# 7.4: Percent - Mathematics

Learning Objectives

• understand the relationship between ratios and percents
• be able to make conversions between fractions, decimals, and percents

## Ratios and Percents

Ratio, Percent
We defined a ratio as a comparison, by division, of two pure numbers or two like denominate numbers. A most convenient number to compare numbers to is 100. Ratios in which one number is compared to 100 are called percents. The word percent comes from the Latin word "per centum." The word "per" means "for each" or "for every," and the word "centum" means "hundred." Thus, we have the following definition.

Percent means “for each hundred," or "for every hundred."

The symbol % is used to represent the word percent.

Sample Set A

The ratio 26 to 100 can be written as 26%. We read 26% as "twenty-six percent."

Sample Set A

The ratio (dfrac{165}{100}) can be written as 165%.

We read 165% as "one hundred sixty-five percent."

Sample Set A

The percent 38% can be written as the fraction (dfrac{38}{100}).

Sample Set A

The percent 210% can be written as the fraction (dfrac{210}{100}) or the mixed number (2dfrac{1)}{100}) or 2.1.

Sample Set A

Since one dollar is 100 cents, 25 cents is (dfrac{25}{100}) of a dollar. This implies that 25 cents is 25% of one dollar.

Practice Set A

Write the ratio 16 to 100 as a percent.

16%

Practice Set A

Write the ratio 195 to 100 as a percent.

195%

Practice Set A

Write the percent 83% as a ratio in fractional form.

(dfrac{83}{100})

Practice Set A

Write the percent 362% as a ratio in fractional form.

(dfrac{362}{100}) or (dfrac{181}{50})

## The Relationship Between Fractions, Decimals, and Percents – Making Conversions

Since a percent is a ratio, and a ratio can be written as a fraction, and a fraction can be written as a decimal, any of these forms can be converted to any other.

Before we proceed to the problems in Sample Set B and Practice Set B, let's summarize the conversion techniques.

 To Convert a Fraction To Convert a Decimal To Convert a Percent To a decimal: Divide the numerator by the denominator To a fraction: Read the decimal and reduce the resulting fraction To a decimal: Move the decimal point 2 places to the left and drop the % symbol To a percent: Convert the fraction first to a decimal, then move the decimal point 2 places to the right and affix the % symbol. To a percent: Move the decimal point 2 places to the right and affix the % symbol To a fraction: Drop the % sign and write the number “over” 100. Reduce, if possible.

Sample Set B

Convert 12% to a decimal.

Solution

(12\% = dfrac{12}{100} = 0.12)

Note that

The % symbol is dropped, and the decimal point moves 2 places to the left.

Sample Set B

Convert 0.75 to a percent.

Solution

(0.75 = dfrac{75}{100} = 75\%)

Note that

The % symbol is affixed, and the decimal point moves 2 units to the right.

Sample Set B

Convert (dfrac{3}{5}) to a percent.

Solution

We see in Example above that we can convert a decimal to a percent. We also know that we can convert a fraction to a decimal. Thus, we can see that if we first convert the fraction to a decimal, we can then convert the decimal to a percent.

(dfrac{3}{5} o egin{array} {r} {.6} {5overline{)3.0}} {underline{3 0}} {0} end{array} ext{ or } dfrac{3}{5} = 0.6 = dfrac{6}{10} = dfrac{60}{100} = 60\%)

Sample Set B

Convert 42% to a fraction.

Solution

(42\% = dfrac{42}{100} = dfrac{21}{50})

or

(42\% = 0.42 = dfrac{42}{100} = dfrac{21}{50})

Practice Set B

Convert 21% to a decimal.

0.21

Practice Set B

Convert 461% to a decimal.

4.61

Practice Set B

Convert 0.55 to a percent.

55%

Practice Set B

Convert 5.64 to a percent.

564%

Practice Set B

Convert (dfrac{3}{20}) to a percent.

15%

Practice Set B

Convert (dfrac{11}{8}) to a percent.

137.5%

Practice Set B

Convert (dfrac{3}{11}) to a percent.

(27.overline{27})%

## Exercises

For the following 12 problems, convert each decimal to a percent.

Exercise (PageIndex{1})

0.25

25%

Exercise (PageIndex{2})

0.36

Exercise (PageIndex{3})

0.48

48%

Exercise (PageIndex{4})

0.343

Exercise (PageIndex{5})

0.771

77.1%

Exercise (PageIndex{6})

1.42

Exercise (PageIndex{7})

2.58

258%

Exercise (PageIndex{8})

4.976

Exercise (PageIndex{9})

16.1814

1,618.14%

Exercise (PageIndex{10})

533.01

Exercise (PageIndex{11})

2

200%

Exercise (PageIndex{12})

14

For the following 10 problems, convert each percent to a deci­mal.

