solve problems with percent lesson 5

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5.2.1: Solving Percent Problems

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Learning Objectives

• Identify the amount, the base, and the percent in a percent problem.
• Find the unknown in a percent problem.

Introduction

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

Parts of a Percent Problem

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

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

• The percent has the percent symbol (%) or the word “percent.” In the problem above, 15% is the percent off the purchase price.
• The base is the whole amount. In the problem above, the whole price of the guitar is $220, which is the base.
• The amount is the number that relates to the percent. It is always part of the whole. In the problem above, the amount is unknown. Since the percent is the percent off , the amount will be the amount off of the price.

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

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

30 is 20% of what number?

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

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

Amount: The amount based on the percent is 30.

Percent=20%

Base=unknown

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

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

What percent of 30 is 3?

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

Solving with Equations

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

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

The percent of the base is the amount.

Percent of the Base is the Amount.

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

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

Write an equation that represents the following problem.

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

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

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

When you have an equation such as \(\ 20 \% \cdot n=30\), you can divide 30 by 20% to find the unknown: \(\ n=30 \div 20 \%\).

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

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

What percent of 72 is 9?

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

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

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

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

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

What is 110% of 24?

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

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

18 is what percent of 48?

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

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

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

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

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

Using Proportions to Solve Percent Problems

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

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

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

30 is 20% of 150.

18 is 125% of what number?

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

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

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

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

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

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

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

The coupon will take $33 off the original price.

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

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

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

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

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

The books cost $30 before tax.

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

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

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

Chapter 6, Lesson 5: Problems Involving Percents

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Solving Percent Problems

Videos to help Grade 6 students solve percent problems. When given a part and the percent, students find the percent of a quantity and solve problems involving finding the whole.

New York State Common Core Math Module 1, Grade 6, Lesson 29

Lesson 29 outcome.

• Students find the percent of a quantity. • Given a part and the percent, students solve problems involving finding the whole.

Lesson 29 Summary

Claim: To find 10% of a number all you need to do is move the decimal to the left once.

Use at least one model to solve each problem (e.g., tape diagram, table, double number line diagram, 10x10 grid).

Claim: If an item is already on sale and then there is another discount taken off the sale price, this is the same as saving the sum of the two discounts from the original price.

Use at least one model to solve each problem (e.g., tape diagram, table, double number line diagram, 10 x 10 grid).

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Free Printable Percent Problems Worksheets for 5th Grade

Math Percent Problems: Discover a collection of free printable worksheets for Grade 5 students, designed to help them master the world of percentages. Dive into these educational resources and enhance their mathematical skills today!

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Percent Problems worksheets for Grade 5 are an essential tool for teachers looking to enhance their students' understanding of math concepts, specifically focusing on percents, ratios, and rates. These worksheets are designed to help students in Grade 5 develop a strong foundation in the subject, as they learn to solve problems involving percents, calculate ratios, and determine rates. By incorporating these worksheets into their lesson plans, teachers can provide their students with ample opportunities to practice and apply their newly acquired skills. Furthermore, these Grade 5 Percent Problems worksheets can be easily adapted to suit the needs of individual learners, ensuring that each student receives the support they require to excel in math.

In addition to Percent Problems worksheets for Grade 5, teachers can also utilize Quizizz as a complementary resource to reinforce their students' understanding of math concepts, such as percents, ratios, and rates. Quizizz offers a wide range of interactive quizzes and games that can be seamlessly integrated into the classroom, providing students with a fun and engaging way to practice their skills. Furthermore, Quizizz allows teachers to track their students' progress, ensuring that they can identify areas where additional support may be needed. By combining the use of Grade 5 Percent Problems worksheets with the interactive features of Quizizz, teachers can create a comprehensive and effective learning experience that caters to the diverse needs of their students, helping them to achieve success in the world of math.

Curriculum  /  Math  /  6th Grade  /  Unit 2: Unit Rates and Percent  /  Lesson 5

Unit Rates and Percent

Lesson 5 of 14

Criteria for Success

Tips for teachers, anchor problems, problem set, target task, additional practice.

