Percent Word Problems Lesson

In our last lesson, we reviewed the six-step method used to solve a word problem that involves a linear equation in one variable.

Six-step method for Solving Word Problems with Linear Equations in One Variable

1. Read the problem and determine what you are asked to find
2. Assign a variable to represent the unknown
   • If more than one unknown exists, we express the other unknowns in terms of this variable
3. Write out an equation that describes the given situation
4. Solve the equation
5. State the answer using a nice clear sentence
6. Check the result
   • We need to make sure the answer is reasonable. In other words, if asked how many students were on a bus, the answer shouldn't be (-4) as we can't have a negative amount of students on a bus.

Solving Percent Word Problems

Another common type of word problem involves percentages. These usually deal with percent increase or percent decrease. Let's look at some examples.
Example 1: Solve each word problem.
Heather works as a cashier for a local grocery store. After one busy weekend shift, she had a total of $2725 in total receipts. This amount included the 9% state and local sales taxes. What was the amount of the tax collected?
Step 1) After reading the problem, it is clear that we need to find the amount of taxes that were collected.
Step 2) We have two unknowns, the pre-tax amount of goods and services and the amount of tax collected.
let x = amount of pre-tax goods and services sold
then .09x = amount of tax collected
Step 3) Write an equation, let's think about what we know.
If we were to sum the amount of pre-tax goods and services sold (x) with the amount of the tax collected (.09x), we would get 2725 (total receipts):
x + .09x = 2725
Step 4) Solve the equation:
x + .09x = 2725
We can multiply both sides by 100 and clear the decimal:
100x + 9x = 272,500
109x = 272,500
x = 2500
Step 5) Since x (2500) represents the amount of pre-tax goods and services sold, we have to subtract this away from the total receipts (2725) to get the amount of the tax collected:
2725 - 2500 = 225
Alternatively, we can plug into the statement above:
.09x = amount of tax collected
.09(2500) = 225
We can state our answer as:
Heather collected $225 in tax during her shift.
Step 6) We can read back through the problem to check our answer. We want to check that the amount of pre-tax goods and services (2500) plus the amount of the tax collected (225) is equal to the total receipts (2725):
2500 + 225 = 2725
2725 = 2725 ✔
Example 2: Solve each word problem.
Becky stayed in a resort for two consecutive days, Tuesday and Wednesday. On Tuesday, Becky paid $230. If the rate on Tuesday is 15% higher than it is on Wednesday, how much did Becky pay for the resort on Wednesday?
Step 1) After reading the problem, it is clear that we need to find the resort fee for Wednesday.
Step 2) We have one unknown, the resort fee for Wednesday.
let x = resort fee for Wednesday
Step 3) Write an equation, let's think about what we know.
We know that the resort fee for Tuesday is 15% higher than Wednesday:
x + .15x = 230
Step 4) Solve the equation:
x + .15x = 230
We can multiply both sides by 100 and clear the decimal:
100x + 15x = 23,000
115x = 23,000
115x/115 = 23,000/115
x = 200
Step 5) Since x (200) represents the resort fee for Wednesday, we can state our answer as:
The resort fee for Wednesday is $200.
Step 6) We can read back through the problem to check our answer. We want to check that the resort fee for Tuesday is 15% higher than Wednesday:
200 + .15 • 200 = 230
200 + 30 = 230
230 = 230 ✔

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