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# 3.7: Percent Problems

Difficulty Level: At Grade Created by: CK-12

## Learning Objectives

At the end of this lesson, students will be able to:

• Find a percent of a number.
• Use the percent equation.
• Find percent of change.

## Vocabulary

Terms introduced in this lesson:

percent
percent equation
rate
total
part
base unit
positive percent change
increase
negative percent change
decrease
mark-up

## Teaching Strategies and Tips

Work out exercises like Examples 1-7 and place the solutions in a table similar to the one below. This frees up board space and provides students with a handy reference to study later.

Example.

Complete the table.

Fraction Decimal Percent
45\begin{align*}\frac{4}{5}\end{align*}
0.12\begin{align*}0.12\end{align*}
2.5%\begin{align*}2.5\%\end{align*}
0.01%\begin{align*}0.01\%\end{align*}
38\begin{align*}\frac{3}{8}\end{align*}
0.001\begin{align*}0.001\end{align*}
43\begin{align*}\frac{4}{3}\end{align*}

Fraction, decimal, and percent conversions can be summed up as:

• decimal \begin{align*}\Rightarrow\end{align*} percent. Multiply by 100\begin{align*}100\end{align*} and affix %\begin{align*}\%\end{align*} symbol. Alternatively, move the decimal point two places right and affix %\begin{align*}\%\end{align*} symbol. See Examples 1 and 2. Additional Example. 1.2=1.2×100100=120%\begin{align*}1.2=\frac{1.2\times100}{100}=120\%\end{align*}.
• percent \begin{align*}\Rightarrow\end{align*} decimal. Divide by 100\begin{align*}100\end{align*} and remove %\begin{align*}\%\end{align*} symbol. Alternatively, move the decimal point two places left and remove %\begin{align*}\%\end{align*} symbol. See Examples 3 and 4. Additional Example. 0.003%=0.003100=0.00003\begin{align*}0.003\%=\frac{0.003}{100}=0.00003\end{align*}.
• fraction \begin{align*}\Rightarrow\end{align*} percent. Represent the fraction as x%\begin{align*}x\%\end{align*} or x/100\begin{align*}x/100\end{align*}. Solve for x\begin{align*}x\end{align*} by cross-multiplying. Alternatively, convert the fraction to a decimal. Then convert decimal to percent as above. See Examples 5 and 6. Additional Example. 25=x100x=40\begin{align*}\frac{2}{5}=\frac{x}{100}\Rightarrow x=40\end{align*}. Therefore, 25=40%\begin{align*}\frac{2}{5}=40\%\end{align*}.
• percent \begin{align*}\Rightarrow\end{align*} fraction. Express percent as a ratio (per 100\begin{align*}100\end{align*}). Reduce fraction. See Example 7. Additional Example. 110%=110100=1110=1110\begin{align*}110\%=\frac{110}{100}=\frac{11}{10}=1\frac{1}{10}\end{align*}.
• fractions \begin{align*}\Rightarrow\end{align*} decimals. Divide numerator by denominator. Use calculator if necessary. Example. .
• decimals \begin{align*}\Rightarrow\end{align*} fractions. Place the digits after the decimal under the appropriate power of ten and reduce. Example. 0.225=2251000=940\begin{align*}0.225=\frac{225}{1000}=\frac{9}{40}\end{align*}.

In Examples 8-11, students setup percent equations to find the percent of a given number.

• Convert the value for R%\begin{align*}R\%\end{align*} in the percent equation R%×Total=Part\begin{align*}R\% \times \;\mathrm{Total} = \;\mathrm{Part}\end{align*} to a decimal before calculating. Rate should be expressed as a decimal in Rate×Total=Part\begin{align*}\;\mathrm{Rate} \times \;\mathrm{Total} = \;\mathrm{Part}\end{align*}.
• Remind students that of means to multiply.

Use Examples 12 and 13 to point out that a positive percent change means an increase in the quantity, and a negative change means a decrease.

