problem solving about percentage

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Course: 7th grade   >   Unit 2

Solving percent problems.

• Equivalent expressions with percent problems
• Percent word problem: magic club
• Percent problems
• Percent word problems: tax and discount
• Tax and tip word problems
• Percent word problem: guavas
• Discount, markup, and commission word problems
• Multi-step ratio and percent problems

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Video transcript

Word Problems on Percentage

Word problems on percentage will help us to solve various types of problems related to percentage. Follow the procedure to solve similar type of percent problems.

Word problems on percentage:

1.  In an exam Ashley secured 332 marks. If she secured 83 % makes, find the maximum marks.

Let the maximum marks be m.

Ashley’s marks = 83% of m

Ashley secured 332 marks

Therefore, 83% of m = 332

⇒ 83/100 × m = 332

⇒ m = (332 × 100)/83

⇒ m =33200/83

Therefore, Ashley got 332 marks out of 400 marks.

2. An alloy contains 26 % of copper. What quantity of alloy is required to get 260 g of copper?

Let the quantity of alloy required = m g

Then 26 % of m =260 g

⇒ 26/100 × m = 260 g

⇒ m = (260 × 100)/26 g

⇒ m = 26000/26 g

⇒ m = 1000 g

3. There are 50 students in a class. If 14% are absent on a particular day, find the number of students present in the class.

Solution:             

Number of students absent on a particular day = 14 % of 50

                                          i.e., 14/100 × 50 = 7

Therefore, the number of students present = 50 - 7 = 43 students.

4. In a basket of apples, 12% of them are rotten and 66 are in good condition. Find the total number of apples in the basket.

Solution:             

Let the total number of apples in the basket be m

12 % of the apples are rotten, and apples in good condition are 66

Therefore, according to the question,

88% of m = 66

⟹ 88/100 × m = 66

⟹ m = (66 × 100)/88

⟹ m = 3 × 25

Therefore, total number of apples in the basket is 75.

5. In an examination, 300 students appeared. Out of these students; 28 % got first division, 54 % got second division and the remaining just passed. Assuming that no student failed; find the number of students who just passed.

The number of students with first division = 28 % of 300

                                                             = 28/100 × 300

                                                             = 8400/100

                                                             = 84

And, the number of students with second division = 54 % of 300

                                                                        = 54/100 × 300

                                                                        =16200/100

                                                                        = 162

Therefore, the number of students who just passed = 300 – (84 + 162)

                                                                           = 54

Questions and Answers on Word Problems on Percentage:

1. In a class 60% of the students are girls. If the total number of students is 30, what is the number of boys?

2. Emma scores 72 marks out of 80 in her English exam. Convert her marks into percent.

Answer: 90%

3. Mason was able to sell 35% of his vegetables before noon. If Mason had 200 kg of vegetables in the morning, how many grams of vegetables was he able to see by noon?

Answer: 70 kg

4. Alexander was able to cover 25% of 150 km journey in the morning. What percent of journey is still left to be covered?

Answer:  112.5 km

5. A cow gives 24 l milk each day. If the milkman sells 75% of the milk, how many liters of milk is left with him?

Answer: 6 l

6.  While shopping Grace spent 90% of the money she had. If she had $ 4500 on shopping, what was the amount of money she spent?

Answer:  $ 4050

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Solving problems with percentages

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

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

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

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

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

x/100 is called the rate.

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

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

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

$$a=r\cdot b$$

$$47\%=0.47a$$

$$=0.47\cdot 34$$

$$a=15.98\approx 16$$

16 of the students wear either glasses or contacts.

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

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

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

$$240-150=90$$

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

$$90=r\cdot 150$$

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

$$0.6=r= 60\%$$

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

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

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

$$1.6-1=0.6$$

$$0.6=60\%$$

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

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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.

Module 1: Whole Numbers, Fractions, Decimals, Percents and Problem Solving

Solving problems using percents, learning outcome.

• Evaluate expressions and word problems involving percents

In this section we will solve percent questions by identifying the parts of the problem. We’ll look at a common application of percent—tips to a server at a restaurant—to see how to set up a basic percent application.

When Aolani and her friends ate dinner at a restaurant, the bill came to [latex]\text{\$80}[/latex]. They wanted to leave a [latex]20\%[/latex] tip. What amount would the tip be?

To solve this, we want to find what amount is [latex]20\%[/latex] of [latex]\$80[/latex]. The [latex]\$80[/latex] is called the base . The percent is the given [latex]20\%[/latex]. The amount of the tip would be [latex]0.20(80)[/latex], or [latex]\$16[/latex] — see the image below. To find the amount of the tip, we multiplied the percent by the base.

A [latex]20\%[/latex] tip for an [latex]\$80[/latex] restaurant bill comes out to [latex]\$16[/latex].

