problem solving percentage worksheets

Percentages Worksheets

Welcome to the percentages math worksheet page where we are 100% committed to providing excellent math worksheets. This page includes Percentages worksheets including calculating percentages of a number, percentage rates, and original amounts and percentage increase and decrease worksheets.

As you probably know, percentages are a special kind of decimal. Most calculations involving percentages involve using the percentage in its decimal form. This is achieved by dividing the percentage amount by 100. There are many worksheets on percentages below. In the first few sections, there are worksheets involving the three main types of percentage problems: calculating the percentage value of a number, calculating the percentage rate of one number compared to another number, and calculating the original amount given the percentage value and the percentage rate.

Most Popular Percentages Worksheets this Week

Percentage Calculations

Calculating the percentage value of a number involves a little bit of multiplication. One should be familiar with decimal multiplication and decimal place value before working with percentage values. The percentage value needs to be converted to a decimal by dividing by 100. 18%, for example is 18 ÷ 100 = 0.18. When a question asks for a percentage value of a number, it is asking you to multiply the two numbers together.

Example question: What is 18% of 2800? Answer: Convert 18% to a decimal and multiply by 2800. 2800 × 0.18 = 504. 504 is 18% of 2800.

• Calculating the Percentage Value (Whole Number Results) Calculating the Percentage Value (Whole Number Results) (Percents from 1% to 99%) Calculating the Percentage Value (Whole Number Results) (Select percents) Calculating the Percentage Value (Whole Number Results) (Percents that are multiples of 5%) Calculating the Percentage Value (Whole Number Results) (Percents that are multiples of 25%)
• Calculating the Percentage Value (Decimal Number Results) Calculating the Percentage Value (Decimal Number Results) (Percents from 1% to 99%) Calculating the Percentage Value (Decimal Number Results) (Select percents) Calculating the Percentage Value (Decimal Number Results) (Percents that are multiples of 5%) Calculating the Percentage Value (Decimal Number Results) (Percents that are multiples of 25%)
• Calculating the Percentage Value (Whole Dollar Results) Calculating the Percentage Value (Whole Dollar Results) (Percents from 1% to 99%) Calculating the Percentage Value (Whole Dollar Results) (Select percents) Calculating the Percentage Value (Whole Dollar Results) (Percents that are multiples of 5%) Calculating the Percentage Value (Whole Dollar Results) (Percents that are multiples of 25%)
• Calculating the Percentage Value (Decimal Dollar Results) Calculating the Percentage Value (Decimal Dollar Results) (Percents from 1% to 99%) Calculating the Percentage Value (Decimal Dollar Results) (Select percents) Calculating the Percentage Value (Decimal Dollar Results) (Percents that are multiples of 5%) Calculating the Percentage Value (Decimal Dollar Results) (Percents that are multiples of 25%)

Calculating what percentage one number is of another number is the second common type of percentage calculation. In this case, division is required followed by converting the decimal to a percentage. If the first number is 100% of the value, the second number will also be 100% if the two numbers are equal; however, this isn't usually the case. If the second number is less than the first number, the second number is less than 100%. If the second number is greater than the first number, the second number is greater than 100%. A simple example is: What percentage of 10 is 6? Because 6 is less than 10, it must also be less than 100% of 10. To calculate, divide 6 by 10 to get 0.6; then convert 0.6 to a percentage by multiplying by 100. 0.6 × 100 = 60%. Therefore, 6 is 60% of 10.

Example question: What percentage of 3700 is 2479? First, recognize that 2479 is less than 3700, so the percentage value must also be less than 100%. Divide 2479 by 3700 and multiply by 100. 2479 ÷ 3700 × 100 = 67%.

• Calculating the Percentage a Whole Number is of Another Whole Number Calculating the Percentage a Whole Number is of Another Whole Number (Percents from 1% to 99%) Calculating the Percentage a Whole Number is of Another Whole Number (Select percents) Calculating the Percentage a Whole Number is of Another Whole Number (Percents that are multiples of 5%) Calculating the Percentage a Whole Number is of Another Whole Number (Percents that are multiples of 25%)
• Calculating the Percentage a Decimal Number is of a Whole Number Calculating the Percentage a Decimal Number is of a Whole Number (Percents from 1% to 99%) Calculating the Percentage a Decimal Number is of a Whole Number (Select percents) Calculating the Percentage a Decimal Number is of a Whole Number (Percents that are multiples of 5%) Calculating the Percentage a Decimal Number is of a Whole Number (Percents that are multiples of 25%)
• Calculating the Percentage a Whole Dollar Amount is of Another Whole Dollar Amount Calculating the Percentage a Whole Dollar Amount is of Another Whole Dollar Amount (Percents from 1% to 99%) Calculating the Percentage a Whole Dollar Amount is of Another Whole Dollar Amount (Select percents) Calculating the Percentage a Whole Dollar Amount is of Another Whole Dollar Amount (Percents that are multiples of 5%) Calculating the Percentage a Whole Dollar Amount is of Another Whole Dollar Amount (Percents that are multiples of 25%)
• Calculating the Percentage a Decimal Dollar Amount is of a Whole Dollar Amount Calculating the Percentage a Decimal Dollar Amount is of a Whole Dollar Amount (Percents from 1% to 99%) Calculating the Percentage a Decimal Dollar Amount is of a Whole Dollar Amount (Select percents) Calculating the Percentage a Decimal Dollar Amount is of a Whole Dollar Amount (Percents that are multiples of 5%) Calculating the Percentage a Decimal Dollar Amount is of a Whole Dollar Amount (Percents that are multiples of 25%)

The third type of percentage calculation involves calculating the original amount from the percentage value and the percentage. The process involved here is the reverse of calculating the percentage value of a number. To get 10% of 100, for example, multiply 100 × 0.10 = 10. To reverse this process, divide 10 by 0.10 to get 100. 10 ÷ 0.10 = 100.

