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CHAPTER 7 Ratios and Proportion

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7-1 Ratios

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Ratio – quotient of two numbers and can be expressed as: 1.As a quotient using a division sign 2.As a fraction 3.As a ratio using a colon

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Write each ratio in simplest form 32:48 25x:20x 9x 2 y:6xy 2

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To write the ratio of two quantities of the same kind: First express the measures in the same unit Then write their ratio

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Write each ratio in simplest form 3 hr: 15 min 9 in: 5 ft 10 cm: 1 m

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7-2 Proportions

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PROPORTION – is an equation that states that two ratios are equivalent. a : b = c : d a = c b d b d

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TERMS – the four numbers a, b, c, and d that are related in the proportion.

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EXTREMES – the first and last terms a : b= c : d a and d are extremes

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MEANS – the second and third terms a : b = c : d b and c are means

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CROSS PRODUCTS – the product of the extremes equals the product of the means. ad= bc

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Find x: x = 12 18 27 Use cross products to write an equation. EXAMPLE 1

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Find x: x-4 = 2 5 3 5 3 Use cross products to write an equation. EXAMPLE 2

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Sam paid $10.50 for 5 blank video tapes. At the same rate, how much would he pay for 12 blank tapes. Write a proportion. Write a proportion. Let x = the cost of 12 blank tapes. EXAMPLE 3

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Fine Photo charges $3 for 2 enlargements. How much does the company charge for 5 enlargements? Write a proportion. Let x = the cost of 5 enlargements. EXAMPLE 4

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7-3 Equations with Fractional Coefficients

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To eliminate the fractional coefficients: Find the LCD Multiply both sides of the equation by the LCD Solve the remaining equation

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EXAMPLE 1 x + x = 10 x + x = 10 2 3 2 3

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EXAMPLE 2 x – x + 2 = 2 x – x + 2 = 2 3 5 3 5

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EXAMPLE 3 2n + n = n + 5 2n + n = n + 5 3 4 3 4

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EXAMPLE 4 x + 1 – x + 2 = 1 x + 1 – x + 2 = 1 3 4 2 3 4 2

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7-4 Fractional Equations

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Definition Fractional equation – an equation in which a variable occurs in a denominator.

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Definition Extraneous root – a root of the transformed equation but not a root of the original equation.

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To Solve: Multiply both sides of the equation by the LCD Solve the remaining equation Check all roots to see that they work in the original equation, and are not extraneous roots

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EXAMPLE 1 3 - 1 = 1 3 - 1 = 1 x 4 12 x 4 12

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EXAMPLE 2 2-x = 4 3-x 9

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EXAMPLE 3 2 - 2 = 1 b 2 – b b - 1 b 2 – b b - 1

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7-5 Percents

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Definition Percent – means hundredths or divided by 100. The symbol for percent is %.

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EXAMPLES 29 percent = 2.6 percent = 637 percent= 0.02 percent=

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Examples Write each number as a percent: 3/51/34.7

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EXAMPLES 15% of 180 is what number? 23 is 25% of what number? What percent of 64 is 48?

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Solving Equations To solve an equation with decimal coefficients, multiply both sides by a power of 10 To solve an equation with decimal coefficients, multiply both sides by a power of 10

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Examples Solve. 1.2x = 36 + 0.4x 94 = 0.15x + 0.08(1000 – x)

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7-6 Percent Problems

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Definition Percent of change 100 100 = change in price original price original price

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EXAMPLE 1 Find the percent increase: Jerry originally paid $600 per month to rent his apartment. It now costs him $650.

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EXAMPLE 2 To attract business, the manager of a musical instruments store decreased the price of an alto saxophone from $500 to $440. What was the percent decrease?

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EXAMPLE 3 Ricardo paid $27 for membership in the Video Club. This was an increase of 8% from last year. What was the price of membership last year?

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EXAMPLE 4 Sheila invested part of $6000 at 6% interest and the rest at 11% interest. Her total annual income from these investments is $460. How much is invested at 6% and how much at 11%.

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7-7 Mixture Problems

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EXAMPLES A health food store sells a mixture of raisins and roasted nuts. Raisins sell for $3.50/kg and nuts sell for $4.75/kg. How many kilograms of each should be mixed to make 20 kg of this snack worth $4.00/kg

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Solution Make a chart

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7-8 Work Problems

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EXAMPLES An installer can carpet a room in 3 hr. An assistant takes 4.5 hr to do the same job. If the assistant helps for 1 hr and then is called away, how long will it take the installer to finish?

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Solution Make a chart

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7-9 Negative Exponents

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DEFINITION If a is a nonzero real number and n is a positive integer, a -n = 1/a n

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EXAMPLES 10 -3 = 5 -4 = X -7 =

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DEFINITION If a is a nonzero real number a 0 = 1. The expression 0 0 has no meaning

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Rules for Exponents b m b n = b m+n b m ÷b n =b m - n b m ÷b n =b m - n (b m ) n =b mn (ab) m = a m b m (a/b) m =a m /b m (a/b) m =a m /b m

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7-10 Scientific Notation

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Scientific Notation To write a positive number in scientific notation, you express it as the product of a number greater than or equal to 1 but less than 10 and an integral power of 10.

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EXAMPLES 58,120,000 = 0.0000072 = 123,134,135 = 12.0233 =

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EXAMPLES 4.95 x 10 4 = 7.63 x 10 -5 = 9.3 x 10 2 = 1.032 x 10 -3 =

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Examples 3.2 x 10 7 2.0 x 10 4 (2.5 x 10 3 )(6.0 x 10 2 )

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