maths.free › Arithmetic › 6. Money Management › Understanding Percent

Understanding Percent

Define and calculate percent.

Learning Objectives

After completing this section, you should be able to:

  1. Define and calculate percent.
  2. Convert between percent, decimal, and fractional values.
  3. Calculate the total, percent, or part.
  4. Solve application problems involving percents.

Define and Calculate Percent

The word percent comes from the Latin phrase per centum, which means “by the hundred.” So any percent is a number divided by 100. Changing a percent to a fraction is to write the percent in its fractional form. To write \(n\)% in its fractional form is to write the percent as the fraction \(\frac{n}{100}\).

Rewriting a Percent as a Fraction

Try it.

Rewrite the following as fractions:

  1. 18%
  2. 84%
  3. 38.7%
  4. 213%
Solution
  1. Using the definition and \(n\) = 18, 18% in fractional form is \(\frac{18}{100}\).
  2. Using the definition and \(n\) = 84, 84% in fractional form is \(\frac{84}{100}\).
  3. Using the definition and \(n\) = 38.7, 38.7% in fractional form is \(\frac{38.7}{100}\).
  4. Using the definition and \(n\) = 213, 213% in fractional form is \(\frac{213}{100}\).

Convert Between Percent, Decimal, and Fractional Values

When any calculation with a percent is to be performed, the form of the percent must be changed, either to its fractional form or its decimal form. We can change a percent into decimal form by dividing the percent by 100 and representing the result as a decimal.

Converting a Percent to Decimal Form

Try it.

Convert the following percents to decimal form:

  1. 17%
  2. 7%
  3. 18.45%
Solution
  1. To convert 17% to its decimal form, divide 17 by 100. This moves the decimal two places to the left, resulting in 0.17. The decimal form of 17% is 0.17.
  2. To convert 7% to its decimal form, divide 7 by 100. This moves the decimal two places to the left, resulting in 0.07. The decimal form of 7% is 0.07.
  3. To convert 18.45% to its decimal form, divide 18.45 by 100. This moves the decimal two places to the left, resulting in 0.1845. The decimal form of 18.45% is 0.1845.

You should notice that, to convert from percent to decimal form, you can simply move the decimal two places to the left without performing the division.

Converting the Decimal Form of a Percent to Percent

Try it.

Convert each of the following to percent:

  1. 0.34
  2. 4.15
  3. 0.0391
Solution
  1. Using the formula and \(x\) = 0.34, we calculate \((0.34\times 100)\%\), which gives us 34%.
  2. Using the formula and \(x\) = 4.15, we calculate \((4.15\times 100)\%\), which gives us 415%.
  3. Using the formula and \(x\) = 0.0391, we calculate \((0.0391\times 100)\%\), which gives us 3.91%.

You should notice that, to convert from decimal form to percent form, you can simply move the decimal two places to the right without performing the multiplication.

Calculate the Total, Percent, or Part

The word “of” is used to indicate multiplication using fractions, as in “one-fourth of 56.” To find “one-fourth of 56” we would multiply 56 by one-fourth. We can think of percents as fractions with a specific denominator—100. So, to calculate “25% of 52,” we multiply 52 by 25%. But, first we need to convert the percent to either fractional form (25/100) or decimal form. Using the decimal form of 25% we have 0.25 × 52, which equals 13.

In this problem, 52 is the total or base, 25 is the percentage, and 13 is the percentage of 52, or the part of 52. This is sometimes referred to as the amount.

Knowing any two of the values in our formula allows us to calculate the third value. In the following example, we know the total and the percent, and are asked to find the percentage of the total.

Finding the Percent of a Total

Try it.

  1. Determine 70% of 3,500
  2. Determine 156% of 720

Solution
  1. The total is \(x\) = 3,500, and the percent is \(n\) = 70. The decimal form of 70% is 0.70. To find the part, or percent of the total, substitute those values into the formula and calculate. \[\begin{array}{lll}\text{part} & = & \text{percent}\times \text{total} \\ & = & 0.70\times 3500 \\ & = & 2450\end{array}\] From this, we say that 70% of 3,500 is 2,450.
  2. The total is \(x\) = 720, and the percent is \(n\) = 156. The decimal form of 156% is 1.56. To find the part, or percent of the total, substitute those values into the formula and calculate. \[\begin{array}{lll}\text{part} & = & \text{percent}\times \text{total} \\ & = & 1.56\times 720 \\ & = & 1,123.2\end{array}\] From this, we say that 156% of 720 is 1,123.2.