Example question: 4066 is 95% of what original amount? To calculate 4066 in the first place, a number was multiplied by 0.95 to get 4066. To reverse this process, divide to get the original number. In this case, 4066 ÷ 0.95 = 4280.

• Calculating the Original Amount from a Whole Number Result and a Percentage Calculating the Original Amount (Percents from 1% to 99%) ( Whole Numbers ) Calculating the Original Amount (Select percents) ( Whole Numbers ) Calculating the Original Amount (Percents that are multiples of 5%) ( Whole Numbers ) Calculating the Original Amount (Percents that are multiples of 25%) ( Whole Numbers )
• Calculating the Original Amount from a Decimal Number Result and a Percentage Calculating the Original Amount (Percents from 1% to 99%) ( Decimals ) Calculating the Original Amount (Select percents) ( Decimals ) Calculating the Original Amount (Percents that are multiples of 5%) ( Decimals ) Calculating the Original Amount (Percents that are multiples of 25%) ( Decimals )
• Calculating the Original Amount from a Whole Dollar Result and a Percentage Calculating the Original Amount (Percents from 1% to 99%) ( Dollar Amounts and Whole Numbers ) Calculating the Original Amount (Select percents) ( Dollar Amounts and Whole Numbers ) Calculating the Original Amount (Percents that are multiples of 5%) ( Dollar Amounts and Whole Numbers ) Calculating the Original Amount (Percents that are multiples of 25%) ( Dollar Amounts and Whole Numbers )
• Calculating the Original Amount from a Decimal Dollar Result and a Percentage Calculating the Original Amount (Percents from 1% to 99%) ( Dollar Amounts and Decimals ) Calculating the Original Amount (Select percents) ( Dollar Amounts and Decimals ) Calculating the Original Amount (Percents that are multiples of 5%) ( Dollar Amounts and Decimals ) Calculating the Original Amount (Percents that are multiples of 25%) ( Dollar Amounts and Decimals )
• Mixed Percentage Calculations with Whole Number Percentage Values Mixed Percentage Calculations (Percents from 1% to 99%) ( Whole Numbers ) Mixed Percentage Calculations (Select percents) ( Whole Numbers ) Mixed Percentage Calculations (Percents that are multiples of 5%) ( Whole Numbers ) Mixed Percentage Calculations (Percents that are multiples of 25%) ( Whole Numbers )
• Mixed Percentage Calculations with Decimal Percentage Values Mixed Percentage Calculations (Percents from 1% to 99%) ( Decimals ) Mixed Percentage Calculations (Select percents) ( Decimals ) Mixed Percentage Calculations (Percents that are multiples of 5%) ( Decimals ) Mixed Percentage Calculations (Percents that are multiples of 25%) ( Decimals )
• Mixed Percentage Calculations with Whole Dollar Percentage Values Mixed Percentage Calculations (Percents from 1% to 99%) ( Dollar Amounts and Whole Numbers ) Mixed Percentage Calculations (Select percents) ( Dollar Amounts and Whole Numbers ) Mixed Percentage Calculations (Percents that are multiples of 5%) ( Dollar Amounts and Whole Numbers ) Mixed Percentage Calculations (Percents that are multiples of 25%) ( Dollar Amounts and Whole Numbers )
• Mixed Percentage Calculations with Decimal Dollar Percentage Values Mixed Percentage Calculations (Percents from 1% to 99%) ( Dollar Amounts and Decimals ) Mixed Percentage Calculations (Select percents) ( Dollar Amounts and Decimals ) Mixed Percentage Calculations (Percents that are multiples of 5%) ( Dollar Amounts and Decimals ) Mixed Percentage Calculations (Percents that are multiples of 25%) ( Dollar Amounts and Decimals )

Percentage Increase/Decrease Worksheets

The worksheets in this section have students determine by what percentage something increases or decreases. Each question includes an original amount and a new amount. Students determine the change from the original to the new amount using a formula: ((new - original)/original) × 100 or another method. It should be straight-forward to determine if there is an increase or a decrease. In the case of a decrease, the percentage change (using the formula) will be negative.

• Percentage Increase/Decrease With Whole Number Percentage Values Percentage Increase/Decrease Whole Numbers with 1% Intervals Percentage Increase/Decrease Whole Numbers with 5% Intervals Percentage Increase/Decrease Whole Numbers with 25% Intervals
• Percentage Increase/Decrease With Decimal Number Percentage Values Percentage Increase/Decrease Decimals with 1% Intervals Percentage Increase/Decrease Decimals with 5% Intervals Percentage Increase/Decrease Decimals with 25% Intervals
• Percentage Increase/Decrease With Whole Dollar Percentage Values Percentage Increase/Decrease Whole Dollar Amounts with 1% Intervals Percentage Increase/Decrease Whole Dollar Amounts with 5% Intervals Percentage Increase/Decrease Whole Dollar Amounts with 25% Intervals
• Percentage Increase/Decrease With Decimal Dollar Percentage Values Percentage Increase/Decrease Decimal Dollar Amounts with 1% Intervals Percentage Increase/Decrease Decimal Dollar Amounts with 5% Intervals Percentage Increase/Decrease Decimal Dollar Amounts with 25% Intervals

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Percent Worksheets

This ensemble of printable percentage worksheets is tailor-made for students of grade 6, grade 7, and grade 8. A plethora of exercises like finding the percent of the shaded region, finding percent of a whole numbers and decimals, comparing quantities, well-researched word problems and a lot more are available here. The pdf worksheets are split into metric and customary units to enable convenient downloads. Access some of these worksheets for free!

