended Math Worksheets

Take a look at some more of our worksheets similar to these.

6th Grade Percentage Word Problems

The sheets in this area are at a harder level than those on this page.

The problems involve finding the percentage of numbers and amounts, as well as finding the amounts when the percentage is given.

• 6th Grade Percent Word Problems
• Percentage Increase and Decrease Worksheets

We have created a range of worksheets based around percentage increases and decreases.

Our worksheets include:

• finding percentage change between two numbers;
• finding a given percentage increase from an amount;
• finding a given percentage decrease from an amount.

Percentage of Money Amounts

Often when we are studying percentages, we look at them in the context of money.

The sheets on this page are all about finding percentages of different amounts of money.

• Money Percentage Worksheets

Percentage of Number Worksheets

If you would like some practice finding the percentage of a range of numbers, then try our Percentage Worksheets page.

You will find a range of worksheets starting with finding simple percentages such as 1%, 10% and 50% to finding much trickier ones.

• Percentage of Numbers Worksheets

Converting Percentages to Fractions

To convert a fraction to a percentage follows on simply from converting a fraction to a decimal.

Simply divide the numerator by the denominator to give you the decimal form. Then multiply the result by 100 to change the decimal into a percentage.

The printable learning fraction page below contains more support, examples and practice converting fractions to decimals.

• Converting Fractions to Percentages

• Convert Percent to Fraction

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Our online percentage practice zone gives you a chance to practice finding percentages of a range of numbers.

You can choose your level of difficulty and test yourself with immediate feedback!

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• Ratio Part to Part Worksheets

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The sheets in this section are at a more basic level than those on this page.

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How to solve 50% of your problems ?: Book help to find a solution to your dealy problems (6x9 inches) Paperback – 23 January 2020

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4.2: Percents Problems and Applications of Percent

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• Page ID 142718

• Morgan Chase
• Clackamas Community College via OpenOregon

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. $ \(2,500\)

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

6. 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?

7. \(45\%\) of the people surveyed answered “yes” to a poll question. If \(180\) people answered “yes”, how many people were surveyed altogether?

5. $ \(19.35\)

6. \(44.8\%\) or \(45\%\)

7. \(400\) 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.

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

9. 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]

8. \(56.2\%\) increase

9. \(12.2\%\) 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.

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

11. 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.

10. \(36.0\%\) decrease

11. \(35.7\%\) 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 \(\PageIndex{1}\)

12. 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.

13. 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.

12. \(0.1875\div25\approx0.8\%\)

13. \(0.13\div10.8\approx1.2\%\)

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.

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

15. 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.

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

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

14. \(4.806\) g; \(5.194\) g

15. \(0.21\div5.000=4.2\%\)

16. \(5.443\) g; \(5.897\) g

17. \(0.24\div5.670\approx4.2\%\)

• 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 ↵

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How to Resolve Most Any Conflict: The Solution

Mastering the five essential steps for conflict resolution..

Posted November 27, 2023 | Reviewed by Michelle Quirk

• What Is Stress?
• Find a therapist to overcome stress
• Miscommunication is inevitable in human relationships and often leads to conflict.
• When fight-or-flight takes hold, conflicts quickly escalate and become more difficult to work through.
• There are five steps that can be learned and applied to reliably resolve most conflicts.

This post is part 2 of a series.

Miscommunication is inevitable in human interactions. Biases, filters, assumptions, expectations, and nonverbal information cause distortions in interpersonal communications, altering them to fit our own point of view. As noted in part 1 , up to 75 percent of all spoken communication is misunderstood, ignored, or forgotten (Guffey & Loewy, 2016; Tankovic, Kapeš & Benazić, 2023).

Miscommunications often lead to conflict. When communicated information is perceived as a threat, our fight-or-flight response is activated. Fight-or-flight shuts down both higher-order thinking and efficient information processing, increasing the possibility of further miscommunication. Once the brain is hijacked by fight-or-flight, conflicts quickly escalate. Learning how to manage communications capable of reducing threat and promoting problem-solving requires attention to several key points.

Managing Communications

To begin with, the earlier you take notice that a conflict exists, the easier it will be to manage it. You don’t need to fully understand what is transpiring to step back. By mutually acknowledging that there seems to be a problem, rather than pushing ahead with a discussion that is becoming heated, everyone becomes empowered, and you are able to start from a place of agreement.

The next goal, before addressing the conflict itself, is to diffuse negative emotions so a conflict does not continue to escalate, induce more fear and anger , or heighten the flight-or-flight response. Creating a sense of safety allows for the diffusion of negative emotions. Some ways this might be accomplished include (but are not limited to) taking a 15-minute break to calm and center, signaling to one another a desire to work together positively, giving voice to how fight-or-flight may have taken hold, and reassuring one another that you both want to resolve the problem positively.

When we do not acknowledge that a stress response has taken hold, the situation can feel dangerous and out of control. Taking a step back allows the stress response to calm so that the focus can shift to a more rational discussion of the conflict itself. Establishing common ground, before attempting to resolve a conflict, also demonstrates respect and compassion, which facilitates trust in the relationship and efficacy in your mutual abilities to resolve the conflict. Check in with one another before you move on to the next step, providing additional time or assurances as needed to build a sense of safety.

