Ratios and Proportions Section 7.1

Objective  Use ratios and proportions.

Key Vocabulary  Ratio  Proportion  Extremes  Means  Cross products

What is a Ratio? ratio A ratio is a comparison of two quantities using division.  It can be expressed as a to b, a : b, or as a fraction a/b where b ≠ 0.  Order is important! Part: Part Part: Whole Whole: Part

Ratio  Example: Your school’s basketball team has won 7 games and lost 3 games. What is the ratio of wins to losses? Because we are comparing wins to losses the first number in our ratio should be the number of wins and the second number is the number of losses. The ratio is

Ratio  Ratios are usually expressed in simplified form. For instance, the ratio of 6:8 is usually simplified to 3:4. Divide out common factors between the numerator and the denominator. A ratio in which the denominator is 1 is called a unit ratio.  Example:

Example 1 The total number of students who participate in sports programs at Central High School is 520. The total number of students in the school is Find the athlete-to-student ratio to the nearest tenth. To find this ratio, divide the number of athletes by the total number of students. Answer: The athlete-to-student ratio is can be written as

Your Turn A.0.3 B.0.5 C.0.7 D.0.8 The country with the longest school year is China with 251 days. Find the ratio of school days to total days in a year for China to the nearest tenth. (Use 365 as the number of days in a year.)

Example 2 Simplify the ratio. a. 60 cm : 200 cm b. 3 ft 18 in. SOLUTION a. 60 cm : 200 cm can be written as the fraction. 60 cm 200 cm Divide numerator and denominator by their greatest common factor, 20. = 60 cm 200 cm 60 ÷ ÷ Simplify. is read as “ 3 to 10.” = 3 10

Example 2 b. Substitute 12 in. for 1 ft. 2 1 Simplify. is read as “ 2 to 1.” = 2 1 Divide numerator and denominator by their greatest common factor, 18. = 36 ÷ ÷ 18 Multiply. = 36 in. 18 in. = 3 · 12 in. 18 in. 3 ft 18 in.

Example 3 In the diagram, AB : BC is 4 : 1 and AC = 30. Find AB and BC. AB + BC = AC Segment Addition Postulate 4x + x = 30 Substitute 4x for AB, x for BC, and 30 for AC. 5x = 30 Add like terms. x = 6 Divide each side by 5. SOLUTION Let x = BC. Because the ratio of AB to BC is 4 to 1, you know that AB = 4x.

Example 3 To find AB and BC, substitute 6 for x. AB = 4x = 4 · 6 = 24BC = x = 6 ANSWER So, AB = 24 and BC = 6.

Example 4 The perimeter of a rectangle is 80 feet. The ratio of the length to the width is 7 : 3. Find the length and the width of the rectangle. SOLUTION The ratio of length to width is 7 to 3. You can let the length l = 7x and the width w = 3x. 2l + 2w = P Formula for the perimeter of a rectangle 2(7x) + 2(3x) = 80 Substitute 7x for l, 3x for w, and 80 for P. 14x + 6x = 80 Multiply. 20x = 80 Add like terms. x = 4 Divide each side by 20.

Example 4 To find the length and width of the rectangle, substitute 4 for x. l = 7x = 7 · 4 = 28w = 3x = 3 · 4 = 12 ANSWER The length is 28 feet, and the width is 12 feet.

Your Turn: ANSWER EF = 16 ; FG = 8 1.In the diagram, EF : FG is 2 : 1 and EG = 24. Find EF and FG. ANSWER length, 24 ft ; width, 18 ft 2.The perimeter of a rectangle is 84 feet. The ratio of the length to the width is 4 : 3. Find the length and the width of the rectangle.

Your Turn:  The perimeter of rectangle ABCD is 60 centimeters. The ratio of AB: BC is 3:2. Find the length and the width of the rectangle

Solution:  Because the ratio of AB:BC is 3:2, you can represent the length of AB as 3x and the width of BC as 2x.

Solution: Statement 2 l + 2w = P 2(3x) + 2(2x) = 60 6x + 4x = 60 10x = 60 x = 6 Reason Formula for perimeter of a rectangle Substitute l, w and P Multiply Combine like terms Divide each side by 10 So, ABCD has a length of 18 centimeters and a width of 12 cm.

Extended Ratios  Ratios can also be used to compare 3 or more quantities, these are called extended ratios.  The extended ratio a:b:c means that the ratio of the first two quantities is a:b, the ratio of the last two quantities is b:c, and the ratio of the first and last quantities is a:c.

