percentage rate base problem solving

Lesson Percentage word problems (Type 3 problems, Finding the Base)

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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.

Finding the Base Number in a Percent Problem Worksheet

Related Topics & Worksheets: Reverse Percentage Percentage Worksheet

Objective: I can find the base number in a percent problem.

Example: 8 is 32% of what number?

Answer: 25  

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Solving Percent Problems

Percent is a great mathematical tool to express quantities and is used extensively in different things – from interest rates, discounts, and taxes to surveys, censuses, etc.

This article is your guide to percent and solving percent problems frequently appearing in major national examinations.

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Table of Contents

What does percent mean.

The word “percent” originated from the Latin phrase per centum, meaning “by hundred.” When we say “percent,” we refer to “parts per 100”. This means that a percent is a fraction with 100 as the denominator. The symbol % is used to indicate a percent.

For example, 3% means three parts per 100 or 3⁄100; 45% means 45⁄100; and 92% means 92⁄100.

Illustrating Percent

Suppose a vendor has 100 biscuits. If 10% of those biscuits are ube-flavored, 10⁄100 or 10 out of 100 biscuits are ube-flavored.

On the other hand, suppose there are 100 students in a school auditorium. If 42% of those students are honor students, 42⁄100 students, or 42 out of 100 students, are honor students.

Expressing Percent as Fraction and Decimal

Since percent means a fraction with 100 as the denominator, we can express a percent as a fraction or a decimal number .

Drop the percent sign and put 100 as the denominator to transform a percent into a fraction. For instance, 25% is simply 25⁄100.

Note that when 25⁄100 is reduced to its lowest terms, you will obtain ¼. This means that 25% is also equivalent to ¼. 

Furthermore, note that when you transform ¼ into its decimal form using the steps we have discussed in the previous reviewer , you will obtain 0.25. Hence, 25% is also equal to 0.25. 

There is an easier way to transform percent into decimals . Drop the percent sign and move the decimal point two places to the left of the given number.

For example, 54% is equivalent to 0.54

Example: Transform 3% to decimal form.

Suppose that your mom prepared ten pieces of your favorite cookies. You are excited to taste those cookies, but you realize that your brother ate 20% of the cookies that your mom prepared. What exactly is the number of cookies eaten by your brother?

To determine the answer to your question above, you must determine 20% of 10. This case involves the application of percentages.

The percentage is the result when you multiply a number by a percent. Returning to your problem about the number of cookies your brother ate, 20% of 10 can be determined if you multiply ten by 20%. The result after you multiply the numbers is called the percentage.

How To Find the Percentage

Follow these steps if you want to find the percentage:

Step 1: Convert the given percent (the one with the % sign) into decimals .

Again, to convert percent into its decimal form, we drop the percent sign and then move the decimal point two places to the left. Thus, 20% = 0.20

Step 2 : Multiply the decimal you have obtained from Step 1 to the given number. The result is the percentage.

To multiply 0.20 by 10, we ignored the decimal point for a while and multiplied the given decimals like whole numbers. We have obtained 0200. Since 0.20 has two decimal places while 10 has none, the final answer should have two decimal places. We count two digits from the right of 0200 and put the decimal point there. Hence, the answer is 02.00, which is equivalent to 2.

Hence, 20% of 10 is 2. This means that out of 10 cookies your mother prepared, 2 of those were eaten by your brother.

Let us have another example.

Example: What is 50% of 120?

Step 1 : Convert the given percent (the one with the % sign) into decimals.

We drop the % sign of 50% and move the decimal point two places to the left.

Thus, 50% = 0.50

Step 2: Multiply the decimal you have obtained from Step 1 to the given number. The result is the percentage.

To multiply 0.50 by 120, we ignored the decimal point for a while and multiplied the given decimals like whole numbers. Through this process, we have obtained 06000. Since 0.50 has two decimal places while 120 has none, the final answer should have two decimal places. We count two digits from the right of 06000 and put the decimal point there. Hence, the answer is 060.00, which is equivalent to 60.

Hence, 50% of 120 is 60.

Simple Tricks in Computing Percentages

We always want to make our computations in mathematics faster and more accurate. For this reason, I will share two tricks you can use when computing percentages.

Trick #1: You can compute some percentages using only mental computation.

If you want to determine the 25%, 50%, 75%, or 100% of a number, you can do so without the help of pen and paper.

• 25% is equivalent to 25⁄100 or ¼. Hence, to find the 25% of a number, divide the given number by 4. Example: 25% of 40 is just 40 ÷ 4 = 10.
• 50% is equivalent to 50⁄100 or ½. Thus, to find the 50% of a number, divide the given number by 2. This means 50% of a number is just half the given number. Example: 50% of 40 is just 40 ÷ 2 = 20.
• 75% is equivalent to 75⁄100 or ¾. Thus, to find the 75% of a number, multiply the given number by three and then divide the result by 4. Example: 75% of 40 is just 40 x 3 = 120 ÷ 4 = 30.
• 100% is equivalent to 100⁄100 or 1. Thus, 100% of a number is the number itself . Example: 100% of 40 is just 40 itself.

