RATIOS, RATES, & PROPORTIONS

1. RATIOS, RATES, & PROPORTIONS MSJC ~ San Jacinto Campus Math Center Workshop Series Janice Levasseur

2. RATIOS • A ratio is the comparison of two quantities with the same unit. • A ratio can be written in three ways: • As a quotient (fraction in simplest form) • As two numbers separated by a colon (:) • As two numbers separated by the word “to” • Note: ratios are “unitless” (no units)

3. Ex: Write the ratio of 25 miles to 40 miles in simplest form. What are we comparing? miles 25 miles to 40 miles Units, like factors, simplify (divide common units out) Simplify The ratio is 5/8 or 5:8 or 5 to 8.

4. Ex: Write the ratio of 12 feet to 20 feet in simplest form. What are we comparing? feet 12 feet to 20 feet Units, like factors, simplify (divide common units out) Simplify The ratio is 3/5 or 3:5 or 3 to 5.

5. Ex: Write the ratio of 21 pounds to 7 pounds in simplest form. What are we comparing? pounds 21 pounds to 7 pounds Units, like factors, simplify (divide common units out) Simplify The ratio is 3/1 or 3:1 or 3 to 1.

6. Your Turn to try a few

7. RATES • A rate is the comparison of two quantities with different units. • A rate is written as a quotient (fraction) in simplest form. • Note: rates have units.

8. Ex: Write the rate of 25 yards to 30 seconds in simplest form. What are we comparing? yards & seconds 25 yards to 30 seconds Units can’t simplify since they are different. Simplify The rate is 5 yards/6 seconds.

9. Ex: Write the rate of 140 miles in 2 hours in simplest form. What are we comparing? miles & hours 140 miles to 2 hours Units can’t simplify since they are different. Simplify The rate is 70 miles/1 hour (70 miles per hour, mph). Notice the denominator is 1 after simplifying.

10. Your Turn to try a few

11. UNIT RATES • A unit rate is a rate in which the denominator number is 1. • The 1 in the denominator is dropped and • often the word “per” is used to make the comparison. Ex: miles per hour  mph miles per gallon  mpg

12. Ex: Write as a unit rate 20 patients in 5 rooms What are we comparing? patients & rooms 20 patients in 5 rooms Units can’t simplify since they are different. Simplify The rate is 4 patients/1room  Four patients per room

13. Ex: Write as a unit rate 8 children in 3 families What are we comparing? Children& families 8 children in 3 families Units can’t simplify since they are different. How do we write the rate with a denominator of 1? Divide top and bottom by 3 The rate is 2 2/3 children/1 family  2 2/3 children per family

14. Your Turn to try a few

15. PROPORTIONS • A proportion is the equality of two ratios or rates. • If a/b and c/d are equal ratios or rates, then a/b = c/d is a proportion. • In any true proportion the cross products are equal: Why? Multiply thru by the LCM Simplify (bd) (bd)  ad = bc  Cross products are equal!

16. x 6 • We will use the property that the cross products are equal for true proportions to solve proportions. Ex: Solve the proportion x 6  72 If the proportion is to be true, the cross products must be equal  find the cross product equation:  7x = (12)(42)  7x = 504  x = 72

17. Ex: Solve the proportion If the proportion is to be true, the cross products must be equal  find the cross product equation:  24 = 3(n – 2)  24 = 3n – 6  30 = 3n x 2  10 = n Check: x 2

18. Ex: Solve the proportion If the proportion is to be true, the cross products must be equal  find the cross product equation:  (5)(3) = 7(n + 1)  15 = 7n + 7  8 = 7n  8/7 = n Check:

19. Your Turn to try a few

20. Ex: The dosage of a certain medication is 2 mg for every 80 lbs of body weight. How many milligrams of this medication are required for a person who weighs 220 lbs? What is the rate at which this medication is given? 2 mg for every 80 lbs Use this rate to determine the dosage for 220-lbs by setting up a proportion (match units)  Let x = required dosage x mg =  2(220) = 80x 220 lbs  440 = 80x  x = 5.5 mg

21. Ex: To determine the number of deer in a game preserve, a forest ranger catches 318 deer, tags them, and release them. Later, 168 deer are caught, and it is found that 56 of them are tagged. Estimate how many deer are in the game preserve. What do we need to find? Let d = deer population size In the original population, how many deer were tagged? 318 From the later catch, what is the tag rate? 56 tagged out of 168 deer We will assume that the initial tag rate and the later catch tag rate are the same

22. Set up a proportion comparing the initial tag rate to the later catch tag rate Initial tag rate = later catch tag rate  (318)(168) = 56d  53,424 = 56d 56 56  d = 954 deer in the reserve

23. Ex: An investment of $1500 earns $120 each year. At the same rate, how much additional money must be invested to earn $300 each year? What do we need to find? Let m = additional money to be invested What is the annual return rate of the investment? $120 for $1500 investment What is the desired return? $300

24. Set up a proportion comparing the current return rate and the desired return rate Initial return rate = desired return rate  120(1500 + m) = (1500)(300)  180,000 + 120m = 450,000  120m = 270,000 Divide by 120 • m = $2250 additional needs to be invest • new investment = $1500 + $2250 = $3750

25. Ex: A nurse is to transfuse 900 cc of blood over a period of 6 hours. What rate would the nurse infuse 300 cc of blood? What do we need to find? The rate of infusion for 300 cc of blood What is the rate of transfusion? 900 cc of blood in 6 hours Set up a proportion comparing the rate of tranfusion to the desired rate of infusion  But to set up the proportion we need to know how long it takes to insfuse 300 cc of blood  Let h = hours required

26. proportion comparing the rate of tranfusion to the desired rate of infusion   900h = (6)(300)  900h = 1800  h = 2 hours Therefore, it will take 2 hours to insfuse 300 cc of blood  New insfusion rate = 300 cc / 2 hours  150 cc/hour is the insfusion rate