Exercise (PageIndex{13})

15%

0.15

Exercise (PageIndex{14})

43%

Exercise (PageIndex{15})

16.2%

0.162

Exercise (PageIndex{16})

53.8%

Exercise (PageIndex{17})

5.05%

0.0505

Exercise (PageIndex{18})

6.11%

Exercise (PageIndex{19})

0.78%

0.0078

Exercise (PageIndex{20})

0.88%

Exercise (PageIndex{21})

0.09%

0.0009

Exercise (PageIndex{22})

0.001%

For the following 14 problems, convert each fraction to a per­cent.

Exercise (PageIndex{23})

(dfrac{1}{5})

20%

Exercise (PageIndex{24})

(dfrac{3}{5})

Exercise (PageIndex{25})

(dfrac{5}{8})

62.5%

Exercise (PageIndex{26})

(dfrac{1}{16})

Exercise (PageIndex{27})

(dfrac{7}{25})

28%

Exercise (PageIndex{28})

(dfrac{16}{45})

Exercise (PageIndex{29})

(dfrac{27}{55})

(49.overline{09})%

Exercise (PageIndex{30})

(dfrac{15}{8})

Exercise (PageIndex{31})

(dfrac{41}{25})

164%

Exercise (PageIndex{32})

(6 dfrac{4}{5})

Exercise (PageIndex{33})

(9 dfrac{9}{20})

945%

Exercise (PageIndex{34})

(dfrac{1}{200})

Exercise (PageIndex{35})

(dfrac{6}{11})

(54.overline{54})%

Exercise (PageIndex{36})

(dfrac{35}{27})

For the following 14 problems, convert each percent to a fraction.

Exercise (PageIndex{37})

80%

(dfrac{4}{5})

Exercise (PageIndex{38})

60%

Exercise (PageIndex{37})

25%

(dfrac{1}{4})

Exercise (PageIndex{38})

75%

Exercise (PageIndex{37})

65%

(dfrac{13}{20})

Exercise (PageIndex{38})

18%

Exercise (PageIndex{37})

12.5%

(dfrac{1}{8})

Exercise (PageIndex{38})

37.5%

Exercise (PageIndex{37})

512.5%

(dfrac{41}{8}) or (5 dfrac{1}{8})

Exercise (PageIndex{38})

937.5%

Exercise (PageIndex{37})

(9.overline{9})%

(dfrac{1}{10})

Exercise (PageIndex{38})

(55.overline{5})%

Exercise (PageIndex{37})

(22.overline{2})%

(dfrac{2}{9})

Exercise (PageIndex{38})

(63.overline{6})%

#### Exercises for Review

Exercise (PageIndex{39})

Find the quotient. (dfrac{40}{54} div 8 dfrac{7}{21}).

(dfrac{2}{9})

Exercise (PageIndex{40})

(dfrac{3}{8}) of what number is (2dfrac{2}{3})?

Exercise (PageIndex{41})

Find the value of (dfrac{28}{15} + dfrac{7}{10} - dfrac{5}{12}).

(dfrac{129}{60}) or (2 dfrac{9}{60} = 2 dfrac{3}{20})

Exercise (PageIndex{42})

Round 6.99997 to the nearest ten thousandths.

Exercise (PageIndex{43})

On a map, 3 inches represent 40 miles. How many inches represent 480 miles?

36 inches

## How to Change Any Number to a Percent, With Examples

Understanding and calculating percentages can help you in many ways: working out the correct tip at a restaurant, knowing how much you are saving on that mega sale or allowing you to interpret data from mathematical and scientific research. In short, learning more about percentages is important for chemistry and all other areas.

A percentage is a way of expressing one number as a portion or share of a whole number, and percentages are always based on their relation to 100, which represents the whole number or object. For example, 75% is the same as 75 out of 100. Any percentage lower than 100 is just part of the whole or total.

Percentages are ratios, they can, therefore, be written as fractions and then decimals. Converting answers from percentages to fractions to decimals can be a good exercise to check the accuracy of your work.You can convert any number to a percent.

## 9.3. Mathematical Functions and Operators

Mathematical operators are provided for many PostgreSQL types. For types without common mathematical conventions for all possible permutations (e.g., date/time types) we describe the actual behavior in subsequent sections.