Trick #2: X% of a number Y is equal to Y% of number X

This trick means we can transfer the % sign to the other number, and the result will be the same.

Example : What is 40% of 25?

Using trick #2, we can transfer the % sign from 40% to 25. Thus, we have 25%. This means 40% of 25 is the same as 25% of 40.

Thus, applying our first trick on finding the 25% of a number, 40 ÷ 4 = 10; hence, 40% of 25 is 10.

Example : What is 92% of 50? 

92% of 50 is the same as 50% of 92. Hence, we can just divide 92 by 2 to obtain the answer, 92 ÷ 2 = 46

Therefore, 92% of 50 is 46.

Base and Rate

The base is the amount you are taking a percent of. Meanwhile, the rate is the percent you are calculating.

For example, if there are 50 students in a classroom and 20% of those students are honor students, it follows that ten students are honor students. 50 is the base since it is the amount we take a percent of. Meanwhile, 20% is the rate since we calculate the percentage. Lastly, 10 is the percentage.

The product of the base and the rate is the percentage .

Percentage = Base × Rate

Example: Determine the percentage, base, and rate if 20% of 90 is 18.

Since 90 x 20% = 90 x 0.20 = 18, 90 is the base, 20% is the rate, and 18 is the percentage.

Calculating Percentage, Base, and Rate

Formula to find the percentage.

The formula to find the percentage, as we have stated, is: 

We can manipulate the mathematical equation above to obtain the formulas for computing the base and the rate:

Formula to Find the Base

Base = Percentage ÷ Rate

Formula to Find the Rate

Rate = Percentage ÷ Base

Example 1: If 10% of a number is 90, what is the number?

We can interpret this question as 10% of ______ = 90. Since “of” is a signal word for multiplication, it also implies 10% x ______ = 90

This means that 10% is the rate while 90 is the percentage. The unknown number is the base. Thus, we need to compute the base.

Using the formula to find the base:

  Base = Percentage ÷ Rate

Base  = 90 ÷ 10%

Convert the given percent into decimal:

Base  = 90 ÷ 0.10

Now that you have already transformed the rate into decimal form, you may divide 90 by 0.10 to obtain the answer.

To perform division with decimal numbers , we need to transform the divisor (0.10) into a whole number by moving two decimal places to the right. Thus, the new divisor is 10. We also move two decimal places for the dividend (90). Thus, the new dividend is 9000.

We now perform long division with our new dividend and divisor:

To find the base, we compute 90 ÷ 0.10 = 900

Hence, the base is 900.

Example 2:  What percent of 720 is 90?

We can translate the question above in this form: _____% of 720 is 90 or _____% x 720 = 90. Therefore, 720 is the base, while 90 is the percentage. The missing number is the rate.

We will now use the formula for finding the rate.

Again, based on the given problem, the percentage is 90 while the base is 720

          Rate = 90 ÷ 720

Notice that the dividend (the first number) is smaller than the divisor (the second number). In this case, you may apply the same steps in transforming fractions into decimal form because  90 ÷ 720 is a proper fraction (i.e., 90⁄720).

Let us divide 90 by 720 using the steps in transforming fractions into decimal form .

We add some zeros and decimal points to proceed with the division process.

We can now divide 900 by 720.

Note that every time the remainder becomes smaller than the divisor, we add zeros to 900 and the remainder to continue the division process.

The quotient we obtained is 0.125. Thus, 0.125 is our rate.

However, the rate must always be expressed with a percent sign. To do this, we multiply 0.125 by 100 or move two decimal places to the right of it and put a percent sign. Thus, 0.125 is equal to 12.5%.

Therefore, the rate is 12.5%

The Percentage, Base, and Rate Triangle

What if you forgot the formula to determine the percentage, base, or rate in a particular problem? Don’t worry because there is a fun way to derive these formulas. 

Shown below is the Percentage, Base, and Rate Triangle . It is a triangle divided into three portions where P (for percentage) is written on the upper portion, and B (for base) and R (for rate) are written on the lower portions. There are also division signs in the triangle’s outer left and outer right parts and a multiplication sign below it.

How To Use the Percentage, Base, and Rate Triangle

Suppose you are looking for the base. You have to cover the B in the triangle and look at the remaining letters and the operation between them. Notice that if you cover B, the remaining letters are P and R, with a division sign between them. This means that to find the base, you must divide P by R.

Next topic:  Ratio and Proportion

Previous topic : Fundamental Operations on Fractions and Decimals

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Written by Jewel Kyle Fabula

in Civil Service Exam , College Entrance Exam , LET , NAPOLCOM Exam , NMAT , PMA Entrance Exam , Reviewers , UPCAT

Jewel Kyle Fabula

Jewel Kyle Fabula is a Bachelor of Science in Economics student at the University of the Philippines Diliman. His passion for learning mathematics developed as he competed in some mathematics competitions during his Junior High School years. He loves cats, playing video games, and listening to music.

