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Solving Formulas for Variables: Methods and Examples

Learn how to solve formulas for variables by isolating the target variable using addition and multiplication. Examples provided.

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Solving Formulas for Variables: Methods and Examples

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  1. Section 2.4 • Formulas and Percents

  2. Objective 1 • Solve a formula for a variable.

  3. Solving a Formula for a Variable • We know that solving an equation is the process of finding the number or numbers that make the equation a true statement. • Formulas contain two or more letters, representing two or more variables. The formula for the perimeter P of a rectangle is where l is the length and w is the width of the rectangle. We say that the formula is solved for P, since P is alone on one side and the other side does not contain a P.

  4. Solving a Formula for a Variable • Solving a formula for a variable means using the addition and multiplication properties of equality to rewrite the formula so that the variable is isolated on one side of the equation.

  5. Solving a Formula for a Variable (cont) • To solve a formula for one of its variables, treat that variable as if it were the only variable in the equation. Think of the other variables as if they were just numbers. Use the addition property of equality to isolate all terms with the specified variable on one side. Then use the multiplication property of equality to get the specified variable alone. The next example shows how to do this.

  6. Area of a Rectangle • The area, A, of a rectangle with length l and width w is given by the formula l w

  7. Perimeter of a Rectangle • The perimeter, P, of a rectangle with length l and width w is given by the formula l w

  8. Solving a Formula for a Variable • Solve the perimeter equation for w. Subtract 2l from both sides. Simplify. Divide both sides by 2. Simplify.

  9. Example • Solve the formula y Subtract b from both sides. Simplify. Divide both sides by m to find x.

  10. Objective 1: Examples • 1a. Solve the formula for

  11. Objective 1: Examples (cont) • 1b.Solve the formula

  12. Objective 1: Examples (cont) • 1c.Solve the formula for

  13. Objective 1: Examples (cont) • 1d.Solve the formula for

  14. Objective 2 • Use the percent formula.

  15. Percents • Percents are the result of expressing numbers as a part of 100. The word percent means per hundred or 1/100. • If 45 of every 100 students take Introductory Algebra, then 45% of the students take Introductory Algebra. As a fraction, it is written

  16. Writing Decimals as Percents • Using the definition of percent, you should be able to write decimals as percents and also be able to write percents as decimals. Here is the rule for writing a decimal as a percent. • Move the decimal point two places to the right. • Attach a percent sign.

  17. Examples • Express 0.47 as a percent. • (since percent means 1/100, both sides here mean “47/100.”) • Express 1.25 as a percent. • When we insert a percent sign, we move the decimal point two places to the right.

  18. Writing Percents as Decimals • Use the following steps to write a percent as a decimal. • Move the decimal point two places to the left. • Remove the percent sign.

  19. Examples • Express 63% as decimal. • Express 150% as decimal.

  20. Percent Formula • A P · B • In the formula, • A PB • B Base Number • P Percent written as a decimal • A The number compared to B A is P percent of B

  21. Example • 8 is what percent of 12? • 8 P· 12 • Rounded to the nearest percent. 8 is P percent of 12

  22. Example • What is 12% of 8? • A 0.12 · 8 • Thus, 12% of 8 is 0.96. What is 12 percent 8 of

  23. Example • 5 is 25% of what number? • 5 0.25 · B • Thus, 5 is 25% is 20. what number? 5 is 25 percent of

  24. Objective 2: Examples • 2a.What is 9% of 50? • Use the formula • A is P percent of B. • 4.5 is 9% of 50.

  25. Objective 2: Examples • 2b. 9 is 60% of what? • Use the formula • A is P percent of B. • 9 is 60% of 15.

  26. Objective 2: Examples • 2c. 18 is what percent of 50? • Use the formula : A is P percent of B.

  27. Objective 2: Examples (cont) • To change 0.36 to a percent, move the decimal point two places to the right and add a percent sign. • 18 is 36% of 50.

  28. Objective 3 • Solve applied problems involving percent change.

  29. Percent Increase and Decrease • Percents are used for comparing changes, such as increases or decreases in sales, population, prices, and production. If a quantity changes, its percent increase or percent decrease can be determined by asking the following question: • The change is what percent of the original amount?

  30. Percent Increase and Decrease • The question is answered using the percent formula as follows: • Percent Increase Percent Decrease • The increase iswhat The decrease is whatpercent of the original percent of the original amount. amount.

  31. Objective 3: Examples • 3a.A television regularly sells for $940. The sale price is $611. Find the percent decrease in the television’s price? • Use the formula : • A is P percent of B.

  32. Objective 3: Examples (cont) • To change 0.35 to a percent, move the decimal point two places to the right and add a percent sign. • There was a 35% decrease.

  33. Objective 3: Examples (cont) • 3b.Suppose you paid $1200 in taxes. During year 1, taxes decrease by 20%. During year 2, taxes increase by 20%. What do you pay in taxes for year 2? How do your taxes for year 2 compare with what you originally paid, namely $1200? If the taxes are not the same, find the percent increase or decrease.

  34. Objective 3: Examples(cont) • First, find the amount that the taxes decreased from the original year to year 1: • Next, subtract this amount of decrease from the original tax amount to obtain the amount paid in year 1. • Amount paid in year 1:

  35. Objective 3: Examples(cont) • Now, find the amount that the taxes increased from year 1 to year 2: • Next, add this amount of increase to the amount paid in year 1 to obtain the amount paid in year 2. • Amount paid in year 2:

  36. Objective 3: Examples(cont) • The taxes for year 2 are less than those originally paid. • Since the taxes are not the same ($1200 original year, $1152 year 2), find the percent decrease.

  37. Objective 3: Examples(cont) • Find the tax decrease:

  38. Objective 3: Examples(cont) • To change 0.04 to a percent, move the decimal point two places to the right and add a percent sign. • The overall tax decrease is 4%.

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