Chapter 4 – Rates, Ratios, and Proportions

1. Chapter 4 – Rates, Ratios, and Proportions Math Skills – Week 5

2. Stuff • Class Project due Mar 16th • Guidelines on class website • Examples next week • Remember you can not miss more than 3 class periods • Equivalent fractions and multiplication / division • Product = multiply • Quotient = divide

3. Outline • Ratio – 4.1 • Rates – 4.2 • Proportions – 4.3

4. 4.1 – Ratios • Units. When we put a unit after a number, we give that number some physical context • 8 Feet • 8 Cars • 8 Tadpoles • Feet, Cars, and Tadpoles are all examples of units

5. 4.1 - Ratios • A Ratio is a comparison of two quantities that have the same units. • We write this comparison as • As a fraction • $6/$8 • As two numbers separated by a colon • $6:$8 • As two numbers separated by the word to • $6 to $8

6. 4.1 - Ratios • To write a ratio in Simplest Form, we write the two numbers so that they have no other common factors other than 1. • Review simplest form. • Steps • Find the prime factorization of each number • Cancel out all of the like quantities • The resulting numbers is the ratio • Example • Write the Ratio 8/10 in simplest form • 4/5

7. 4.1 - Ratios • Examples • Write the comparison $6 to $8 as a ratio in simplest form using a fraction, a colon, and the word to. • $6/$8 = 6/8 = ¾ • $6:$8 = 6:8 = 3:4 • $6 to $8 = 6 to 8 = 3 to 4 • Write the comparison 18 quarts to 6 quarts as a ratio in simplest form using a fraction, a colon, and the word to. • 18 quarts/6 quarts = 18/6 = 3/1 • 18 quarts:6 quarts = 18:6 = 3:1 • 18 quarts to 6 quarts = 18 to 6 = 3 to 1 • Example 4 pg.176

8. 4.1 - Ratios • Class Examples • Write the comparison of 20 pounds to 24 pounds as a ratio in simplest form using a fraction, a colon, and the word to. • 20 pounds/24 pounds = 20/24 = 5/6 • 20 pounds:24 pounds = 20:24 = 5:6 • 20 pounds to 24 pounds = 20 to 24 = 5 to 6 • Write the comparison of 64 miles to 8 miles as a ratio in simplest form using a fraction, a colon, and the word to. • 64 miles/8 miles= 64/8 = 8/1 • 64 miles:8 miles = 64:8 = 8:1 • 64 miles to 8 miles= 64 to 8 = 8 to 1

9. 4.2 - Rates • A rate is a comparison of two quantities that have different units. • Note: Rates are always written as fractions • Example • A distance runner ran 26 miles in 4 hours. The distance to time rate is: • 26 miles / 4 hours = 13 miles / 2 hours • Write 6 roof supports for every 9 feet as a rate in simplest form. • 6 roof supports / 9 feet = 2 roof supports / 3 feet

10. 4.2 - Rates • Class Example • Write “15 pounds of fertilizer for 12 trees” • 15 pounds / 12 trees = 5 pounds / 4 trees

11. 4.2 - Rates • A unit rate, is a rate that has 1 in the denominator • $3.25 / 1 pound or $3.25/pound is read as “$3.25 per pound” • To write a unit rate: • Steps • Divide the number in the numerator by the number in the denominator of the rate. Ex: On a trip, I traveled 344 miles before my car ran out of gas. My tank holds 16 gallons of gas. What is the unit rate that I traveled? • 344 miles/16 gallons = 21.5 miles/gallon

12. 4.2 - Rates • Example • Write “300 feet in 8 seconds” as a unit rate • 300 feet/8 seconds = 3.75 feet/second • Pg. 180 You Try It #3 • Class Example • Write “260 miles in 8 hours” as a unit rate • 260 miles/8 hours = 32.5 miles/hour

13. 4.3 - Proportions • A proportion is an expression of the equality of two ratios or rates. • If I say 50 Miles/4 gallons = 25 miles/2 gallons is this a true statement? • In order to say Yes or No, Things to check: • The units in the numerator and denominator must be the same for both rates/ratios. • Check that one ratio is the same as the other multiplied by 1 (written in fractional form) • 25/2 x 2/2 = 50/4, thus this is a true proportion

14. 4.3 - Proportions • To determine if a proportion is true. • Steps • Method 1 • Write each fraction in simplest form. • If the fractions are equal, we prove that the proportion is true • Example • Is the proportion 3/6 = ½ a true proportion? • 3/6 in simplest form is 1/2 , ½ = ½ thus this is a true proportion

15. 4.3 - Proportions • The cross product of a proportion is defined as shown below. 2 8 x = 3 12 x 2 x 12 = 24 3 x 8 = 24

16. 4.3 - Proportions • For a true proportion, the cross products of the proportion are always equal • To determine if a proportion is true: • Method 2 (Preferred) • Steps • Write the cross products of the proportion. • If the cross products are equal, we say the proportion is true. • Example: • Is the proportion 3/6 = ½ a true proportion? • Cross products are 3 x 2 = 6 and 6 x 1. Thus the proportion is true.

17. 4.3 - Proportions • In Summary: • A proportion is true if: • The fractions are equal when reduced to simplest form OR • The cross products are equal • A proportion is not true if: • The fractions are not equal when reduced to simplest form OR • The cross products are not equal

18. 4.3 - Proportions • Examples • Is 5/8 = 10/16 a true proportion? • Yes. 5 x 16 = 80 and 8 x 10 = 80, cross products are equal so this is a true proportion. • Is 62 miles/4 gallons = 33 miles/2 gallons a true proportion? • Nope. 62 x 2 = 132 and 33 x 4 = 124, cross products are not equal so this is not a true proportion. • Class examples • Is 6/10 = 9/15 a true proportion? • True. 6 x 15 = 90, and 9 x 10 = 90. • Is $32/6 hours = $90/8 hours a true proportion? • Not true. 32 x 8 = 256 and 90 x 6 = 540

19. 4.3 - Proportions • To solve a proportion, we need to find the missing number. • Think… • “What does n have to be in order for this proportion to be true?” 2 n n = 8 x = 3 12 x 2 x 12 = 24 3 x n = 24

20. 4.3 - Proportions • Examples • Solve n/12 = 25/60 • 60 x n = 12 x 25  60 x n = 300, 300/60 n=5. • Solve 4/9 = n/16 • 9 x n = 4 x 16  9 x n = 64  n = 64/9  n ≈ 7.1 • Pg.186 YouTryIt 8,9 • Class Examples • Solve 15/20 = 12/n • 15 x n = 20 x 12  240 = n x 15  n = 240/15  n = 16 • Solve n/12 = 4/1 • 1 x n = 12 x 4  1 x n = 48  n = 48/1  n = 48