Table 9-2 shows the available mathematical operators.

Table 9-2. Mathematical Operators

The bitwise operators are also available for the bit string types bit and bit varying, as shown in Table 9-3. Bit string operands of &, |, and # must be of equal length. When bit shifting, the original length of the string is preserved, as shown in the table.

Table 9-3. Bit String Bitwise Operators

Table 9-4 shows the available mathematical functions. In the table, dp indicates double precision. Many of these functions are provided in multiple forms with different argument types. Except where noted, any given form of a function returns the same data type as its argument. The functions working with double precision data are mostly implemented on top of the host system's C library accuracy and behavior in boundary cases may therefore vary depending on the host system.

Table 9-4. Mathematical Functions

Function Return Type Description Example Result
abs (x) (same as x) absolute value abs(-17.4) 17.4
cbrt (dp) dp cube root cbrt(27.0) 3
ceil (dp or numeric) (same as input) smallest integer not less than argument ceil(-42.8) -42
degrees (dp) dp radians to degrees degrees(0.5) 28.6478897565412
exp (dp or numeric) (same as input) exponential exp(1.0) 2.71828182845905
floor (dp or numeric) (same as input) largest integer not greater than argument floor(-42.8) -43
ln (dp or numeric) (same as input) natural logarithm ln(2.0) 0.693147180559945
log (dp or numeric) (same as input) base 10 logarithm log(100.0) 2
log (b numeric, x numeric) numeric logarithm to base b log(2.0, 64.0) 6.0000000000
mod (y, x) (same as argument types) remainder of y/x mod(9,4) 1
pi () dp "π" constant pi() 3.14159265358979
pow (a dp, b dp) dp a raised to the power of b pow(9.0, 3.0) 729
pow (a numeric, b numeric) numeric a raised to the power of b pow(9.0, 3.0) 729
random () dp random value between 0.0 and 1.0 random()
round (dp or numeric) (same as input) round to nearest integer round(42.4) 42
round (v numeric, s integer) numeric round to s decimal places round(42.4382, 2) 42.44
setseed (dp) int32 set seed for subsequent random() calls setseed(0.54823) 1177314959
sign (dp or numeric) (same as input) sign of the argument (-1, 0, +1) sign(-8.4) -1
sqrt (dp or numeric) (same as input) square root sqrt(2.0) 1.4142135623731
trunc (dp or numeric) (same as input) truncate toward zero trunc(42.8) 42
trunc (v numeric, s integer) numeric truncate to s decimal places trunc(42.4382, 2) 42.43

Finally, Table 9-5 shows the available trigonometric functions. All trigonometric functions take arguments and return values of type double precision.

## 7.4: Percent - Mathematics

Introducing Proportional Relationships

• Representing Proportional Relationships with Tables
• Representing Proportional Relationships with Equations
• Comparing Proportional and Nonproportional Relationships
• Representing Proportional Relationships with Graphs
• Let's Put it to Work

### Unit 4

Proportional Relationships and Percentages

• Proportional Relationships with Fractions
• Percent Increase and Decrease
• Applying Percentages
• Let's Put it to Work

### Unit 5

Rational Number Arithmetic

• Interpreting Negative Numbers
• Adding and Subtracting Rational Numbers
• Multiplying and Dividing Rational Numbers
• Four Operations with Rational Numbers
• Solving Equations When There Are Negative Numbers
• Let's Put It to Work

### Unit 6

Expressions, Equations, and Inequalities

• Representing Situations of the Form $px+q=r$ and $p(x+q)=r$
• Solving Equations of the Form $px+q=r$ and $p(x+q)=r$ and Problems That Lead to Those Equations
• Inequalities
• Writing Equivalent Expressions
• Let's Put it to Work

### Unit 7

Angles, Triangles, and Prisms

• Angle Relationships
• Drawing Polygons with Given Conditions
• Solid Geometry
• Let's Put It to Work

### Unit 8

• Probabilities of Single Step Events
• Probabilities of Multi-step Events
• Sampling
• Using Samples
• Let's Put it to Work

### Unit 9

IM 6–8 Math was originally developed by Open Up Resources and authored by Illustrative Mathematics®, and is copyright 2017-2019 by Open Up Resources. It is licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0). OUR's 6–8 Math Curriculum is available at https://openupresources.org/math-curriculum/.

The second set of English assessments (marked as set "B") are copyright 2019 by Open Up Resources, and are licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0).

Spanish translation of the "B" assessments are copyright 2020 by Illustrative Mathematics, and are licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0).