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Percent of the Shaded Region: Blocks

Engage in this series of base 10 blocks worksheets that contain nine problems per page. Discern and count the shaded squares to determine the percentage of the shaded area.

• Download the set

Percent of the Shaded Region: Shapes

Keenly observe the shaded region of the shapes provided. Find the percentage of the shaded area in each problem.

Percent of a Whole Number

Each 6th grade worksheet contains 14 problems calculating the percentage of whole numbers. Use the answer keys to verify your responses.

Percent: Units of Measurement with Word Problems

Calculate the amount for each base value in these worksheets that contain units of measurement. Also, solve the percent word problems based on interesting real-life scenarios.

Percent of a Decimal Number

Find the value that constitutes an equivalent percentage for each decimal and round them to the nearest hundredth.

Comparing Quantities: with Word Problems

Work out the percentages. Then, compare them and insert the appropriate <, > or = symbols in the boxes. Word problems are also furnished to help learners grasp the concept of comparing quantities.

Percent on a Number Line

Based on the number line models provided, fill in the boxes with either the appropriate percentages or numbers. These printable worksheets form a great visual aid for 6th grade and 7th grade students in understanding percentages.

Find the Value of the Unknowns

Read each question carefully to find the unknown percentages, base values or amounts. Round your answers to the nearest hundredth.

Percent of Increase or Decrease: with Word Problems

Based on the original amount, find the ratio of change in quantity. Then, calculate the percentage of increase or decrease. Each worksheet includes a number of word problems suitable for 7th grade and 8th grade students.

Find the New Amount: Percent of Increase or Decrease

Find the increase or decrease in amount using the given percentage. Add or subtract the amount so derived to determine the change in quantity. A few word problems are incorporated here for variety.

Percent Error

There's no room for error in our enigmatic pdf worksheets on calculating the percent error in an estimate! Practice finding the difference between a guess and the exact value in the form of percentage in this section.

Convert between Fractions, Decimals, Percent Worksheets

Engage this set of printable worksheets that include an assortment of exercises based on conversion between fractions, decimals and percent.

(30 Worksheets)

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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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Percent Worksheets

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What is a percentage?

A percentage is a number or ratio representing a fraction of 100, and it is denoted by the symbol %.

How to calculate the percentage of a number?

Calculating the percentage of a number can be very easy. The number that you need to convert into percentage can be given to you in two ways, fraction and decimal.

1. Percentages and decimals

The decimal format is easier to calculate. Converting a decimal is simple as all you need to do is multiply it with 100. Example: To covert 0.77 to a percent, simply multiply 0.77 by 100, to get a result of 77%

2. Percentages and fractions

If you are given a fraction, first you need to divide the top number with the bottom number and then multiply it with a 100 to get the percentage. 

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Worksheets about percentages (for example, find what percentage 78 is of 123)

Worksheets about converting percents into decimals (for example, write 56% as a decimal or write 0.392 as a percent)

• Percent - the basic concept of percent
• Percentage of a number using mental math
• How to calculate a given percentage of a number
• How to calculate an unknown percentage
• Basics of percent of change

Percentage of number worksheets generator

Tip: Set the "step" for the percentage in the generator to be 10 or 25, and limit the range for the base to multiples of ten to create percent problems you can solve in your head, such as "Find 25% of 60 or find 60% of 900".

Key to Percents Workbooks

Key to Percents first emphasizes mental computation and estimation skills--since most work with percents is done without pencil and paper. Then students are taught to solve percent problems using equal fractions and decimal multiplication. Finally, percents are used to solve word problems in a variety of applications. Key to Percents assumes only a knowledge of fraction and decimal computation. Book 1 covers Percent Concepts. Book 2 covers Percents and Fractions. Book 3 covers Percents and Decimals.

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Percents Worksheets

Math worksheets: percents.

These worksheets provide practice in common calculations involving percents , including changing decimals to and from percents, finding percentages of numbers and fining how many percent a number is of another number.

Sample Grade 6 Percents Worksheet

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Percent Worksheets

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Percentage Increase and Decrease Worksheets

Welcome to our Percentage Increase and Decrease Worksheets page. In this area, we have a selection of worksheets involving finding percentage increases and decreases and working out percentage change.

There are also links to our handy Percentage Increase Calculator and also our How to Find Percentage Change support page to help you with these skills.

For full functionality of this site it is necessary to enable JavaScript.

Here are the instructions how to enable JavaScript in your web browser .

We have a wide range of worksheets on this page to help you learn and practice working with percentage increases and decreases.

Before you learn about percentage increases and decreases, you need to know how to find the percentage of a number.

The worksheets available on this page include:

• finding the final amount after a percentage increase or decrease;
• finding the percentage change between two different amounts;
• solving percentage increase and decrease word problems.
• solving percentage change word problems

Quicklinks to ...