In the previous example, we knew the total and the percent and found the part using our formula. We may instead know the percent and the part, but not the total. We can use our formula again to solve for the total.

Similarly, the percent can be found if the total and the percent of the total (the part) are known. This will result in the decimal form of the percent, so it must be converted to percent form.

Condensed — the full section is in OpenStax Contemporary Mathematics.

Solve Application Problems Involving Percents

Percents are frequently used in finance, research, science experiments, and even casual conversation. Understanding these types of values helps when consuming media or discussing finances, for instance. Effectively working with and interpreting numbers and percents will help you become an informed consumer of this information.

In most cases, working through what is presented requires you to identify that you are indeed working with a question of percents, which two of the three values that are related through percents are known, and which of the three values you need to find.

Retention Rate at College

Try it.

Justine applies to a medium size university outside her hometown and finds out that the retention rate (percent of students who return for their sophomore year) for the 2021 academic year at the university was 84%. During a visit to the registrar’s office, she finds out that 1,350 people had enrolled in academic year 2021. How many students from the academic year 2021 are returning for the 2022 academic year?

Solution

The percent of students who will return for the 2022 academic year (the retention rate) is 84%. The total number of students who enrolled in the 2021 academic year was 1,350. This means the percent is known and the total is known. From this, we can determine the number of students who will return (percent of the total) for the 2022 academic year using the formula \[\text{part}=\text{percent}\times \text{total}\]. Substituting into the formula and calculating, we find that the number of students that are returning is

\[\begin{array}{lll}\text{part} & = & \text{percent}\times \text{total} \\ & = & 0.84\times 1,350 \\ & = & 1,134\end{array}\]

So 1,134 students will return for the 2022 academic year.

Percent of Chemistry Majors

Try it.

Cameron enrolls in a calculus class. In this class of 45 students, there are 18 chemistry majors. What percent of the class are chemistry majors?

Solution

In this situation, the percent is to be determined. We know the total number of students, 45, and the part of the students that are chemistry majors, 18. Using that information and the formula \[\text{part}=\text{percent}\times \text{total}\], the percent can be found. Substituting and solving, we have

\[\begin{array}{lll}18 & = & \text{percent}\times 45 \\ \frac{18}{45} & = & \frac{\text{percent}\times 45}{45} \\ 0.4 & = & \frac{\text{percent}\times 45}{45} \\ 0.4 & = & \text{percent}\end{array}\]

Converting the 0.4 from decimal form, we find that 40% of the students in the calculus class are chemistry majors.

Condensed — the full section is in OpenStax Contemporary Mathematics.

Key Concepts

  • Know what a percent is as a fraction, a decimal, and as a part of the whole.
  • Use the percent equation to find any of the three values that are related by the equation.
  • Apply the percent equation in applications.

Video

  • Finding Percent of a Total
  • Finding the Total from the Percent and the Part
  • Finding the Percent When the Total and the Part Are Known

Practice (9)

Try each one on paper first. Reveal the answer to check; verified ones can be opened in the solver for every step.

  1. Rewrite the following as fractions:

    1. 18%
    2. 84%
    3. 38.7%
    4. 213%
    ಉತ್ತರವನ್ನು ತಿಳಿಸಿ
    1. Using the definition and \(n\) = 18, 18% in fractional form is \(\frac{18}{100}\).
    2. Using the definition and \(n\) = 84, 84% in fractional form is \(\frac{84}{100}\).
    3. Using the definition and \(n\) = 38.7, 38.7% in fractional form is \(\frac{38.7}{100}\).
    4. Using the definition and \(n\) = 213, 213% in fractional form is \(\frac{213}{100}\).

  2. Convert the following percents to decimal form:

    1. 17%
    2. 7%
    3. 18.45%
    ಉತ್ತರವನ್ನು ತಿಳಿಸಿ
    1. To convert 17% to its decimal form, divide 17 by 100. This moves the decimal two places to the left, resulting in 0.17. The decimal form of 17% is 0.17.
    2. To convert 7% to its decimal form, divide 7 by 100. This moves the decimal two places to the left, resulting in 0.07. The decimal form of 7% is 0.07.
    3. To convert 18.45% to its decimal form, divide 18.45 by 100. This moves the decimal two places to the left, resulting in 0.1845. The decimal form of 18.45% is 0.1845.

  3. Convert each of the following to percent:

    1. 0.34
    2. 4.15
    3. 0.0391
    ಉತ್ತರವನ್ನು ತಿಳಿಸಿ
    1. Using the formula and \(x\) = 0.34, we calculate \((0.34\times 100)\%\), which gives us 34%.
    2. Using the formula and \(x\) = 4.15, we calculate \((4.15\times 100)\%\), which gives us 415%.
    3. Using the formula and \(x\) = 0.0391, we calculate \((0.0391\times 100)\%\), which gives us 3.91%.

