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2.3: Percentages (How Does It All Relate?)

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    Your class just wrote its first math quiz. You got 13 out of 19 questions correct, or \(\dfrac{13}{19}\). In speaking with your friends Sandhu and Illija, who are in other classes, you find out that they also wrote math quizzes; however, theirs were different. Sandhu scored 16 out of 23, or \(\dfrac{16}{23}\), while Illija got 11 out of 16, or \(\dfrac{11}{16}\). Who achieved the highest grade? Who had the lowest? The answers are not readily apparent, because fractions are difficult to compare.

    Now express your grades in percentages rather than fractions. You scored 68%, Sandhu scored 70%, and Illija scored 69%. Notice you can easily answer the questions now. The advantage of percentages is that they facilitate comparison and comprehension.

    Converting Decimals to Percentages

    A percentage is a part of a whole expressed in hundredths. In other words, it is a value out of 100. For example, 93% means 93 out of 100, or \(\dfrac{93}{100}\).

    The Formula


    Formula 2.1

    How It Works

    Assume you want to convert the decimal number 0.0875 into a percentage. This number represents the \(\boldsymbol{dec}\) variable in the formula. Substitute into Formula 2.1:

    \[\%=0.0875 \times 100=8.75 \%\nonumber \]

    Important Notes

    You can also solve this formula for the decimal number. To convert any percentage back into its decimal form, you need to perform a mathematical opposite. Since a percentage is a result of multiplying by 100, the mathematical opposite is achieved by dividing by 100. Therefore, to convert 81% back into decimal form, you take \(81 \% \div 100=0.81\).

    Things To Watch Out For

    Your Texas Instruments BAII Plus calculator has a % key that can be used to convert any percentage number into its decimal format. For example, if you press 81 and then %, your calculator displays 0.81.

    While this function works well when dealing with a single percentage, it causes problems when your math problem involves multiple percentages. For example, try keying 4% + 3% = into the calculator using the % key. This should be the same as \(0.04 + 0.03 = 0.07\). Notice, however, that your calculator has 0.0412 on the display.

    Why is this? As a business calculator, you BAII Plus is programmed to take portions of a whole. When you key 3% into the calculator, it takes 3% of the first number keyed in, which was 4%. As a formula, the calculator sees \(4 \%+(3 \% \times 4\%)\). This works out to \(0.04 + 0.0012 = 0.0412\).

    To prevent this from happening, your best course of action is not to use the % key on your calculator. It is best to key all percentages as decimal numbers whenever possible, thus eliminating any chance that the % key takes a portion of your whole. Throughout this textbook, all percentages are converted to decimals before calculations take place.

    Paths To Success

    When working with percentages, you can use some tricks for remembering the formula and moving the decimal point.

    Remembering the Formula. When an equation involves only multiplication of all terms on one side with an isolated solution on the other side, use a mnemonic called the triangle technique. In this technique, draw a triangle with a horizontal line through its middle. Above the line goes the solution, and below the line, separated by vertical lines, goes each of the terms involved in the multiplication. The figure to the right shows how Formula 2.1 would be drawn using the triangle technique.


    To use this triangle, identify the unknown variable, which you then calculate from the other variables in the triangle:

    1. Anything on the same line gets multiplied together. If solving for %, then the other variables are on the same line and multiplied as \(\boldsymbol{dec} \times 100\).
    2. Any pair of items with one above the other is treated like a fraction and divided. If solving for \(\boldsymbol{dec}\), then the other variables are above/below each other and are divided as \(\dfrac{\%}{100}\).

    Moving the Decimal. Another easy way to work with percentages is to remember that multiplying or dividing by 100 moves the decimal over two places.

    1. If you are multiplying by 100, the decimal position moves two positions to the right.\[0.73 \times 100=0.\underbrace{\overrightarrow{7}}_{\text {2 positions}}\underbrace{\overrightarrow{3}}_{\text { to the right}}=73 \% \nonumber \]
    2. If you are dividing by 100, the decimal position moves two positions to the left.\[73\% \div 100=\underbrace{\overleftarrow{7}}_{\text {2 positions}}\underbrace{\overleftarrow{3}}_{\text { to the left}}\cdot = 0.73 \nonumber \]
    Example \(\PageIndex{1}\): Working with Percentages

    Convert (a) and (b) into percentages. Convert (c) back into decimal format.

