6.3 Solving Proportions Using Cross Products

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Presentation transcript:

6.3 Solving Proportions Using Cross Products LESSON Solving Proportions Using Cross Products Every pair of ratios has two cross products. A cross product of two ratios is the product of the numerator of one ratio and the denominator of the other ratio. 3 5 6 10 , 2 3 6 11 , Ratios: Cross products: 3 • 10 5 • 6 2 • 11 3 • 6 Notice that for the ratios and , the ratios are equal and their cross products are also equal. For the ratios and , the ratios are not equal, and neither are their cross products. 3 5 6 10 2 11 You can use cross products to tell whether two ratios form a proportion. If the cross products are equal, then the ratios form a proportion.

6.3 Solving Proportions Using Cross Products 1 LESSON Solving Proportions Using Cross Products EXAMPLE 1 Determining if Ratios Form a Proportion Tell whether the ratios form a proportion. 12 20 32 50 , 12 20 32 50 = ? Write proportion. 12 • 50 20 • 32 = ? Form cross products. 600 = 640 Multiply. ANSWER The ratios do not form a proportion.

6.3 Solving Proportions Using Cross Products LESSON Solving Proportions Using Cross Products You can use the multiplication property of equality to demonstrate an important property about the cross products of a proportion. a b c d = Given. 1 bd 1 a b c d = • 1 Multiply each side by bd. Divide out common factors. ad = cb Simplify. This result proves the cross products property.

6.3 Solving Proportions Using Cross Products Cross Products Property LESSON Solving Proportions Using Cross Products Cross Products Property Words The cross products of a proportion are equal. Given that = , you know that 2 • 15 = 5 • 6. 2 5 6 15 Numbers If = , where b = 0 and d = 0, then ad = bc. a b c d Algebra You can use the cross products property to solve proportions.

6.3 Solving Proportions Using Cross Products 2 LESSON Solving Proportions Using Cross Products EXAMPLE 2 Writing and Solving a Proportion Hair Growth Human hair grows about 0.7 centimeter in 2 weeks. How long does hair take to grow 14 centimeters? SOLUTION 14 x 0.7 2 = Length of hair grown Number of weeks 0.7 • x = 2 • 14 Cross products property 0.7x = 28 Multiply. 0.7x 28 = = 0.7 Divide each side by 0.7. x = 40 Simplify. ANSWER Hair takes about 40 weeks to grow 14 centimeters.

6.3 Solving Proportions Using Cross Products LESSON Solving Proportions Using Cross Products Methods for Solving a Proportion SUMMARY To solve the proportion = , use one of the following: 5 12 x 36 Equivalent ratios 5 12 x 36 5 12  3 15 36  3 Algebra 5 12 x 36 = 36 • 36 • Multiply each side by 36. 15 = x Simplify. Cross products 5 • 36 = 12x Cross products property 15 = x Divide each side by 12.