    1. Determine 70% of 3,500
    2. Determine 156% of 720

    ಉತ್ತರವನ್ನು ತಿಳಿಸಿ
    1. The total is \(x\) = 3,500, and the percent is \(n\) = 70. The decimal form of 70% is 0.70. To find the part, or percent of the total, substitute those values into the formula and calculate. \[\begin{array}{lll}\text{part} & = & \text{percent}\times \text{total} \\ & = & 0.70\times 3500 \\ & = & 2450\end{array}\] From this, we say that 70% of 3,500 is 2,450.
    2. The total is \(x\) = 720, and the percent is \(n\) = 156. The decimal form of 156% is 1.56. To find the part, or percent of the total, substitute those values into the formula and calculate. \[\begin{array}{lll}\text{part} & = & \text{percent}\times \text{total} \\ & = & 1.56\times 720 \\ & = & 1,123.2\end{array}\] From this, we say that 156% of 720 is 1,123.2.

    1. What is the total if 35% of the total is 70?
    2. What is the total if 10% of the total is 4,000?

    ಉತ್ತರವನ್ನು ತಿಳಿಸಿ
    1. Step 1: The percent is 35, which in decimal form is 0.35. We were given that 35% of the total is 70, so the part is 70. We are to find the total. Substituting into the formula, we have \[\begin{array}{lll}\text{part} & = & \text{percent}\times \text{total} \\ 70 & = & 0.35\times \text{total}\end{array}\]
      Step 2: To find the total, we solve the equation for the total. \[\begin{array}{lll}70 & = & 0.35\times \text{total} \\ \frac{70}{0.35} & = & \frac{0.35\times \text{total}}{0.35} \\ 200 & = & \frac{0.35\times \text{total}}{0.35} \\ 200 & = & \text{total}\end{array}\]
      From this we see that 200 is the total, or, that 35% of 200 is 70.
    2. Step 1: The percent is 10, which in decimal form is 0.1. We were given that 10% of the total is 4,000, so the part is 4,000. Substituting into the formula, we have \[\begin{array}{lll}\text{part} & = & \text{percent}\times \text{total} \\ 4,000 & = & 0.1\times \text{total}\end{array}\]
      Step 2: To find the total, we solve the equation for the total. \[\begin{array}{lll}4,000 & = & 0.1\times \text{total} \\ \frac{4,000}{0.1} & = & \frac{0.1\times \text{total}}{0.1} \\ 40,000 & = & \frac{0.1\times \text{total}}{0.1} \\ 40,000 & = & \text{total}\end{array}\]
      From this we see that 40,000 is the total, or that 10% of 40,000 is 4,000.

    1. What percent of 500 is 175?
    2. What percent of 228 is 155?

    ಉತ್ತರವನ್ನು ತಿಳಿಸಿ
    1. Step 1: The total is 500, the percent of the total is 175. Substituting into the formula, we have \[\begin{array}{lll}\text{part} & = & \text{percent}\times \text{total} \\ 175 & = & \text{percent}\times 500\end{array}\]
      Step 2: To find the percent, we solve the equation for the percent. \[\begin{array}{lll}175 & = & \text{percent}\times 500 \\ \frac{175}{500} & = & \frac{\text{percent}\times 500}{500} \\ 0.35 & = & \frac{\text{percent}\times 500}{500} \\ 0.35 & = & \text{percent}\end{array}\]
      We see the percent in decimal form is 0.35. Converting from the decimal form yields 35%. We say that 175 is 35% of 500.
    2. Step 1: The total is 228, the percent of the total is 155. Substituting into the formula, we have \[\begin{array}{lll}\text{part} & = & \text{percent}\times \text{total} \\ 155 & = & \text{percent}\times 228\end{array}\]
      Step 2: To find the percent, we solve the equation for the percent. \[\begin{array}{lll}155 & = & \text{percent}\times 228 \\ \frac{155}{228} & = & \frac{\text{percent}\times 228}{228} \\ 0.6798 & = & \frac{\text{percent}\times 228}{228} \\ 0.6798 & = & \text{percent}\end{array}\]
      We see the percent is 0.6798 (rounded to four decimal places). Converting from the decimal form yields 67.98%. We say that 155 is 67.98% of 228.