• How to find the Percentage Increase and Decrease
• Percentage Increase Calculator
• Percentage Increase and Decrease Examples
• Percentage Change Worksheets
• Percentage Increase, Decrease and Change Worksheets
• More related Math resources

Find the Percentage Increase and Decrease Online Quiz

How to find the percentage change.

The percentage change is when we know the initial value and the final value and we need to find the percentage increase or decrease.

We have created a support page to help you understand how to find the percentage change.

There are some worked examples and clear step-by-step instructions on this page.

• How to Calculate Percentage Change

How to Find the Percentage Increase & Decrease

If we know the initial amount, and we need to find the final amount after a percentage increase or decrease, we just need to follow these two simple steps:

• Step 1) Work out the percentage of the original amount.
• Step 2a) If it is a percentage increase , then add this amount to the original amount.
• Step 2b) If it is a percentage decrease , then subtract this amount from the original amount.

We have now found our final amount after the percentage increase or decrease.

Percentage Increase and Decrease Worked Examples

Example 1) a laptop usually costs $750. it is reduced by 15% in a sale. what is the sale price.

Step 1) Work out the percentage.

15% of $750 = $112.50

Step 2) Subtract the amount from the original price.

$750 - $112.50 = $637.50

The sale price of the laptop is $637.50.

Example 2) I have $6000 which I leave in a savings account for a year. The interest rate is 3%. How much money will I have at the end of a year?

3% of $6000 = $180

Step 2) Add the amount to the original price.

$6000 + $180 = $6180

At the end of a year, I have $6180.

Example 3) A bag of rice weighs 40 ounces. A larger bag weighs 35% more. How much does the larger bag weigh?

35% of 40 = 14

40 + 14 = 54

The larger bag weighs 54 ounces.

Example 3) An uncut version of a film lasts 150 minutes. The released version lasts 18% less time. How long does the released version last?

18% of 150 = 27

150 - 27 = 123

The released version lasts 123 minutes.

Here you will find a selection of worksheets on percentage increase and decrease.

We have split the worksheets into 3 different sections:

• Set 1) Percentage Increase and Decrease - finding the final amount when a percentage and an initial amount are given;
• Set 2) Percentage Change worksheets - finding the percentage change between an intial amount and a final amount are given;
• Set 3) Percentage Increase / Decrease / Change - finding the final amount after percentage increases and decreases, as well as the percentage change.

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

Set 1) Percentage Increase and Decrease Worksheets

These worksheets involve finding a final amount, when an initial amount and a percentage increase or decrease are given.

• Percentage Increase and Decrease Worksheet 1
• PDF version
• UK Version PDF
• Percentage Increase and Decrease Worksheet 2
• Percentage Increase and Decrease Worksheet 3
• Percentage Increase and Decrease Worksheet 4
• Percentage Increase and Decrease Word Problems 1

Set 2) Percentage Change Worksheets

• Find the Percentage Change Sheet 1
• Find the Percentage Change Sheet 2
• Percentage Change Word Problems Sheet 1
• Percentage Change Word Problems Sheet 2

Set 3) Percentage Increase, Decrease & Change Worksheets

• Percentage Increase, Decrease & Change Sheet 1
• Percentage Increase, Decrease & Change Sheet 2
• Percentage Increase, Decrease & Change Word Problems 1
• Percentage Increase, Decrease & Change Word Problems 2

More Recommended Math Resources

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

This is the calculator to use if you want to find a percentage increase or decrease.

There are two different calculators on the page:

• one calculator will work out the percentage change given an initial and final amount;
• the other calculator will find the final amount given a percentage increase (or decrease) and an initial amount.

The calculator will not just find the answer for you, but also show you the working out for each step.

Reverse Percentage Calculator

Sometimes you are told what the percentage increase/decrease is, and you are also told the final number.

Your task is then to find the original number before the percentage increase or decrease.

Our reverse percentage calculator will help you to find the original number.

• Reverse Percentages Calculator

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.

• Percentage Word Problems 5th Grade
• 6th Grade Percent Word Problems

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

Online Percentage Practice Zone

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!

• Online Percentage Practice

Our quizzes have been created using Google Forms.

At the end of the quiz, you will get the chance to see your results by clicking 'See Score'.

This will take you to a new webpage where your results will be shown. You can print a copy of your results from this page, either as a pdf or as a paper copy.

For incorrect responses, we have added some helpful learning points to explain which answer was correct and why.

We do not collect any personal data from our quizzes, except in the 'First Name' and 'Group/Class' fields which are both optional and only used for teachers to identify students within their educational setting.

We also collect the results from the quizzes which we use to help us to develop our resources and give us insight into future resources to create.

For more information on the information we collect, please take a look at our Privacy Policy

We would be grateful for any feedback on our quizzes, please let us know using our Contact Us link, or use the Facebook Comments form at the bottom of the page.

This quick quiz tests your knowledge and skill at finding percentage increasee and decreases from different values.

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• How to Print support

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Percentage Word Problem Worksheets

Percentages can be calculated from fractions and decimals. Although there are many steps to calculate a percentage, it can be simplified a bit. Multiplication is used if you work with a decimal, and division is used to convert a mixed number to a percentage.

The word percentage means 100 percent. For example, 10 percent means 10 out of 100. This can be written as 10 or 10% or as a fraction of 10/100, or as a decimal such as .10. It can look at numbers written in different formats and choose them as potential percentages can help students prepare for tests.