    1. \(\dfrac{3}{8}\)
    2. 1.3187
    3. 12.399%


    For questions (a) and (b), you need to convert these into percentage format. For question (c), you need to convert it back to decimal format.

    What You Already Know

    1. This is a fraction to be converted into a decimal, or \(\boldsymbol{dec}\).
    2. This is \(\boldsymbol{dec}\).
    3. This is %.

    How You Will Get There

    1. Convert the fraction into a decimal to have \(\boldsymbol{dec}\). Then apply Formula 2.1: \(\% = \boldsymbol{dec} \times 100\) to get the percentage.
    2. As this term is already in decimal format, apply Formula 2.1: \(\% = \boldsymbol{dec} \times 100\) to get the percentage.
    3. This term is already in percentage format. Using the triangle technique, calculate the decimal number through \(\boldsymbol{dec}=\dfrac{\%}{100}\).


    1. \(\dfrac{3}{8}=3 \div 8=0.375\) \(\%=0.375 \times 100\) \(\% = 37.5\%\)
    2. \(\%=1.3187 \times 100\) \(\% = 131.87\%\)
    3. \(\boldsymbol{dec}=\dfrac{12.399 \%}{100}=0.12399\)

    In percentage format, the first two numbers are 37.5% and 131.87%. In decimal format, the last number is 0.12399.

    Rate, Portion, Base

    In your personal life and career, you will often need to either calculate or compare various quantities involving fractions. For example, if your income is $3,000 per month and you can't spend more than 30% on housing, what is your maximum housing dollar amount? Or perhaps your manager tells you that this year's sales of $1,487,003 are 102% of last year's sales. What were your sales last year?

    The Formula


    Formula 2.2

    How It Works

    Assume that your company has set a budget of $1,000,000. This is the entire amount of the budget and represents your base. Your department gets $430,000 of the budget—this is your department’s part of the whole and represents the portion. You want to know the relationship between your budget and the company’s budget. In other words, you are looking for the rate.


    • Apply Formula 2.2, where rate \(=\dfrac{\$ 430,000}{\$ 1,000,000}\).
    • Your budget is 0.43, or 43%, of the company's budget.

    Important Notes

    There are three parts to this formula. Mistakes commonly occur through incorrect assignment of a quantity to its associated variable. The table below provides some tips and clue words to help you make the correct assignment.

    Variable Key Words Example
    Base of If your department can spend 43% of the company's total budget of $1,000,000, what is your maximum departmental spending?
    Portion is, are If your department can spend 43% of the company's total budget of $1,000,000, what is your maximum departmental spending?
    Rate %, percent, rate If your department can spend 43% of the company's total budget of $1,000,000, what is your maximum departmental spending?

    Things To Watch Out For

    In resolving the rate, you must express all numbers in the same units—you cannot have apples and oranges in the same sequence of calculations. In the above example, both the company's budget and the departmental budget are in the units of dollars. Alternatively, you would not be able to calculate the rate if you had a base expressed in kilometres and a portion expressed in metres. Before you perform the rate calculation, express both in kilometres or both in metres.

    Paths To Success


    Formula 2.2 is another formula you can use the triangle technique for. You do not need to memorize multiple versions of the formula for each of the variables. The triangle appears to the right.

    Be very careful when performing operations involving rates, particularly in summing or averaging rates.

    Summing Rates

    Summing rates requires each rate to be a part of the same whole or base. If Bob has 5% of the kilometres traveled and Sheila has 6% of the oranges, these are not part of the same whole and cannot be added. If you did, what does the 11% represent? The result has no interpretation. However, if there are 100 oranges of which Bob has 5% and Sheila has 6%, the rates can be added and you can say that in total they have 11% of the oranges.

