 Motivation Problem

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Solving Percentage Problems

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Basic Percent Equation

# Basic Percent Equation

As you read the examples, ask yourself if you understand how to restate each problem as a simple percent relationship. In this equation, the base is the number of which we are taking a percentage and the amount is the value that results from taking the percent of the base. This means that in any percent problem, there are three basic values to be concerned about: the percent, the base, and the resulting amount. A percentage problem may ask us to find any one of these three values.

The Basic Percent Equation is the basic relationship that we need to learn to understand. We need to know how to identify which number is the base and which number is the amount?

Example: Suppose you go out to dinner at a restaurant. After dinner when you pay your bill, you decide to give your server a 15% tip. If the total bill (before tipping) is \$20.00, then how much should you leave as a tip?

We can restate the problem as: 15% of the total bill of \$20 is the tip.
The base in this case is the total bill of \$20.00, since this is the value we are taking a percentage of. We solve for the tip which is the resulting amount.

(percent) × (base) = (amount)

(percent) × (bill)   =       (tip)

(15%)     ×  (\$20) =       x

where x represents the amount of the tip.

Next, we convert the percent to either fraction or decimal form and then multiply:

x = (0.15)(20)

x = 3

You would tip the server \$3.

Example: You live in a city that charges 6% sales tax on all purchases. If you go to a store and purchase \$30 worth of merchandise, what is your total bill?

We can restate the sales tax portion of the problem as: 6% of the \$30 worth of merchandise is the sales tax.
Next, we compute the tax on the purchase using the Basic Percent Equation . We do not know the amount of sales tax, so we let x represent the amount of sales tax in the equation and solve for x.

(percent) × (base) = (amount)

(6%) × (\$30) = (amount of tax)

We compute

(0.06)(30) = x

1.80 = x

The amount of tax is \$1.80. Notice that this does not give the total bill. It only gives the amount of tax paid on the purchase. To compute the total bill, we add the amount of tax on to the cost of the merchandise.

Since \$30.00 + \$1.80 = \$31.80, the total bill is \$31.80.

We may also solve the problem in a single equation.

Second Method for the above problem: You live in a city that charges 6% sales tax on all purchases.   If you go to a store and purchase \$30 worth of merchandise, what is your total bill?

Note that you will pay 100% of the cost plus 6% for sales tax, so you will pay 106% of the cost of the merchandise.

We restate the problem as: 106% of the \$30 worth of merchandise is the total cost.

Next, we compute the tax on the purchase using the Basic Percent Equation . We do not know the total cost, so we let x represent the amount of the total cost in the equation and solve for x.

(percent) × (base) = (amount)

(106%) × (\$30) = (amount of the total cost)

We compute

(1.06)(30) = x

31.80 = x

The total bill is \$31.80.

For the next examples, formulate each part for yourself before checking.

Example: Suppose that you take a quiz and get 15 correct out of 20. What is your percent score on the quiz?

We restate the problem as: restate Again, we will use the variable x for the unknown percent.

Translating into the Basic Percent Equation: (percent) × (base) = (amount). translate Solve the equation: solve At this point, we have several choices for changing the result to a percent. Here are two ways: Two Methods We now have the solution. answer Example: Suppose that you work in the quality control department of a factory. During one afternoon, you check 80 components and find that 12 of them are defective.   What percent of the components that you checked were functioning properly?

We restate the problem as: restate Again, we work this problem using our Basic Percent Equation . We are looking for the percent, so we use x for the percent value in the equation. Since the base is what we are taking the percent of, the base in this situation would be the 80 components that were checked.   We want to know what percent of the 80 components worked properly. Since 12 of the components were defective, we note that the number of parts that work properly is 80 – 12 = 68.

Using the Basic Percent Equation we get: translate Solving the equation we get: solve We now have the solution. answer Example: In a recent survey, 75% of the people surveyed were concerned about the economy. If 60 of the people surveyed were concerned about the economy, how many people took part in the survey? Restate Translate Solve Answer ## Self Check Problem

A store owner wants to make a 30% profit based on the owner's cost. If the owner made a profit of \$18, what was the cost of the item for the owner?

Solution ### Joke or Quote

Mathematics is made of 50 percent formulas, 50 percent proofs, and 50 percent imagination.