Benefits of Percentage Word Problem Worksheets

Cuemath's interactive math worksheets consist of visual simulations to help your child visualize the concepts being taught, i.e., "see things in action and reinforce learning from it." The percentage word problem worksheets follow a step-by-step learning process that helps students better understand concepts, recognize mistakes, and possibly develop a strategy to tackle future problems and In the Percent Problems Worksheet, we will also practice different types of questions about calculating percentage word problems.

Download Percentage Word Problem Worksheet PDFs

These 150-foot-high sails could help solve shipping’s climate problem

To cut costs and carbon emissions, cargo ships are putting a new spin on an ancient technology: the sail.

These aren’t the sailboats of yore. Modern sails look more like airplane wings, smokestacks or balloons, and they use artificial intelligence to catch the wind with little help from mariners who long ago forgot the art of hoisting a mainsail.

Sails can reduce an existing ship’s fuel consumption - and greenhouse emissions - by something like 10 or 20 percent, according to maritime experts, making them an attractive option for ship owners looking to cut costs or comply with environmental regulations.

Ships burn some of the world’s dirtiest fuels and generate roughly 3 percent of global emissions, a share that’s only expected to rise over time, according to the United Nations. The European Union created a cap-and-trade system for shipping emissions earlier this year, and the U.N. International Maritime Organization is finalizing its own emissions rules now that would penalize the owners of dirty vessels.

Rather than sending those dirty vessels to the scrapyard, companies can install sails to clean up some of their emissions and extend their ships’ lives. And as the industry eventually moves toward alternative fuels that are low-carbon but high-cost, saving money on fuel will become even more important.

There are now 39 large commercial ships with sails, according to the International Windship Association, an industry group that represents sailmakers, ship owners and ship designers. That’s a drop in the bucket compared to the roughly 100,000 cargo ships plying the seas, but the technology seems poised to take off as sails move from test projects to real-world use. Sailmakers are building new factories to meet the expected demand.

“We’re at an inflection point,” said Matthew Collette, a professor of naval architecture and marine engineering at the University of Michigan. “We’re going to see this coming very quickly to a larger number of ships.”

Here are some of the strange sails that may one day push your online orders across the seas.

Ships with wings

One of the most versatile sail designs looks and works a lot like an airplane wing.

“All we’ve done is taken that wing and put it vertically,” said John Cooper, CEO of BAR Technologies, a company that manufactures this type of sail, “so instead of creating lift, we’re creating thrust.”

On an airplane, wind flows over the wings and creates air pressure differences that push the plane up. On a ship, the wings works the same way - except they’re angled to push the ship forward.

The wings come with sensors that measure weather conditions, and they automatically change their angle and shape to catch the wind. These sails can typically be used for most of a voyage, but they fold down to the deck if the wind blows faster than 30 knots, or when the ship is docking or loading cargo.

Last year, BAR Technologies installed two of its wings on a 43,000-ton ship designed to carry dry bulk cargo such as grains, coal or minerals. Over its first six months, the ship saved 14 percent on fuel as it crisscrossed the Atlantic, Pacific and Indian oceans, according to the ship’s owner.

The return of the rotor sail

At first glance, rotor sails may look like smokestacks rising from the deck, but they’re actually tall, rotating cylinders that use wind to push a ship forward.

When the wind is blowing at the right angle, an electrical motor spins the rotor sails, speeding up the air flow on one side of the sail and slowing it down on the other. That creates an air pressure difference that pushes the ship forward.

Norsepower, the biggest rotor sail manufacturer, says they typically help ships save 8 to 10 percent on fuel.

Rotor sails are more than a century old; German inventor Anton Flettner patented the idea in 1922 and an experimental cargo ship fitted with rotor sails crossed the Atlantic in 1926. But Norsepower, which is based in Finland, says the sails have come a long way since then, thanks to lightweight composite materials and AI systems that adjust to the wind to make the sails more efficient.

“We can make a much better sail than Mr. Flettner did in the 1920s,” said Tuomas Riski, Norsepower’s CEO.

The Michelin Man sail

Michelin is developing a more experimental inflatable sail which, appropriately, looks a lot like the company’s mascot.

Michelin’s design works similarly to a classic sail, made to catch the wind and redirect its power forward. The main difference is that its sail is made of inflatable fabric instead of a canvas sheet, and its mast can retract down to the deck. The sail can change its size depending on wind conditions.

Like the other sails, Michelin’s product operates by itself. “It has to be fully automated because today’s sailors have no time and no particular knowledge about sails,” said Gildas Quemeneur, who is leading the project.

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A.I.’s Original Sin

A times investigation found that tech giants altered their own rules to train their newest artificial intelligence systems..

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

[MUSIC PLAYING]

Today, a “Times” investigation shows how as the country’s biggest technology companies race to build powerful new artificial intelligence systems, they bent and broke the rules from the start.

My colleague Cade Metz on what he uncovered.

It’s Tuesday, April 16th.

Cade, when we think about all the artificial intelligence products released over the past couple of years, including, of course, these chatbots we’ve talked a lot about on the show, we so frequently talk about their future their future capabilities, their influence on society, jobs, our lives. But you recently decided to go back in time to AI’s past, to its origins to understand the decisions that were made, basically, at the birth of this technology. So why did you decide to do that?

Because if you’re thinking about the future of these chatbots, that is defined by their past. The thing you have to realize is that these chatbots learn their skills by analyzing enormous amounts of digital data.

So what my colleagues and I wanted to do with our investigation was really focus on that effort to gather more data. We wanted to look at the type of data these companies were collecting, how they were gathering it, and how they were feeding it into their systems.