In Example 14, teachers are encouraged to work out the calculations for Mark-up, Final retail price, 20%\begin{align*}20\%\end{align*} discount, and 25%\begin{align*}25\%\end{align*} discount as a way to motivate the same calculations done algebraically in the next step.

Is the order in which we calculate discounts and sales tax significant? In other words, should stores subtract the discount first and then add the tax on the new total or should the total amount be taxed first and then have the discount subtracted from that? or does it matter? Assume the discount is a percent and not a fixed discount.

Solution. Consider an example:

Original price of the item =\$12.50\begin{align*}= \12.50\end{align*}

Discount =35%\begin{align*}= 35\%\end{align*}

Sales tax rate in the county of purchase =7.75%\begin{align*}= 7.75\%\end{align*}

Tax First, Discount Second

12.50+0.0775(12.50)=13.4713.470.35(13.47)=8.76\begin{align*}12.50 + 0.0775 (12.50) = 13.47\\ 13.47 - 0.35 (13.47) = 8.76\end{align*}

Discount First, Tax Second

12.500.35(12.50)=8.138.13+0.0775(8.13)=8.76\begin{align*}12.50 - 0.35 (12.50) = 8.13\\ 8.13 + 0.0775 (8.13) = 8.76\end{align*}

The steps above can be repeated algebraically. Let:

p\begin{align*}p\end{align*} = original price of the item

d\begin{align*}d\end{align*} = discount

t\begin{align*}t\end{align*} = sales tax rate in the county of purchase

Tax First, Discount Second

p+t(p)=\begin{align*}p + t (p) =\end{align*} amount with tax

p+t(p)d(p+t(p))=\begin{align*}p + t (p) - d (p + t (p)) =\end{align*} final amount with discount

Discount First, Tax Second

pd(p)=\begin{align*}p - d (p) =\end{align*} amount after discount

pd(p)+t(pd(p))=\begin{align*}p - d (p) + t (p - d (p)) =\end{align*} final amount with discount

Simplifying the two expressions for the final amount results in identical expressions (careful when distributing).

Conclusion: There is no difference in the total amount to be paid if the tax is added to the total first, followed by the discount, or the discount applied to the total first, followed by the tax.

General Tips:

• Give students time to consider the possibilities on their own.
• Have students choose their own numbers for original price, discount, and sales tax rate. This can be done in groups or individually. Calculators are recommended.
• Asking around the classroom, it is suspicious that everyone’s final two calculations are the same. Use this to formulate a conjecture.
• Ask students to turn in their reasoning process as an assignment.
• As the above algebraic argument is completely variable driven (no numbers), teachers are advised to show each step.
• Students reason in various ways. Some common responses are:

“Add the tax first, otherwise you are cheating the government out of its tax.”

“Taking the discount first decreases the bill, and so the tax will not be as great.”

“Tax should be added on last, as the discounted price is the true price of the item – since that is how much the item is being sold for.”

“Doing the tax first increases the amount to be paid and so the discount will be larger.” According to students, this translates to a smaller price to be paid.

“The quantities must be equal since they have seen it being done in both ways.”

As a class, discuss the validity of some of these responses.

## Error Troubleshooting

Despite the %\begin{align*}\%\end{align*} symbol in Example 4, students see the decimal and incorrectly move it two places right. This error is common for percents less than 1\begin{align*}1\end{align*} (i.e., 0.5%\begin{align*}0.5\%\end{align*}) and percents greater than 100\begin{align*}100\end{align*} (i.e., \begin{align*}110\%\end{align*}).

Express \begin{align*}0.01\%\end{align*} as a decimal.

Solution: Move the decimal two places left and remove the \begin{align*}\%\end{align*} symbol. \begin{align*}0.01\%=0.0001\end{align*}.

Express \begin{align*}120.25\end{align*} as a percent.

Solution: Move the decimal two places right and affix the \begin{align*}\%\end{align*} symbol. \begin{align*}120.25=1.2025\%\end{align*}.

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