Solve challenging problems involving unit rate.

Common Core Standards

Core standards.

The core standards covered in this lesson

Ratios and Proportional Relationships

6.RP.A.2 — Understand the concept of a unit rate a/b associated with a ratio a:b with b ≠ 0, and use rate language in the context of a ratio relationship. Expectations for unit rates in this grade are limited to non-complex fractions. For example, "This recipe has a ratio of 3 cups of flour to 4 cups of sugar, so there is 3/4 cup of flour for each cup of sugar." "We paid $75 for 15 hamburgers, which is a rate of $5 per hamburger."

6.RP.A.3 — Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.

6.RP.A.3.B — Solve unit rate problems including those involving unit pricing and constant speed. For example, if it took 7 hours to mow 4 lawns, then at that rate, how many lawns could be mowed in 35 hours? At what rate were lawns being mowed?

The essential concepts students need to demonstrate or understand to achieve the lesson objective

• Determine a solution pathway and strategy to solve complex problems involving rate (MP.1).
• Organize information and workspace to keep track of the solution pathway.

Suggestions for teachers to help them teach this lesson

Students have a variety of strategies at their hands to model and solve complex problems involving rate. The problems are open to students taking different approaches; students should clearly organize and explain their strategies so that others, who may have taken a different approach, can follow their line of thinking (MP.4).

Lesson Materials

• Calculators (1 per student) — See note from lesson 1.

Unlock features to optimize your prep time, plan engaging lessons, and monitor student progress.

Problems designed to teach key points of the lesson and guiding questions to help draw out student understanding

Chris and David run along a bike path toward a pond. Chris can run 3 miles in 30 minutes, and David can run 5 miles in 60 minutes. They both start running at the same time at the start of the bike path, shown below.

a.   If both Chris and David run at their current rates, how long will it take each one to get to the pond?

b.   Who will be closer to the pond after 90 minutes? How much farther ahead will this person be in front of the other person? 

Guiding Questions

A set of suggested resources or problem types that teachers can turn into a problem set

Give your students more opportunities to practice the skills in this lesson with a downloadable problem set aligned to the daily objective.

A task that represents the peak thinking of the lesson - mastery will indicate whether or not objective was achieved

Market Place is selling chicken for $4.50 per pound. Stop and Buy is selling 5 pounds of chicken for $23.75. You need to buy 8 pounds of chicken. At these rates, which store is cheaper? How much cheaper is it?

Student Response

An example response to the Target Task at the level of detail expected of the students.

The following resources include problems and activities aligned to the objective of the lesson that can be used for additional practice or to create your own problem set.

• Include a variety of review problems from earlier in the unit, including some error analysis problems.
• Illustrative Mathematics Friends Meeting on Bicycles — Challenge: This is a lengthy and increasingly challenging problem with multiple ways to solve it.
• Dan Meyer's Three-Act Math Shower v. Bath
• Dan Meyer's Three-Act Math Split Time

Topic A: Defining Rate & Solving Rate Problems

Investigate and use rate in real-world situations.

6.RP.A.3 6.RP.A.3.B

Define rate and unit rate, and find rates from situations involving ratios. 

6.RP.A.2 6.RP.A.3.B

Find unit rates and use them to solve problems.

Compare situations using unit rates, including speed, price, and work problems.

6.RP.A.2 6.RP.A.3 6.RP.A.3.B

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Topic B: Measurement Unit Conversions

Reason with measurement units and solve unit conversion problems using different strategies.

Solve measurement unit conversion problems.

Topic C: Percent

Define percent as a rate per 100. Find percentages of 100 and 1.

Solve percent problems using benchmark fractions and percentages, with and without written work.

Convert between fractions, percent, and decimals.

Find a percent of a quantity. 

Find the whole quantity, given a part and percent.

Solve percent problems, including finding percent when given a part and a whole. 

Solve multi-step problems involving rate and percent.

6.RP.A.2 6.RP.A.3 6.RP.A.3.C 6.RP.A.3.D

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