And when you all undertake this line of reporting, what do you end up finding?

We found that three major players in this race OpenAI, Google, and Meta as they were locked into this competition to develop better and better artificial intelligence, they were willing to do almost anything to get their hands on this data, including ignoring, and in some cases, violating corporate rules and wading into a legal gray area as they gathered this data.

Basically, cutting corners.

Cutting corners left and right.

OK, let’s start with OpenAI, the flashiest player of all.

The most interesting thing we’ve found, is that in late 2021, as OpenAI, the startup in San Francisco that built ChatGPT, as they were pulling together the fundamental technology that would power that chatbot, they ran out of data, essentially.

They had used just about all the respectable English language text on the internet to build this system. And just let that sink in for a bit.

I mean, I’m trying to let that sink in. They basically, like a Pac-Man on a old game, just consumed almost all the English words on the internet, which is kind of unfathomable.

Wikipedia articles by the thousands, news articles, Reddit threads, digital books by the millions. We’re talking about hundreds of billions, even trillions of words.

So by the end of 2021, OpenAI had no more English language texts that they could feed into these systems, but their ambitions are such that they wanted even more.

So here, we should remember that if you’re gathering up all the English language text on the internet, a large portion of that is going to be copyrighted.

So if you’re one of these companies gathering data at that scale, you are absolutely gathering copyrighted data, as well.

Which suggests that, from the very beginning, these companies, a company like OpenAI with ChatGPT, is starting to break, bend the rules.

Yes, they are determined to build this technology thus they are willing to venture into what is a legal gray area.

So given that, what does OpenAI do once it, as you had said, runs out of English language words to mop up and feed into this system?

So they get together, and they say, all right, so what are other options here? And they say, well, what about all the audio and video on the internet? We could transcribe all the audio and video, turn it into text, and feed that into their systems.

Interesting.

So a small team at OpenAI, which included its president and co-founder Greg Brockman, built a speech-recognition technology called Whisper, which could transcribe audio files into text with high accuracy.

And then they gathered up all sorts of audio files, from across the internet, including audio books, podcasts —

— and most importantly, YouTube videos.

Hmm, of which there’s a seemingly endless supply, right? Fair to say maybe tens of millions of videos.

According to my reporting, we’re talking about at least 1,000,000 hours of YouTube videos were scraped off of that video sharing site, fed into this speech recognition system in order to produce new text for training OpenAI’s chatbot. And YouTube’s terms of service do not allow a company like OpenAI to do this. YouTube, which is owned by Google, explicitly says you are not allowed to, in internet parlance, scrape videos en masse from across YouTube and use those videos to build a new application.

That is exactly what OpenAI did. According to my reporting, employees at the company knew that it broke YouTube terms of service, but they resolved to do it anyway.

So, Cade, this makes me want to understand what’s going on over at Google, which as we have talked about in the past on the show, is itself, thinking about and developing its own artificial intelligence model and product.

Well, as OpenAI scrapes up all these YouTube videos and starts to use them to build their chatbot, according to my reporting, some employees at Google, at the very least, are aware that this is happening.

Yes, now when we went to the company about this, a Google spokesman said it did not know that OpenAI was scraping YouTube content and said the company takes legal action over this kind of thing when there’s a clear reason to do so. But according to my reporting, at least some Google employees turned a blind eye to OpenAI’s activities because Google was also using YouTube content to train its AI.

So if they raise a stink about what OpenAI is doing, they end up shining a spotlight on themselves. And they don’t want to do that.

I guess I want to understand what Google’s relationship is to YouTube. Because of course, Google owns YouTube. So what is it allowed or not allowed to do when it comes to feeding YouTube data into Google’s AI models?

It’s an important distinction. Because Google owns YouTube, it defines what can be done with that data. And Google argues that it has a right to that data, that its terms of service allow it to use that data. However, because of that copyright issue, because the copyright to those videos belong to you and I, lawyers who I’ve spoken to say, people could take Google to court and try to determine whether or not those terms of service really allow Google to do this. There’s another legal gray area here where, although Google argues that it’s OK, others may argue it’s not.

Of course, what makes this all so interesting is, you essentially have one tech company Google, keeping another tech company OpenAI’s dirty little secret about basically stealing from YouTube because it doesn’t want people to know that it too is taking from YouTube. And so these companies are essentially enabling each other as they simultaneously seem to be bending or breaking the rules.

What this shows is that there is this belief, and it has been there for years within these companies, among their researchers, that they have a right to this data because they’re on a larger mission to build a technology that they believe will transform the world.

And if you really want to understand this attitude, you can look at our reporting from inside Meta.

And so what does Meta end up doing, according to your reporting?

Well, like Google and other companies, Meta had to scramble to build artificial intelligence that could compete with OpenAI. Mark Zuckerberg is calling engineers and executives at all hours pushing them to acquire this data that is needed to improve the chatbot.

And at one point, my colleagues and I got hold of recordings of these Meta executives and engineers discussing this problem. How they could get their hands on more data where they should try to find it? And they explored all sorts of options.

They talked about licensing books, one by one, at $10 a pop and feeding those into the model.

They even discussed acquiring the book publisher Simon & Schuster and feeding its entire library into their AI model. But ultimately, they decided all that was just too cumbersome, too time consuming, and on the recordings of these meetings, you can hear executives talk about how they were willing to run roughshod over copyright law and ignore the legal concerns and go ahead and scrape the internet and feed this stuff into their models.

