CHAPTER 7 Ratios and Proportion 7 1 Ratios

CHAPTER 7 Ratios and Proportion

7 -1 Ratios

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

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

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

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

7 -2 Proportions

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

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

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

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

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

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

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

EXAMPLE 3 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. Let x = the cost of 12 blank tapes.

EXAMPLE 4 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.

7 -3 Equations with Fractional Coefficients

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

EXAMPLE 1 x + x = 10 2 3

EXAMPLE 2 x–x+2=2 3 5

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

EXAMPLE 4 x+1–x+2=1 3 4 2

7 -4 Fractional Equations

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

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

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

EXAMPLE 1 3 - 1= 1 x 4 12

EXAMPLE 2 2 -x = 4 3 -x 9

EXAMPLE 3 2 - 2 =1 b 2 – b b - 1

7 -5 Percents

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

EXAMPLES 29 percent = 2. 6 percent = 637 percent= 0. 02 percent=

Examples Write each number as a percent: 3/5 1/3 4. 7

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

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

Examples Solve. 1. 2 x = 36 + 0. 4 x 94 = 0. 15 x + 0. 08(1000 – x)

7 -6 Percent Problems

Definition Percent of change 100 = change in price original price

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

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?

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?

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%.

7 -7 Mixture Problems

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

Solution Make a chart

7 -8 Work Problems

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?

Solution Make a chart

7 -9 Negative Exponents

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

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

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

Rules for Exponents bmbn = bm+n bm ÷bn =bm - n (bm)n =bmn (ab)m = ambm (a/b)m =am/bm

7 -10 Scientific Notation

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.

EXAMPLES 58, 120, 000 = 0. 0000072 = 123, 134, 135 = 12. 0233 =

EXAMPLES 4. 95 x 104 = 7. 63 x 10 -5 = 9. 3 x 102 = 1. 032 x 10 -3=

Examples 3. 2 x 107 2. 0 x 104 (2. 5 x 103)(6. 0 x 102)

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