The Illustrative Mathematics name and logo are not subject to the Creative Commons license and may not be used without the prior and express written consent of Illustrative Mathematics.

## Percentage Calculator

Use Alcula's percentage calculator to compute percentages and answer questions such as:

• How much is 7% of 25000?
• What percentage of 10000 is 120?
• 250 is 8 percent of what amount?
• How much is 12000+8%

In the calculator window, choose the question you need answered and enter the 2 quantities that you already know. The calculated result will automatically display on the right of the question you chose, along with the answers to all the other questions.

To calculate percentage change, use one of the three calculators at the bottom.

The percentage increase calculator calculates the chosen percentage from the initial quantity and adds it to the initial quantity to calculate the quantity after the increase.

Similarly, the percentage decrease calculator subtracts the chosen percentage of the initial quantity from the initial quantity.

Finally, the percent change calculator takes as input an initial quantity and a final quantity and calculates the difference as a percentage.

## Quartiles

Another related idea is Quartiles, which splits the data into quarters:

### Example: 1, 3, 3, 4, 5, 6, 6, 7, 8, 8

The numbers are in order. Cut the list into quarters:

In this case Quartile 2 is half way between 5 and 6:

The Quartiles also divide the data into divisions of 25%, so:

• Quartile 1 (Q1) can be called the 25th percentile
• Quartile 2 (Q2) can be called the 50th percentile
• Quartile 3 (Q3) can be called the 75th percentile

### Example: (continued)

For 1, 3, 3, 4, 5, 6, 6, 7, 8, 8:

• The 25th percentile = 3
• The 50th percentile = 5.5
• The 75th percentile = 7

## Practice questions

1. A survey of American car buyers indicates that if a person buys a Ford, there is a 60% chance that their next purchase will be a Ford, while owners of a GM will buy a GM again with a probability of 0.80. Express the buying habits of these consumers in a transition matrix.

2. A hockey player decides to either shoot the puck (S) or pass it to a teammate (P) according to the following transition matrix.

a. If the player shot on the first play, what is the probability that he will pass on the third play?

b. What is the long-term shoot vs. pass distribution of this player?

3. The local police department conducts a campaign to reduce the rates of texting and driving in the community. The effects of the campaign are summarized in the transition matrix below:

If 35% of people in the community reported texting and driving before the campaign:

a. What is the percentage of people in the community that reported texting and driving after the campaign?

b. If the campaign were to be repeated multiple times, what is the long-range trend in terms of the lowest rate that texting and driving can be reduced to in this community?

4. A large company conducted a training program with their employees to reduce the incidence of slips, trips and falls in the workplace. About 15% of workers reported a slip, trip or fall accident the previous year (year 1). After the training program (year 2), 75% of those who previously reported an accident reported no further accidents, while 5% of those who didn’t report a previous accident reported one this year.

a. Create a transition matrix for this scenario.

b. If the company employs 8500 workers, how many slip, trip and fall accidents were reported in year 2?

c. If the program continued for another year, how many accidents would be reported in year 3?

d. If the training program were to be repeated for many years, what is the lowest prevalence of slip, trip or fall accidents that could be achieved?

## 3 Simple Ways to Calculate Percentages (Math)

How to calculate percentages is easier than you think. Learning this can help you to easily calculate tips at restaurants and how to use percentages to easily calculate sales prices when shopping.

If you’re not sure how to perform any of those handy calculations, or if you’re just in need of a general percentage refresher, check out our guide on how to calculate percentages below.

### 1. Calculating the Percentage of a Whole

To calculate a percentage, the whole amount must be known. This is in addition to the percentage or portion amount. You may be asked “what percentage of W is P,” where W is the whole amount and P is the portion amount. Or the question may be “how much is X percent of W,” where X represents a percentage figure.

##### 1. What is a percentage?

A percentage is a way to express a number as a part of a whole. To calculate a percentage, we look at the whole as equal to 100%. For example, say you have 10 apples (=100%). If you eat 2 apples, then you have eaten 2/10 × 100% = 20% of your apples and you are left with 80% of your original apples.

The term “percent” in English comes from the Italian per cento or the French pour cent, which literally mean per hundred.

##### 2. What is the value of the whole?

For instance, let’s say we have a jar containing 1199 red marbles and 485 blue marbles, making it 1684 marbles in total. In this case, 1684 makes up a whole jar of marbles and will be set equal to 100%.

##### 3. Turn the value into percentage

Let’s say we want to find out the percentage of the jar that is taken up by the 485 blue marbles.

