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1.9: Solving Problems Using Percents

  • Page ID
    45768
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    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 \text{\$80}. They wanted to leave a 20\% tip. What amount would the tip be?

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

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

    The figure shows a customer copy of a restaurant receipt with the amount of the bill, $80, and the amount of the tip, $16. There is a group of bills totaling $16.

     

    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 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 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, 15\% 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 $220, 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.

    \text{percent}\cdot{\text{base}}=\text{amount}

    Example

    Find 50\% of 20

    Solution:

    First identify each piece of the problem:

    percent: 50\% or .5

    base: 20

    amount: unknown

    Now plug them into your equation \text{percent}\cdot{\text{base}}=\text{amount}

    .5\cdot{20}= ?

    .5\cdot{20}= 10

    Therefore, 10 is the amount or part that is 50\% of 20.

    Example

    What is 25\% of 80?

    [reveal-answer q=”813233″]Show Answer[/reveal-answer]
    [hidden-answer a=”813233″]

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

    [/hidden-answer]

    Try It

    [ohm_question]80094[/ohm_question]

    Thumbnail for the embedded element "Find the Percent of a Number"

    A YouTube element has been excluded from this version of the text. You can view it online here: http://pb.libretexts.org/afm-2/?p=78

    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.

    \Large{\frac{\text{amount}}{\text{base}}}\normalsize=\text{percent}

    Example

    What percent of 320 is 80?

    Solution:

    First identify each piece of the problem:

    percent: unknown

    base: 320

    amount: 80

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

    \large\frac{80}{320}\normalsize=?

    \large\frac{80}{320}\normalsize=.25

    Therefore, 80 is 25\% of 320.

    TRY IT

    [ohm_question]80097[/ohm_question]

    Thumbnail for the embedded element "Use the Percent Equation to Find a Percent"

    A YouTube element has been excluded from this version of the text. You can view it online here: http://pb.libretexts.org/afm-2/?p=78

    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.

    \Large{\frac{\text{amount}}{\text{percent}}}\normalsize=\text{base}

    EXample

    60 is 40\% of what number?

    Solution:

    First identify each piece of the problem:

    percent:40\% or .4

    base: unknown

    amount: 60

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

    (60)\div(.4)=?

    (60)\div(.4)=150

    Therefore, 60 is 40\% of 150.

    Example

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

    [reveal-answer q=”731314″]Show Answer[/reveal-answer]
    [hidden-answer a=”731314″]

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

    base = 30000\div(.15)=$200000

    The non-profit receives $200000 a year in donations

    [/hidden-answer]

    TRY IT

    [ohm_question]157022[/ohm_question]

    Thumbnail for the embedded element "Use a Percent Equation to Solve for a Base or Whole Amount"

    A YouTube element has been excluded from this version of the text. You can view it online here: http://pb.libretexts.org/afm-2/?p=78

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

    try it

    [ohm_question]146672[/ohm_question]

    try it

    [ohm_question]146692[/ohm_question]

    try it

    [ohm_question]146693[/ohm_question]

    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.

    example

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

    Solution

    What are you asked to find? the amount of the tip
    What formula/equation should you use? \text{percent}\cdot{\text{base}}=\text{amount}
    Substitute in the correct values. (.18)\cdot{68.50}
    Solve.  (.18)\cdot{68.50}=12.33
    Write a complete sentence that answers the question. The couple should leave a tip of \text{\$12.33}.

    try it

    [ohm_question]146694[/ohm_question]

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

    Thumbnail for the embedded element "Percent Application - Tipping"

    A YouTube element has been excluded from this version of the text. You can view it online here: http://pb.libretexts.org/afm-2/?p=78

    example

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

    The figures shows the nutrition facts for cereal.
    [reveal-answer q=”744443″]Show Answer[/reveal-answer]
    [hidden-answer a=”744443″]

    Solution

    What are you asked to find? the total daily amount of potassium recommended (whole)
    What formula/equation should you use? \Large{\frac{\text{amount}}{\text{percent}}}\normalsize=\text{base}
    Substitute in the correct values. \Large{\frac{85}{.02}}
    Solve. \Large{\frac{85}{.02}}\normalsize=4250
    Write a complete sentence that answers the question. The amount of potassium that is recommended is 4,250 mg.

    [/hidden-answer]

    try it

    [ohm_question]146697[/ohm_question]

    [ohm_question]146702[/ohm_question]

    try it

    [ohm_question]146703[/ohm_question]

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