They acknowledged that they might be sued over this. But they talked about how OpenAI had done this before them. That they, Meta were just following what they saw as a market precedent.

Interesting, so they go from having conversations like, should we buy a publisher that has tons of copyrighted material suggesting that they’re very conscious of the kind of legal terrain and what’s right and what’s wrong. And instead say, nah, let’s just follow the OpenAI model, that blueprint and just do what we want to do, do what we think we have a right to do, which is to kind of just gobble up all this material across the internet.

It’s a snapshot of that Silicon Valley attitude that we talked about. Because they believe they are building this transformative technology, because they are in this intensely competitive situation where money and power is at stake, they are willing to go there.

But what that means is that there is, at the birth of this technology, a kind of original sin that can’t really be erased.

It can’t be erased, and people are beginning to notice. And they are beginning to sue these companies over it. These companies have to have this copyrighted data to build their systems. It is fundamental to their creation. If a lawsuit bars them from using that copyrighted data, that could bring down this technology.

We’ll be right back.

So Cade, walk us through these lawsuits that are being filed against these AI companies based on the decisions they made early on to use technology as they did and the chances that it could result in these companies not being able to get the data they so desperately say they need.

These suits are coming from a wide range of places. They’re coming from computer programmers who are concerned that their computer programs have been fed into these systems. They’re coming from book authors who have seen their books being used. They’re coming from publishing companies. They’re coming from news corporations like, “The New York Times,” incidentally, which has filed a lawsuit against OpenAI and Microsoft.

News organizations that are concerned over their news articles being used to build these systems.

And here, I think it’s important to say as a matter of transparency, Cade, that your reporting is separate from that lawsuit. That lawsuit was filed by the business side of “The New York Times” by people who are not involved in your reporting or in this “Daily” episode, just to get that out of the way.

I’m assuming that you have spoken to many lawyers about this, and I wonder if there’s some insight that you can shed on the basic legal terrain? I mean, do the companies seem to have a strong case that they have a right to this information, or do companies like the “Times,” who are suing them, seem to have a pretty strong case that, no, that decision violates their copyrighted materials.

Like so many legal questions, this is incredibly complicated. It comes down to what’s called fair use, which is a part of copyright law that determines whether companies can use copyrighted data to build new things. And there are many factors that go into this. There are good arguments on the OpenAI side. There are good arguments on “The New York Times” side.

Copyright law says that can’t take my work and reproduce it and sell it to someone. That’s not allowed. But what’s called fair use does allow companies and individuals to use copyrighted works in part. They can take snippets of it. They can take the copyrighted works and transform it into something new. That is what OpenAI and others are arguing they’re doing.

But there are other things to consider. Does that transformative work compete with the individuals and companies that supplied the data that owned the copyrights?

And here, the suit between “The New York Times” company and OpenAI is illustrative. If “The New York Times” creates articles that are then used to build a chatbot, does that chatbot end up competing with “The New York Times?” Do people end up going to that chatbot for their information, rather than going to the “Times” website and actually reading the article? That is one of the questions that will end up deciding this case and cases like it.

So what would it mean for these AI companies for some, or even all of these lawsuits to succeed?

Well, if these tech companies are required to license the copyrighted data that goes into their systems, if they’re required to pay for it, that becomes a problem for these companies. We’re talking about digital data the size of the entire internet.

Licensing all that copyrighted data is not necessarily feasible. We quote the venture capital firm Andreessen Horowitz in our story where one of their lawyers says that it does not work for these companies to license that data. It’s too expensive. It’s on too large a scale.

Hmm, it would essentially make this technology economically impractical.

Exactly, so a jury or a judge or a law ruling against OpenAI, could fundamentally change the way this technology is built. The extreme case is these companies are no longer allowed to use copyrighted material in building these chatbots. And that means they have to start from scratch. They have to rebuild everything they’ve built. So this is something that, not only imperils what they have today, it imperils what they want to build in the future.

And conversely, what happens if the courts rule in favor of these companies and say, you know what, this is fair use. You were fine to have scraped this material and to keep borrowing this material into the future free of charge?

Well, one significant roadblock drops for these companies. And they can continue to gather up all that extra data, including images and sounds and videos and build increasingly powerful systems. But the thing is, even if they can access as much copyrighted material as they want, these companies may still run into a problem.

Pretty soon they’re going to run out of digital data on the internet.

That human-created data they rely on is going to dry up. They’re using up this data faster than humans create it. One research organization estimates that by 2026, these companies will run out of viable data on the internet.

Wow. Well, in that case, what would these tech companies do? I mean, where are they going to go if they’ve already scraped YouTube, if they’ve already scraped podcasts, if they’ve already gobbled up the internet and that altogether is not sufficient?

What many people inside these companies will tell you, including Sam Altman, the chief executive of OpenAI, they’ll tell you that what they will turn to is what’s called synthetic data.

And what is that?

That Is data generated by an AI model that is then used to build a better AI model. It’s AI helping to build better AI. That is the vision, ultimately, they have for the future that they won’t need all this human generated text. They’ll just have the AI build the text that will feed future versions of AI.

So they will feed the AI systems the material that the AI systems themselves create. But is that really a workable solid plan? Is that considered high-quality data? Is that good enough?

If you do this on a large scale, you quickly run into problems. As we all know, as we’ve discussed on this podcast, these systems make mistakes. They hallucinate . They make stuff up. They show biases that they’ve learned from internet data. And if you start using the data generated by the AI to build new AI, those mistakes start to reinforce themselves.