##### 4. Put the two values into a fraction

In our example, we need to find out what percent 485 (number of blue marbles) is of 1684 (total number of marbles). Therefore the fraction, in this case, is 485/1684.

##### 5. Convert the fraction to a decimal.

To turn 485/1684 to a decimal, divide 485 by 1684. This comes to 0.288.

##### 6. Convert the decimal into a percent

Multiply the result obtained in the step above by 100. For this example, 0.288 multiplied by 100 equals 28.8 or 28.8%.

Formula: 0.288 x 100 = 28.8 or 28.8%

A simple way to multiply a decimal by 100 is to move the decimal to the right two places.

### 2. Reverse Percentage

You may come across a question that will ask you to work backward and find the original price of something after the price has increased. If you are given a quantity after a percentage increase or decrease, you may need to find the original amount.

##### 1. When to do reverse percentage?

Sometimes you’re given the percentage of an amount and need to know the numerical value of the percent. Examples include calculating taxes, tips, and loan interest.

## Lesson 7 Summary

We can use a double number line diagram to show information about percent increase and percent decrease:

The initial amount of cereal is 500 grams, which is lined up with 100% in the diagram. We can find a 20% increase to 500 by adding 20% of 500:

egin500+(0.2)oldcdot 500 &= (1.20)oldcdot 500&=600end

In the diagram, we can see that 600 corresponds to 120%.

If the initial amount of 500 grams is decreased by 40%, we can find how much cereal there is by subtracting 40% of the 500 grams:

egin500−(0.4)oldcdot 500 &= (0.6)oldcdot 500&=300end

So a 40% decrease is the same as 60% of the initial amount. In the diagram, we can see that 300 is lined up with 60%.

To solve percentage problems, we need to be clear about what corresponds to 100%. For example, suppose there are 20 students in a class, and we know this is an increase of 25% from last year. In this case, the number of students in the class last year corresponds to 100%. So the initial amount (100%) is unknown and the final amount (125%) is 20 students.

Looking at the double number line, if 20 students is a 25% increase from the previous year, then there were 16 students in the class last year.

## How to calculate percentages

Learn how to calculate percentages in this easy lesson! When you're asked to calculate an (unknown) percentage ("What percentage. "), you need to first write the fraction PART/TOTAL, and then simply write that fraction as a decimal and as a percentage. See many examples below.

The concepts and ideas of this lesson are also explained in this video:

 What percentage of the height of a 15-ft tree is a 3-ft sapling? A choir has 22 women and 18 men. Find what percentage of the choir&rsquos members are men. One pair of jeans costs $25 and another costs$28. How many percent is the price of cheaper jeans of the price of the more expensive jeans?

Look carefully at the questions above. Notice that the problems don&rsquot tell you the percentage in other words, there is no number in the problem written as x%. Instead, they ask you to find it!

Asking &ldquoWhat percentage?&rdquo or "How many percent?" is the same as asking &ldquoHow many hundredth parts?&rdquo

We can solve these questions in a two-part process:

1. First find out the part that is being asked for as a fraction. The denominator probably won&rsquot be 100.
2. Convert that fraction to a decimal. Then you can easily convert the decimal to a percentage!

Example 1. A choir has 22 women and 18 men. Find what percentage of the choir&rsquos members are men.

1. Find out what part (fraction) of the choir&rsquos members are men. That is 18/40, or 9/20.
2. Write 9/20 as a percent. Use equivalent fractions: 9/20 = 45/100 = 45%.

Example 2. One pair of jeans costs $25 and the other costs$28. How many percent is the price of cheaper jeans of the price of the more expensive jeans?

1. Write what part the cheaper price is of the more expensive price. The answer is 25/28.
2. Write 25/28 as a percentage. A calculator gives 25/28 = 0.8928. Rounded to the nearest whole percent, that&rsquos 89%.

1. a. What percentage of a 15-ft tree is a little 3-ft sapling?

b. How many percent is $12 of$16?

2. Find how many percent the smaller object&rsquos height is of the taller object&rsquos height.

3. A 2-year old child measures 32 inches tall and weighs 24 pounds. A 10-year old child measures 52 inches tall and weighs 96 pounds.

a. How many percent is the smaller child&rsquos age of the older child&rsquos age?

b. How many percent is the smaller child&rsquos height of the older child&rsquos height?

4. Write the percentages into the sectors in the circle graphs Think of fractions!

5. The circle graph at the right gives the angle measure of each sector of the circle. Find what percentage each sector is of the whole circle, and write the percentage in the sector. Remember, the whole circle is 360°.