Example 5 In ΔEFG, the ratio of the measures of the angles is 5:12:13, and the perimeter is 90 centimeters. Find the measures of the angles. Just as the ratio or 5:12 is equivalent to or 5x:12x, the extended ratio 5:12:13 can be written as 5x:12x:13x. Write and solve an equation to find the value of x. ___ 5 12 ______ 5x5x 12x

Example 5 Answer: So, the measures of the angles are 5(6) or 30, 12(6) or 72, and 13(6) or 78. 5x + 12x + 13x= 180Triangle Sum Theorem 30x= 180Combine like terms. x= 6Divide each side by 30.

Your Turn A.30, 50, 70 B.36, 60, 84 C.45, 60, 75 D.54, 90, 126 The ratios of the angles in ΔABC is 3:5:7. Find the measure of the angles.

Important Ratios Find the first 13 terms in the following sequence: 1, 1, 2, 3, 5, 8, … Fibonacci Sequence This is called the Fibonacci Sequence!

Important Ratios What happens when you take the ratios of two successive Fibonacci numbers, larger over smaller? What number do you approach? ETC!

Important Ratios

The Golden Ratio The Golden Ratio = …

The Golden Ratio  The ancient Greeks considered the Golden Ratio when used in shapes the most aesthetically pleasing to the eye.  The Greeks used the Golden Ratio to do everything from making a pentagram, to constructing a building, to combing their hair.

Cartoon

What’s a Proportion?  A proportion is an equation stating that two ratios are equal.  Example:

Proportions  If the ratio of a/b is equal to the ratio c/d; then the following proportion can be written:  The values a and d are the extremes. The values b and c are the means. When the proportion is written as a:b = c:d, the extremes are in the first and last positions. The means are in the two middle positions.  = MeansExtremes

Properties of Proportions CROSS PRODUCT PROPERTY 1. The product of the extremes equals the product of the means. If

Solving a Proportion What’s the relationship between the cross products of a proportion? 360  12.4  150 They’re equal!

Solving a Proportion To solve a proportion involving a variable, simply set the two cross products equal to each other. Then solve! 275  x15  25

More Proportion Properties

Example 6 A. Answer: y = 27.3 Original proportion Cross Products Multiply. Divide each side by 6. y = 27.3

Example 6 B. Answer: x = –2 Original proportion Cross Products Simplify. Add 30 to each side. Divide each side by 24.

Your Turn A.b = 0.65 B.b = 4.5 C.b = –14.5 D.b = 147 A.

Your Turn A.n = 9 B.n = 8.9 C.n = 3 D.n = 1.8 B.

Example 7 5 · 6 = 3(y + 2) Cross product property 30 = 3y + 6 Multiply and use distributive property. 30 – 6 = 3y + 6 – 6 Subtract 6 from each side. 24 = 3y Simplify. 8 = y Simplify. Solve the proportion. = y SOLUTION Write original proportion. = y Divide each side by 3. = 3y3y

Example 7 Solve a Proportion CHECK Check your solution by substituting 8 for y. === y + 2 6

Your Turn: ANSWER Solve the proportion. 1. = x 2. = 15 y = m

PROPORTIONS CAN BE USED TO MAKE PREDICTIONS

Example 8 PETS Monique randomly surveyed 30 students from her class and found that 18 had a dog or a cat for a pet. If there are 870 students in Monique’s school, predict the total number of students with a dog or a cat. Write and solve a proportion that compares the number of students who have a pet to the number of students in the school.

Example 8 18 ● 870= 30xCross Products Property 15,660= 30xSimplify. 522= xDivide each side by 30. ←Students who have a pet ←total number of students Answer: Based on Monique's survey, about 522 students at her school have a dog or a cat for a pet.

Your Turn A.324 students B.344 students C.405 students D.486 students Brittany randomly surveyed 50 students and found that 20 had a part-time job. If there are 810 students in Brittany's school, predict the total number of students with a part-time job.

Equivalent Ratios SSimplify the following ratios: 4 to 8 10 to 8 8 to 10 Step 1 – Write the ratio as a fraction Step 2 – Simplify the fraction (Find the greatest common factor (GCF) of both numbers and divide the numerator and denominator by the GCF). Step 3 – Write the equivalent ratio in the same form as the question 4=4= 4 / 4 = 1 = 1 to / 4 2 GCF = 4

Equivalent Ratios can be formed by multiplying the ratio by any number.  For example, the ratio 2 : 3 can also be written as 4 : 6 (multiply original ratio by by 2) 6 : 9 (multiply original ratio by by 3) 8 : 12 (multiply original ratio by by 4) The ratio 2 : 3 can be expressed as 2x to 3x (multiply the original ratio by any number x)

Equivalent Proportions  Equivalent forms of a proportion all have the same cross product.  The following proportions are equivalent.  Examples:

Assignment  Pg. 361 – 363 #1 – 55 odd