    Averaging Rates

    Simple averaging of rates requires each rate to be a measure of the same variable with the same base. If 36% of your customers are female and 54% have high income, the average of 45% is meaningless since each rate measures a different variable. Recall that earlier in this chapter you achieved 68% on your test and Sandhu scored 70%. However, your test involved 19 questions and Sandhu’s involved 23 questions. These rates also cannot be simply averaged to 69% on the reasoning that (68% + 70%) / 2 = 69%, since the bases are not the same. When two variables measure the same characteristic but have different bases (such as the math quizzes), you must use a weighted-average technique, which will be discussed in Section 3.1.

    When can you average rates? Hypothetically, assume Sandhu achieved his 70% by writing the same test with 19 questions. Since both rates measure the same variable and have the same base, the simple average of 69% is now calculable.

    Exercise \(\PageIndex{1}\): Give It Some Thought

    Consider the following situations and select the best answer without performing any calculations.

    1. If the rate is 0.25%, in comparison to the base the portion is
      1. just a little bit smaller than the base.
      2. a lot smaller than the base.
      3. just a little bit bigger than the base.
      4. a lot bigger than the base.
    2. If the portion is $44,931 and the base is $30,000, the rate is
      1. smaller than 100%.
      2. equal to 100%.
      3. larger than 100% but less than 200%.
      4. larger than 200%.
    3. If the rate is 75% and the portion is $50,000, the base is
      1. smaller than $50,000.
      2. larger than $50,000.
      3. the same as the portion and equal to $50,000.
    1. b (0.25% is 0.0025, resulting in a very small portion)
    2. c (the portion is larger than the base, but not twice as large)
    3. b (the portion represents 75% of the base, meaning the base must be larger)
    Example \(\PageIndex{2}\): Rate, Portion, Base

    Solve for the unknown in the following three scenarios.

    1. If your total income is $3,000 per month and you can't spend more than 30% on housing, what is the maximum amount of your total income that can be spent on housing?
    2. Your manager tells you that 2014 sales are 102% of 2013 sales. The sales for 2014 are $1,487,003. What were the sales in 2013?
    3. In Calgary, total commercial real estate sales in the first quarter of 2008 were $1.28 billion. The industrial, commercial, and institutional (ICI) land sector in Calgary had sales of $409.6 million. What percentage of commercial real estate sales is accounted for by the ICI land sector?


    1. You are looking for the maximum amount of your income that can be spent on housing.
    2. You need to figure out the sales for 2013.
    3. You must determine the percentage of commercial real estate sales accounted for by the ICI land sector in Calgary.

    What You Already Know

    1. Look for key words in the question: "what is the maximum amount" and "of your total income." The total income is the base, and the maximum amount is the portion.
      • Base = $3,000 Rate = 30% Portion = maximum amount
    2. Look for key words in the question: "sales for 2014 are $1,487,003" and "of 2013 sales." The 2014 sales is the portion, and the 2013 sales is the base.
      • Portion = $1,487,003 Rate = 102% Base = 2013 sales
    3. Look for key words in the question: "of commercial real estate sales" and "are accounted for by the ICI land sector." The commercial real estate sales are the base, and the ICI land sector sales are the portion.
      • Base=$1.28 billion Portion=$409.6 million Rate=percentage

    How You Will Get There

    1. Apply Formula 2.2, but rearrange using the triangle technique to have Portion = Rate \(\times\) Base.
    2. Apply Formula 2.2, but rearrange using the triangle technique to have Base=Portion/Rate.
    3. Apply Formula 2.2, Rate=Portion/Base.


    1. \(\text { Portion }=30 \% \times \$ 3,000=0.3 \times \$ 3,000=\$ 900\)
    2. \(\text { Base }=\dfrac{\$ 1,487,003}{102 \%}=\dfrac{\$ 1,487,003}{1.02}=\$ 1,457,846.08\)
    3. \(\text { Rate }=\dfrac{\$ 409.6 \text { million }}{\$ 1.28 \text { billion }}=\dfrac{\$ 409,600,000}{\$ 1,280,000,000}=0.32 \text { or } 32 \%\)


    1. The maximum you can spend on housing is $900 per month.
    2. 2013 sales were $1,457,846.08.
    3. The ICI land sector accounted for 32% of commercial real estate sales in Calgary for the first quarter of 2008.

    Contributors and Attributions

    This page titled 2.3: Percentages (How Does It All Relate?) is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Jean-Paul Olivier via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.