Active Listening

Once a conducive atmosphere has been established, the goal shifts to understanding each other’s point of view. It is not yet time to begin trying to solve the problem. Understanding the other’s perspective is a critical step, only possible with active listening . Active listening differs from passive listening, where information is absorbed and processed unilaterally with no opportunity for questions, clarification, or feedback. We are not able to fully comprehend what is being communicated with passive listening. Miscommunication happens when you interpret, without seeking clarification, and listen passively, simply awaiting your turn to speak. Active listening is when you pay full attention to another’s point of view, without judgment, and, when they finish speaking, clarifying to make sure you understand their perspective before switching to stating your point of view.

There are numerous forms of active listening including restating, paraphrasing, asking open-ended questions, and using "I" statements. Further, use your body language and gestures to show you’re engaged. You can do this by angling your body toward the other(s), sustaining eye contact, and nodding. Simply saying “I understand” and moving on to stating your point of view will not do the trick. This is still passive as it does not allow the other person insight into what you have heard, nor offer opportunities for clarification. Often, one person believes there is a significant relational conflict and the other may not even recognize that a misunderstanding has occurred or may judge it as slight and inconsequential. Active listening requires that you restate what you believe you have heard and then ask if you are correct, providing the time and opportunity for clarification and correction. Active listening allows all voices to be heard and truly understood. The goal of this step is for each person to prove that they understand the viewpoint of the other. Do not move on to the next step until all parties agree that what is being restated is, in fact, the intention of the communication.

Once the conflict has been clearly defined, from all perspectives, it is time to move on to the next step. The goal here is to list all the conditions that must be met for a solution to be acceptable to everyone involved, even if the conditions appear to be contradictory. There is no limit to how many pre-conditions are set, only that they be specific and realistic. When all of the pre-conditions are listed, you are assured of understanding the problem as well as is required for a true solution to be generated. This step is frequently skipped. However, listing the essential preconditions, before searching for possible solutions, allows you to move ahead together, with shared goals . It increases the precision of the solution and decreases the probability of crystalizing a desired outcome early on, before the situation is entirely understood. Reducing investment in particular solutions opens everyone up to alternatives and increases the likelihood of a win-win scenario (Likert & Likert, 1978). It is noteworthy that many conflicts can be resolved with these four steps alone. Once all perspectives are clarified and preconditions are stated, a solution often naturally evolves. So, it really pays to put in a considerable and conscious effort in establishing common ground at the outset of communications.

Generating Possible Solutions

Only now it is time to generate possible solutions. Consider the advantages and disadvantages of each possible solution and be open to compromise. By working together to creatively brainstorm potential solutions, all parties will feel respected. Once you choose a solution, clearly define all the terms and conditions for implementation and make sure everyone understands their responsibilities. Set times to check in with one another to make sure the solution remains sound and is accomplishing what you had hoped for.

Of course, some conflicts do not resolve with this process. There are dynamics that can get in the way including a lack of honesty or commitment to the process, power imbalances, unrealistic expectations, insufficient time, or an unwillingness to compromise. Some people are more transactional, confrontational, or focused on winning rather than creating mutual resolution. Conflicts can also be sustained by factors such as inequality, resource scarcity, political factors, lack of trust, cultural issues, and emotional entrenchment in a respective position. In some cases, there has been too much damage, in which case forgiveness and reconciliation must be established before moving toward conflict resolution.

Solidarity—that is, creating a commonality of understanding, feelings, and purpose—is a powerful force for conflict resolution. When collaboration , empathy, trust, and shared commitment are implemented, almost all conflicts can be resolved in equitable ways that promote healing and resilience .

Likert, R., & Likert, J. G. (1978). A Method for Coping With Conflict in Problem-Solving Groups. Group & Organization Studies , 3(4), 427–434. https://doi.org/10.1177/105960117800300406

Paxton, A., Roche, J. M., Ibarra, A. & Tanenhaus, M. K. (2021). Predictions of Miscommunication in Verbal Communication During Collaborative Joint Action. J Speech Lang Hear Res. , 64 (2), pp. 613–627. doi: 10.1044/2020_JSLHR-20-00137.

Leigh W. Jerome, Ph.D. , is a clinical psychologist, artist, and the founder and executive director of the non-profit art forum Relational Space.

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To Solve Big Problems, Make Everyone Feel Included in Your Mission

• Rajiv J. Shah

It’s human nature to want to work towards something bigger than a paycheck.

It’s human nature to want to spend our days working on something more than a paycheck, and the best leaders find ways to connect their teammates’ work to something bigger. That starts by making everyone feel included in the mission. If your team feels connected to the mission, then making big bets and solving big problems will always be within your reach. In this article, the author discusses how several prominent leaders instilled their teams with a sense of the organization’s larger purpose, and how you can do the same.

No matter their rank, teammates want to believe that what they do, day in and day out, matters. Yet meaning can be incredibly hard to find. Every job has its good days and bad, its high points and low, its crowning achievements and mundane expense reports. Drudgery can fill too many days, while inspiration can fill too few.

• RS Rajiv J. Shah is president of The Rockefeller Foundation. From 2009-2015, he served as administration to the U.S. Agency for International Development. Shah’s new book is Big Bets: How Large-Scale Change Really Happens .

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