Lesson 60: Direct and Indirect Variation

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Presentation transcript:

Lesson 60: Direct and Indirect Variation

When the statement of a problem says that A varies directly as B or that A is directly proportional to B, the equation A = kB Is implied.

When the statement says that A varies inversely as B or that A is inversely proportional to B, the equation A = k/B Is implied.

The constant k is called the constant of proportionality The constant k is called the constant of proportionality. Note that k is always in the numerator. In direct variation, both variables are in the numerator; and in inverse variation, one variable is in the numerator and the other is in the denominator. The key to working variation problems is recognizing the equation implied by the statement. The following examples should be helpful.

Statement: The number of boys varied directly as the number of girls Statement: The number of boys varied directly as the number of girls. Equation?

Answer: B = kG

Statement: The price varied inversely as the number. Equation?

Answer: P = k/N

Statement: The resistance is directly proportional to the length Statement: The resistance is directly proportional to the length. Equation?

Answer: R = kL

Statement: The number of revolutions per minute is inversely proportional to the number of teeth. Equation?

Answer: RPM = k/T

Statement: The water produced varied directly as the amount of hydrogen burned. Equation?

Answer: W = kH

Direct and indirect variation problems are four-step problems Direct and indirect variation problems are four-step problems. The first step is recognizing that the words varies directly (is directly proportional to) and varies inversely (is inversely proportional to) imply equations of the forms A = kB and A = k/B

The next step is to find k The next step is to find k. in order to find k, the problem must give sample values of A and B. The third step is to replace k in the equation with the proper number. The last steps are to reread the problem, make the final substitution, and then solve the equation.

Example: The number of boys in every classroom of a school varied directly as the number of girls. In one room there are 8 boys and 2 girls. If there are 5 girls in another room, how many boys are in this room?

Answer: B = kG (8) = k(2) k = 4 B = 4G B = 4(5) 20 boys

Example: The number of revolutions per minute (RPM) varies inversely as the number of teeth in the gear. If 40 teeth result in 100 RPM, what would be the RPM if the gear had 30 teeth?

Answer: RPM = k/N 100 = k/(40) k = 4000 RPM = 4000/N RPM = 4000/30 RPM = 133 1/3

Example: The number of clowns was directly proportional to the number of performers. If there were 40 clowns when there were 20,000 performers, how many clowns would there be if there were 12,000 performers?

Answer: C = kP 40 = k(20,000) k = 1/500 C = 1/500P C = 1/500(12,000) 24 Clowns

HW: Lesson 60 #1-30