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§ 2.4

§ 2.4. Formulas and Percents. Solving a Formula for a Variable. We know that solving an equation is the process of finding the number or number that makes the equation a true statement.

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§ 2.4

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  1. §2.4 Formulas and Percents

  2. Solving a Formula for a Variable We know that solving an equation is the process of finding the number or number that makes 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 2l +2w = P 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. Blitzer, Introductory Algebra, 5e – Slide #2 Section 2.4

  3. 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. 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. Blitzer, Introductory Algebra, 5e – Slide #3 Section 2.4

  4. Area of a Rectangle The area, A, of a rectangle with length l and width w is given by the formula A = lw. l w l Blitzer, Introductory Algebra, 5e – Slide #4 Section 2.4

  5. Perimeter of a Rectangle The perimeter, P, of a rectangle with length l and width w is given by the formula P = 2l + 2w. l w Blitzer, Introductory Algebra, 5e – Slide #5 Section 2.4

  6. Solving a formula for a variable Solve the perimeter equation for l. 2w + 2l = P 2w + 2l – 2w = P – 2w Subtract 2w from both sides. 2l = P – 2w Simplify. Divide both sides by 2. Simplify. Blitzer, Introductory Algebra, 5e – Slide #6 Section 2.4

  7. Solving a Formula for a Variable EXAMPLE Solve the formula y = mx + b for m SOLUTION y = mx + b Think of m saying, “I really want to be alone.” y – b = mx + b – b Subtract b from both sides. y – b = mx Perform the addition. b – b = 0. Divide both sides by m to find x. Blitzer, Introductory Algebra, 5e – Slide #7 Section 2.4

  8. Percents • Percents are the result of expressing numbers as a part of 100. The word percent means perhundred 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 Blitzer, Introductory Algebra, 5e – Slide #8 Section 2.4

  9. 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. Why does this work? Think about it… Blitzer, Introductory Algebra, 5e – Slide #9 Section 2.4

  10. Writing Decimals as Percents EXAMPLE Express 0.47 as a percent. 0.47 = 47% (since percent means 1/100, both sides here just say “47/100.”) Express 1.25 as a percent. 1.25 = 125% When you insert a percent sign, you must move your decimal two places to the right. Blitzer, Introductory Algebra, 5e – Slide #10 Section 2.4

  11. Writing Percents as Decimals And now, going the opposite way… Percents to Decimals • Move the decimal point two places to the left. • Remove the percent sign. Blitzer, Introductory Algebra, 5e – Slide #11 Section 2.4

  12. Writing Percents as Decimals EXAMPLE Express 63 % as a decimal. 63 % = .63 Express 150% as a decimal. 150% = 1.50 Two to the left, take it out, two to the right, put it in – Got it? Just recall the definition, and it all makes perfect sense! Blitzer, Introductory Algebra, 5e – Slide #12 Section 2.4

  13. 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. B is P percent Of A Blitzer, Introductory Algebra, 5e – Slide #13 Section 2.4

  14. Using the Percent Formula EXAMPLE 8 is what percent of 12? 8 = P · 12 8 = P12 8 is P percent Of 12 Blitzer, Introductory Algebra, 5e – Slide #14 Section 2.4

  15. Using the Percent Formula EXAMPLE What is 12% of 8? A = .12 · 8 A = .12 (8) A = .96 Thus, 12% of 8 is .96. Of 8 What is 12 percent Blitzer, Introductory Algebra, 5e – Slide #15 Section 2.4

  16. Using the Percent Formula EXAMPLE 5 is 25 percent of what? 5 = .25 · B 5 = .25B 5 is 25 percent Of What? Blitzer, Introductory Algebra, 5e – Slide #16 Section 2.4

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