In problems in which you find a percent of a number, the missing term is the part. You will be given the % which is always 100.

The following examples all ask you to find a percent of a number. The missing term is the part (the “is” part). Look at the examples carefully so you’ll recognize the wording.

• What is [latex]14\%\text{ of }60[/latex]?[latex]\dfrac{\textit{is}}{\textit{of}}=\dfrac{\%}{100}\longrightarrow\dfrac{\textit{N}}{60}=\dfrac{14}{100}[/latex]
• Find [latex]10\%\text{ of }27[/latex].[latex]\dfrac{\textit{is}}{\textit{of}}=\dfrac{\%}{100}\longrightarrow\dfrac{\textit{P}}{27}=\dfrac{10}{100}[/latex]
• [latex]5\%\text{ of }15[/latex] is                    .[latex]\dfrac{\%}{100}=\dfrac{\textit{is}}{\textit{of}}\longrightarrow\dfrac{5}{100}=\dfrac{\textit{X}}{15}[/latex]
• [latex]75\%\text{ of }12 =[/latex]                    .[latex]\dfrac{\%}{100}=\dfrac{\textit{is}}{\textit{of}}\longrightarrow\dfrac{75}{100}=\dfrac{\textit{K}}{12}[/latex]

In percent problems, the number after the word of usually represents the whole.

Percents Greater Than or Equal to 100%

Remember that [latex]100\% = 1[/latex].

100% of anything is the whole thing. If you spend 100% of your pay cheque, you spend the whole thing. If you get 100% on a test, you have the whole thing correct.

If you have more than 100%, you have more than the whole thing. If you spend 110% of your paycheque, you spent more than you earned, and you may be in trouble! It is hard to get more than 100% on a test unless the instructor has given bonus marks for extra questions. You may hear of percents more than 100% in increases, such as costs of housing or inflation. For example, “The Browns just sold their house and made a 200% profit.” This means they got back what they paid and two times more!

Taxes

The amount of tax to be paid is calculated by finding a percent of a number. The tax rate is usually given as a percent. The basic proportion for these problems is:

Please note that the tax rates used in the questions in this book are for the year 2010 and are subject to change.

The British Columbia Harmonized Sales Tax (HST) is 12%. In B.C., the provincial portion of the harmonized sales tax does not have to be paid on children’s clothes, food, books, gasoline and diesel fuel, and other special items.

Income tax is charged at different percentages according to the amount of a person’s taxable income. The first $28,000 of taxable income is taxed at 17%. Note that other tax rules and charges may apply in real situations.

Cross-Border Shopping

The Canadian (CAN) and American (US) dollars are not equal in value. The exchange rate (the value of one Canadian dollar compared to a dollar from another country) changes often; the current rate is usually available from banks, on the news, in the newspapers and on a web site. In the winter of 2010, the Canadian dollar was around $0.92 of an American dollar (ratio is [latex]$1.00\text{ CAN}:$0.92\text{ US}[/latex]), so CAN money was valued at 92% of US money.

To find the value of one US dollar in Canadian funds, use this proportion:

[latex]\dfrac{$1\text{ CAN}}{$0.92\text{ US}}=\dfrac{\textit{N}\text{ CAN}}{$1\text{ US}}[/latex]

[latex]\textit{N}=$1\div$0.92=$1.086[/latex], so US money was valued at 109% of CAN money.

Note that the proportion changes as the exchange rate changes.

What if you buy in the United States?

• Change the US cost to the Canadian equivalent (multiply by 109%).
• If you have more than the purchases allowed (call the Canada Border Service Agency for information), the Canadian Customs charge duty on the Canadian value of your purchases. The percent of the duty (the rate) varies according to what the item is, where it was made and the duty rates of the day. For example, duty on poultry is 12.5%, on non-US cotton 25%, and on liquor 110%!
  • Duty is gradually being eliminated under the Canada-US Free Trade Agreement. If an item is made in North America, there is no duty charged because of NAFTA (North American Free Trade Agreement)
• HST (12%) is charged on the duty and on the Canadian value of the purchases (which includes any US sales taxes).

Look at this example (assume $1.00 CAN = $0.92 US).

