Linear Equations and Inequalities in One Variable

Formulas and Percents

This section explores how to manipulate algebraic formulas to solve for a specific variable and how to apply percent concepts in various mathematical and real-world contexts. Mastery of these skills is essential for solving equations, interpreting mathematical models, and handling applied problems involving percentages.

Solving a Formula for a Variable

• Definition: Solving a formula for a variable means rewriting the equation so that the chosen variable is isolated on one side. The goal is not to find a numerical value, but to express the variable in terms of the others.

• Method:

  • Treat all other variables as constants.

  • Use the addition and multiplication properties of equality to isolate the desired variable.

• Example 1: Solve the area formula for length in .

  • Divide both sides by :

  • Result:

• Example 2: Solve the perimeter formula for .

  • Subtract from both sides:

  • Divide both sides by 2:

• Example 3: Solve for .

  • Subtract from both sides:

  • Divide both sides by :

• Example 4: Solving a formula containing a fraction for a variable follows the same principles: clear denominators if necessary, then isolate the variable.

Percent Review

• Definition: A percent is a ratio that compares a number to 100. The symbol % means "per hundred."

• Fraction and Decimal Forms:

  • 45% = = 0.45

Writing Percents as Decimals

• Move the decimal point two places to the left and drop the percent sign.

• Examples:

  • 63% = 0.63

  • 150% = 1.50

Writing Decimals as Percents

• Move the decimal point two places to the right and add a percent sign.

• Examples:

  • 0.47 = 47%

  • 1.25 = 125%

Percent Formula

• The percent formula relates three quantities: the part (), the percent ( as a decimal), and the base ():

• Where:

  • = the part (the number compared to )

  • = percent (written as a decimal)

  • = base (the reference number)

Types of Percent Problems

• There are three basic types of percent problems, all solved using :

• Type 1: What number is percent of ?

  • Example: What number is 9% of 50?

• Type 2: is percent of what number?

  • Example: 9 is 60% of what number?

• Type 3: is what percent of ?

  • Example: 18 is what percent of 50?

Percent Increase or Decrease

• Percents are used to compare changes in quantities, such as increases or decreases in prices, populations, or sales.

• To find the percent increase or decrease, use the formula:

Percent Increase:

Percent Decrease:

• Convert the result to a percent by moving the decimal point two places to the right and adding a percent sign.

Example: Finding Percent Decrease

• A television regularly sells for $940. The sale price is $611. Find the percent decrease.

• Decrease:

• Percent decrease:

Example: Successive Percent Changes

• Suppose you paid $1200 in taxes. Year 1: taxes decrease by 20%. Year 2: taxes increase by 20%.

• Year 1:

• Year 2:

• Overall change: decrease

• Percent decrease:

Additional info: Successive percent changes do not simply add or subtract; each change is based on the new amount, not the original.