The systems start to get trapped in these cul-de-sacs where they end up not getting better but getting worse.

What you’re really saying is, these AI machines need the unique perfection of the human creative mind.

Well, as it stands today, that is absolutely the case. But these companies have grand visions for where this will go. And they feel, and they’re already starting to experiment with this, that if you have an AI system that is sufficiently powerful, if you make a copy of it, if you have two of these AI models, one can produce new data, and the other one can judge that data.

It can curate that data as a human would. It can provide the human judgment, So. To speak. So as one model produces the data, the other one can judge it, discard the bad data, and keep the good data. And that’s how they ultimately see these systems creating viable synthetic data. But that has not happened yet, and it’s unclear whether it will work.

It feels like the real lesson of your investigation is that if you have to allegedly steal data to feed your AI model and make it economically feasible, then maybe you have a pretty broken model. And that if you need to create fake data, as a result, which as you just said, kind of undermines AI’s goal of mimicking human thinking and language, then maybe you really have a broken model.

And so that makes me wonder if the folks you talk to, the companies that we’re focused on here, ever ask themselves the question, could we do this differently? Could we create an AI model that just needs a lot less data?

They have thought about other models for decades. The thing to realize here, is that is much easier said than done. We’re talking about creating systems that can mimic the human brain. That is an incredibly ambitious task. And after struggling with that for decades, these companies have finally stumbled on something that they feel works that is a path to that incredibly ambitious goal.

And they’re going to continue to push in that direction. Yes, they’re exploring other options, but those other options aren’t working.

What works is more data and more data and more data. And because they see a path there, they’re going to continue down that path. And if there are roadblocks there, and they think they can knock them down, they’re going to knock them down.

But what if the tech companies never get enough or make enough data to get where they think they want to go, even as they’re knocking down walls along the way? That does seem like a real possibility.

If these companies can’t get their hands on more data, then these technologies, as they’re built today, stop improving.

We will see their limitations. We will see how difficult it really is to build a system that can match, let alone surpass the human brain.

These companies will be forced to look for other options, technically. And we will see the limitations of these grandiose visions that they have for the future of artificial intelligence.

OK, thank you very much. We appreciate it.

Glad to be here.

Here’s what else you need to know today. Israeli leaders spent Monday debating whether and how to retaliate against Iran’s missile and drone attack over the weekend. Herzi Halevi, Israel’s Military Chief of Staff, declared that the attack will be responded to.

In Washington, a spokesman for the US State Department, Matthew Miller reiterated American calls for restraint —

^MATTHEW MILLER^ Of course, we continue to make clear to everyone that we talked to that we want to see de-escalation that we don’t want to see a wider regional war. That’s something that’s been —

— but emphasized that a final call about retaliation was up to Israel. ^MATTHEW MILLER^ Israel is a sovereign country. They have to make their own decisions about how best to defend themselves. What we always try to do —

And the first criminal trial of a former US President officially got underway on Monday in a Manhattan courtroom. Donald Trump, on trial for allegedly falsifying documents to cover up a sex scandal involving a porn star, watched as jury selection began.

The initial pool of 96 jurors quickly dwindled. More than half of them were dismissed after indicating that they did not believe that they could be impartial. The day ended without a single juror being chosen.

Today’s episode was produced by Stella Tan, Michael Simon Johnson, Muge Zaidi, and Rikki Novetsky. It was edited by Marc Georges and Liz O. Baylen, contains original music by Diane Wong, Dan Powell, and Pat McCusker, and was engineered by Chris Wood. 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.

• April 22, 2024   •   24:30 The Evolving Danger of the New Bird Flu
• April 19, 2024   •   30:42 The Supreme Court Takes Up Homelessness
• April 18, 2024   •   30:07 The Opening Days of Trump’s First Criminal Trial
• April 17, 2024   •   24:52 Are ‘Forever Chemicals’ a Forever Problem?
• April 16, 2024   •   29:29 A.I.’s Original Sin
• April 15, 2024   •   24:07 Iran’s Unprecedented Attack on Israel
• April 14, 2024   •   46:17 The Sunday Read: ‘What I Saw Working at The National Enquirer During Donald Trump’s Rise’
• April 12, 2024   •   34:23 How One Family Lost $900,000 in a Timeshare Scam
• April 11, 2024   •   28:39 The Staggering Success of Trump’s Trial Delay Tactics
• April 10, 2024   •   22:49 Trump’s Abortion Dilemma
• April 9, 2024   •   30:48 How Tesla Planted the Seeds for Its Own Potential Downfall
• April 8, 2024   •   30:28 The Eclipse Chaser

Hosted by Michael Barbaro

Featuring Cade Metz

Produced by Stella Tan ,  Michael Simon Johnson ,  Mooj Zadie and Rikki Novetsky

Edited by Marc Georges and Liz O. Baylen

Original music by Diane Wong ,  Dan Powell and Pat McCusker

Engineered by Chris Wood

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

A Times investigation shows how the country’s biggest technology companies, as they raced to build powerful new artificial intelligence systems, bent and broke the rules from the start.

Cade Metz, a technology reporter for The Times, explains what he uncovered.

On today’s episode

Cade Metz , a technology reporter for The New York Times.

Background reading

How tech giants cut corners to harvest data for A.I.

What to know about tech companies using A.I. to teach their own A.I.

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.

Cade Metz writes about artificial intelligence, driverless cars, robotics, virtual reality and other emerging areas of technology. More about Cade Metz

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