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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%

Video lessons

A skirt cost $35 regulary in a shop. At a sale the price of the skirtreduces with 30%. How much will the skirt cost after the discount?

Solve "54 is 25% of what number?"

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Definition:

The percent, base, and rate are connected with one another in terms of computation. To find the percentage , multiply the base by the rate. Remember that the rate must be changed from a percent to a decimal before multiplying can be done. Rate times base equals percentage.

PERCENTAGE (P=BxR) –  The result obtained when a number is multiplied by a percent.

BASE (B=P/R) –  The whole in a problem. The amount you are taking a percent of.

RATE (R=P/B) –  The ratio of amount to the base. It is written as a percent.

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Percent , Rate , Base

Understanding percent , rate , and base is essential in various mathematical and real-life contexts. In this study guide, we will cover the basics of percent , rate , and base , and provide examples to help you grasp these concepts.

Percent means "per hundred" and is denoted by the symbol "%". It is used to express a number as a fraction of 100. For example, 25% is equivalent to the fraction 25/100 or the decimal 0.25.

A rate is a special ratio in which the two terms are in different units . For example, miles per hour (mph) is a rate . It compares the distance traveled to the time taken. Rates are often expressed using the word "per" or the symbol "/", such as 60 miles per hour or 60 mph.

The base is the original value in a percent problem. It is the whole or the original amount before a percentage is calculated. For example, if you're calculating 20% of 80, then 80 is the base .

Key Formulas

The following formulas are essential when dealing with percent , rate , and base :

Percent = (Part / Whole) * 100

Rate = (Part / Base )

Base = (Part / Rate )

Let's work through a few examples to illustrate these concepts:

Example 1: Calculating Percent

If you scored 35 out of 50 on a test, what is your score as a percentage?

Percent = (35 / 50) * 100 = 70%

Example 2: Calculating Rate

If a car travels 300 miles in 5 hours , what is its speed in miles per hour ?

Rate = 300 miles / 5 hours = 60 mph

Example 3: Finding the Base

If 15 is 20% of a number, what is the original number?

Base = 15 / 0.20 = 75

When studying percent , rate , and base , it's helpful to practice converting between fractions , decimals , and percentages . Additionally, working through real-life problems involving discounts, taxes, and tips can improve your understanding of these concepts.

Remember to use the key formulas and units to guide your problem-solving process. Understanding the relationship between percent , rate , and base will also make it easier to solve problems in various scenarios.

By mastering percent , rate , and base , you'll develop a valuable skill set for handling a wide range of mathematical and practical situations.

Good luck with your studies!

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Grade 6 Mathematics Module: Finding the Percentage, Base and Rate in a Given Problem

This Self-Learning Module (SLM) is prepared so that you, our dear learners, can continue your studies and learn while at home. Activities, questions, directions, exercises, and discussions are carefully stated for you to understand each lesson.

Each SLM is composed of different parts. Each part shall guide you step-by-step as you discover and understand the lesson prepared for you.

Pre-tests are provided to measure your prior knowledge on lessons in each SLM. This will tell you if you need to proceed on completing this module or if you need to ask your facilitator or your teacher’s assistance for better understanding of the lesson. At the end of each module, you need to answer the post-test to self-check your learning. Answer keys are provided for each activity and test. We trust that you will be honest in using these.

Please use this module with care. Do not put unnecessary marks on any part of this SLM. Use a separate sheet of paper in answering the exercises and tests. And read the instructions carefully before performing each task.

If you have any questions in using this SLM or any difficulty in answering the tasks in this module, do not hesitate to consult your teacher or facilitator.

This module was designed and written with you in mind. It is here to help you master the lessons on Finding the Percentage, Base and Rate or Percent in given problems. The scope of this module permits it to be used in many different learning situations. The language used recognizes your diverse vocabulary level. The lessons are arranged to follow the standard sequence of the course. But the order in which you read them can be changed to correspond with the textbook you are now using.

The module is divided into three lessons, namely:

• Lesson 1 – Finding the Percentage in a Given Problem
• Lesson 2 – Finding the Rate in a Given Problem
• Lesson 3 – Finding the Base in a Given Problem

After going through this module, you are expected to:

1. identify the percentage, rate and base in a given problem;

2. find the base, percentage or rate or percent in a given problem; and

3. solve routine and non-routine problems involving the percentage, rate and base using appropriate strategies and tools.

Grade 6 Mathematics Quarter 2 Self-Learning Module: Finding the Percentage, Base and Rate in a Given Problem

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Determining Percentage, Base, & Rate

Cathy Gonzaga

This serves as additional worksheet to develop skill in identifying and solving for the percentage, base or rate in a given situation

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