Pieces of a Percent Problem

Percent problems involve three quantities:  the base amount (the whole), the percent , and the amount (a part of the whole or partial amount).

The amount is a percent of the base.

Let’s look at another example:

Jeff has a Guitar Strings coupon for [latex]15\%[/latex] off any purchase of [latex]$100[/latex] or more. He wants to buy a used guitar that has a price tag of [latex]$220[/latex] on it. Jeff wonders how much money the coupon will take off the original [latex]$220[/latex] price. Problems involving percents will have some combination of these 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, [latex]15\%[/latex] is the percent off the purchase price. The base is the whole amount or original amount. In the problem above, the “whole” price of the guitar is [latex]$220[/latex], which is the base. The amount is the unknown and what we will need to calculate.

There are thee cases: a missing amount, a missing percent or a missing base. Let’s take a look at each possibility.

Solving for the Amount

When solving for the amount in a percent problem, you will multiply the percent (as a decimal or fraction) by the base. Typically we choose the decimal value for percent.

[latex]\text{percent}\cdot{\text{base}}=\text{amount}[/latex]

Find [latex]50\%[/latex] of [latex]20[/latex]

First identify each piece of the problem:

percent: [latex]50\%[/latex] or [latex].5[/latex]

base: [latex]20[/latex]

amount: unknown

Now plug them into your equation [latex]\text{percent}\cdot{\text{base}}=\text{amount}[/latex]

[latex].5\cdot{20}= ?[/latex]

[latex].5\cdot{20}= 10[/latex]

Therefore, [latex]10[/latex] is the amount or part that is [latex]50\%[/latex] of [latex]20[/latex].

What is [latex]25\%[/latex] of [latex]80[/latex]?

The base is [latex]80[/latex] and the percent is [latex]25\%[/latex], so amount [latex]= 80(0.25) = 20[/latex]

Solving for the Percent

When solving for the percent in a percent problem, you will divide the amount by the base. The equation above is rearranged and the percent will come back as a decimal of fraction you can report in the form asked of you.

[latex]\Large{\frac{\text{amount}}{\text{base}}}\normalsize=\text{percent}[/latex]

What percent of [latex]320[/latex] is [latex]80[/latex]?

percent: unknown

base: [latex]320[/latex]

amount: [latex]80[/latex]

Now plug the values into your equation [latex]\Large{\frac{\text{amount}}{\text{base}}}\normalsize=\text{percent}[/latex]

[latex]\large\frac{80}{320}\normalsize=?[/latex]

[latex]\large\frac{80}{320}\normalsize=.25[/latex]

Therefore, [latex]80[/latex] is [latex]25\%[/latex] of [latex]320[/latex].

Solving for the Base

When solving for the base in a percent problem, you will divide the amount by the percent (as a decimal or fraction). The equation above is rearranged and you will find the base after plugging in the values.

[latex]\Large{\frac{\text{amount}}{\text{percent}}}\normalsize=\text{base}[/latex]

[latex]60[/latex] is [latex]40\%[/latex] of what number?

percent:[latex]40\%[/latex] or [latex].4[/latex]

base: unknown

amount: [latex]60[/latex]

Now plug the values into your equation [latex]\Large{\frac{\text{amount}}{\text{percent}}}\normalsize=\text{base}[/latex]

[latex](60)\div(.4)=?[/latex]

[latex](60)\div(.4)=150[/latex]

Therefore, [latex]60[/latex] is [latex]40\%[/latex] of [latex]150[/latex].

An article says that [latex]15\%[/latex] of a non-profit’s donations, about [latex]$30,000[/latex] a year, comes from individual donors.  What is the total amount of donations the non-profit receives?

The percent is [latex]15\%[/latex], and [latex]$30,000[/latex] is the amount (or part of the whole). We are looking for the base.

base  = [latex]30000\div(.15)=$200000[/latex]

The non-profit receives [latex]$200000[/latex] a year in donations

Here are a few more percent problems for you to try.

Many applications of percent occur in our daily lives, such as tips, sales tax, discount, and interest. To solve these applications we’ll translate to a basic percent equation, just like those we solved in the previous examples in this section. Once you translate the sentence into a percent equation, you know how to solve it.

Dezohn and his girlfriend enjoyed a dinner at a restaurant, and the bill was [latex]\text{\$68.50}[/latex]. They want to leave an [latex]\text{18%}[/latex] tip. If the tip will be [latex]\text{18%}[/latex] of the total bill, how much should the tip be?

In the next video we show another example of finding how much tip to give based on percent.

The label on Masao’s breakfast cereal said that one serving of cereal provides [latex]85[/latex] milligrams (mg) of potassium, which is [latex]\text{2%}[/latex] of the recommended daily amount. What is the total recommended daily amount of potassium?

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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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