  7. Justine applies to a medium size university outside her hometown and finds out that the retention rate (percent of students who return for their sophomore year) for the 2021 academic year at the university was 84%. During a visit to the registrar’s office, she finds out that 1,350 people had enrolled in academic year 2021. How many students from the academic year 2021 are returning for the 2022 academic year?

    ಉತ್ತರವನ್ನು ತಿಳಿಸಿ

    The percent of students who will return for the 2022 academic year (the retention rate) is 84%. The total number of students who enrolled in the 2021 academic year was 1,350. This means the percent is known and the total is known. From this, we can determine the number of students who will return (percent of the total) for the 2022 academic year using the formula \[\text{part}=\text{percent}\times \text{total}\]. Substituting into the formula and calculating, we find that the number of students that are returning is

    \[\begin{array}{lll}\text{part} & = & \text{percent}\times \text{total} \\ & = & 0.84\times 1,350 \\ & = & 1,134\end{array}\]

    So 1,134 students will return for the 2022 academic year.

  8. Cameron enrolls in a calculus class. In this class of 45 students, there are 18 chemistry majors. What percent of the class are chemistry majors?

    ಉತ್ತರವನ್ನು ತಿಳಿಸಿ

    In this situation, the percent is to be determined. We know the total number of students, 45, and the part of the students that are chemistry majors, 18. Using that information and the formula \[\text{part}=\text{percent}\times \text{total}\], the percent can be found. Substituting and solving, we have

    \[\begin{array}{lll}18 & = & \text{percent}\times 45 \\ \frac{18}{45} & = & \frac{\text{percent}\times 45}{45} \\ 0.4 & = & \frac{\text{percent}\times 45}{45} \\ 0.4 & = & \text{percent}\end{array}\]

    Converting the 0.4 from decimal form, we find that 40% of the students in the calculus class are chemistry majors.

  9. Mariel makes a 20% commission on every sale she makes. One week, her commission check is for $153.00. What were her total sales that week?

    ಉತ್ತರವನ್ನು ತಿಳಿಸಿ

    In this problem, Mariel’s total sales is to be determined. We know the percent she earns is 20%. We also know that her sales commission was $153.00, which is the percent of the total. Using this information and the formula \[\text{part}=\text{percent}\times \text{total}\] we can find Mariel’s total sales. The decimal form of 20% is 0.2. The part, or percent of the total, is 153. Substituting and solving, we obtain

    \[\begin{array}{lll}\text{part} & = & \text{percent}\times \text{total} \\ 153 & = & 0.2\times \text{total} \\ \frac{153}{0.2} & = & \frac{0.2\times \text{total}}{0.2} \\ 765 & = & \frac{0.2\times \text{total}}{0.2} \\ 765 & = & \text{total}\end{array}\]

    Mariel’s total sales were $765.00.

Symbols used here

\pm
plus or minus
Both signs at once: x = 3 ± 2 means 5 and 1.
\neq
not equal
The two sides are different.
\leq,\ \geq
less/greater than or equal
Inequalities that allow equality; < and > exclude it.
\approx
approximately equal
Equal to the precision shown, not exactly.
\sqrt{x},\ \sqrt[n]{x}
square root, n-th root
The non-negative number whose square (n-th power) is x.
a \bmod n
remainder
What is left after dividing a by n.
\%
per cent
Per hundred: 15% = 15/100.
a : b,\ \frac{a}{b}
ratio, fraction
a for every b; a divided by b.

How to: Understanding Percent

  1. Define and calculate percent.
  2. Convert between percent, decimal, and fractional values.
  3. Calculate the total, percent, or part.
  4. Solve application problems involving percents.
  5. 18%
  6. 84%
  7. 38.7%
  8. 213%

Questions people ask

Why does the order of operations matter?

Because 2 + 3 × 4 would otherwise be two different numbers. The convention (brackets, exponents, multiplication and division, addition and subtraction) exists so every reader gets the same value from the same expression.

How do I check an arithmetic answer?

Estimate first (round every number and compute roughly), then compare. If the estimate and the exact answer disagree by more than a little, one of them is wrong. The solver shows every operation, so you can find which line went astray.

Why are fractions harder than decimals?

They are not harder, they are more exact: 1/3 is a precise number, 0.333 is an approximation. Fractions need a common denominator to add, which is the one extra step people trip on.

ನಿಮ್ಮದೇ ಆದದ್ದನ್ನು ಪ್ರಯತ್ನಿಸಿ

Parts of this page are adapted from OpenStax Contemporary Mathematics (CC BY-NC-SA 4.0). Condensed and re-explained here; errors are ours.

ಇನ್ನಷ್ಟು Arithmetic

Order of operationsFractionsPercentagesPowers and roots