[latex]\begin{equation}\begin{split} \text{Men's leather shoes, US price} &\hspace{2cm} &$64.80 \\ \\ \text{US sales tax }6\% &\hspace{2cm} &3.89 \\ \text{US total cost} &\hspace{2cm} &68.69 \\ \\ \text{Equivalent cost in CAN funds (US cost }\times 109\%\text{)} \\ \\ \dfrac{\textit{N}}{68.69}=\dfrac{109}{100} &\hspace{2cm} &$74.87 \\ \\ \text{Duty on leather shoes is }22.8\% &\hspace{2cm} &$17.07 \\ \\ \text{HST (}12\%\text{) on CAN value plus duty} \\ \\ 12\%\text{ of }$74.19\text{ and }16.92\text{ together} &\hspace{2cm} &$11.03 \end{split}\end{equation}[/latex]

The total cost of a pair of leather shoes priced at $64.80 in the United States will be the American price in Canadian funds + duty + HST = $102.97 CAN.

Increases and Decreases, Discounts and Mark-ups

Increases (amount changing to more) and decreases (amount changing to less) are often given as a percent. For example,

• The Insurance Corporation of B.C. increased car insurance rates by 3.3% in March 2007.
• The number of acute care beds at the local hospital has decreased by 28% in the last year.
• The new work contract provides a 4% wage increase in the first year and a 2½% increase in the second year.

The amount of an increase or decrease is calculated by finding a percent of a number. When the percent of an increase or decrease is given, the proportion is:

[latex]\dfrac{\text{amount of increase or decrease}}{\text{whole amount}}=\dfrac{\text{increase or decrease }\%}{100}[/latex]

Discounts are a form of decrease. The discount is the amount taken off a price; it is the price reduction.

Sale prices (discounted prices) may be advertised as:

• “All items 20% off.”
• “Everything in stock reduced by 25% to 50%.”
• “33⅓% savings!”
• “2% discount for cash.”

Decrease and discount problems may need to be solved in several steps. Sometimes the problems ask for:

• the amount of the decrease (may be called the “saving”) (1 step).
• the amount left after the decrease (2 steps).

Increases and mark-ups are calculated in the same way as decreases and discounts. However, an increase or mark-up is added to the original amount.

Mark-ups are the amount added to the cost price before an item is resold. Many factors must be considered when businesses decide on the percent of the mark-up:

• all costs of operating a business
• the profit wanted
• the community the business is in
• the competition the business has

For example, the mark-up on leather shoes may be 45%, but on running shoes it may be 60%. Kitchen appliances might have a 42% mark-up, while lawn mowers might have a 55% mark-up.

Wage Increase

Having a wage increase at work is always a good thing! Often the raise will be given as a percentage. That means that everyone will see more money on their pay cheque, but they will each have a different amount because they all get paid a different amount to start with.

Commission and Tips

Salespeople may receive a commission as part or all of their pay. The business owner pays the salesperson an agreed-upon percent of the selling price of the product.

• Real estate agents are paid by commission on their sales.
• Car and truck salespeople may be paid a small salary per month, but their main income is the commission on the vehicles they sell.

Tips are appreciation payments for service. The customer gives tips directly to the worker. Taxi drivers, waiters, bellhops and chambermaids in hotels often receive a minimal hourly wage. A large part of their earnings is from tips. In restaurants, expect to leave at least a 15% tip for adequate service.

To calculate the amount of a commission (or a tip), find the percentage of the total amount using the proportion:

The commission is the part.
The total amount is the whole.

Commission problems often have several steps. You may have to:

• add together several items to find the total of sales.
• subtract a base amount for which salespeople do not receive a commission.
• add the amount of commission to the basic wage to find out how much the person earned.

In a real situation, we would probably round the amount of the bill to the nearest dollar and then calculate the tip.

Topic A: Self-Test

Mark       /11                  Aim        9/11

1. Answer the following.
   (4 marks)
   1. [latex]6\%\text{ of }30 =[/latex]
   2. [latex]\tfrac{1}{4}\%\text{ of }48[/latex] is?
   3. Find [latex]131\%\text{ of }400[/latex].
   4. Write the general proportion that can be used to solve percent problems.

    

2. Solve the problems. (2 marks each, except c)
   (7 marks)
   1. The $125,000 house is insured for 85% of its value against fire damage. How much money should the owner receive if the house is destroyed by fire?
   2. Charlotte sells leisure clothes in her small town for a large national company. She receives $500 a month, and a 20% commission on all monthly sales over $1 000. What are her monthly earnings if her total sales are $2,300?
   3. The barbeque was originally priced at $599 but Jack bought it during a 35% off, end-of-season sale.
      1. What was the sale price? (1 mark)
      2. Calculate the harmonized sales tax (12%). (1 mark)
      3. Give the total cost of Jack’s barbeque. (1 mark)