1. Ratio and Proportions

2. Ratio of a to b The quotient a/b if a and b are 2 quantities that are measured in the same units can also be written as a:b. * b cannot = 0, because the denominator cannot be 0. Always write ratios in simplified form! (reduce the fraction!)

3. Ratios A ratio is a comparison of numbers that can be expressed as a fraction. If there were 18 boys and 12 girls in a class, you could compare the number of boys to girls by saying there is a ratio of 18 boys to 12 girls. You could represent that comparison in three different ways: –18 to 12 –18 : 12 – 3 18 12

4. Ratios The ratio of 18 to 12 is another way to represent the fraction All three representations are equal. –18 to 12 = 18:12 = The first operation to perform on a ratio is to reduce it to lowest terms –18:12 = –18:12 = 3:2 4 18 12 18 12 18 12 3 2 3 2 3 2 ÷ 6

5. Example: simplify.

6. Ratios A basketball team wins 16 games and loses 14 games. Find the reduced ratio of: –Wins to losses – 16:14 = = –Losses to wins – 14:16 = = –Wins to total games played – 16:30 = = The order of the numbers is critical 6 1614 8 7 1416 7 8 1630 815

7. Example: the perimeter of an isosceles  is 56in. The ratio of LM:MN is 5:4. find the lengths of the sides of the . L M N 5x5x 4x4x 5x +5x+ 4x=56 x=4 ) ( 14x=56

8. Ex: the measure of the  s in a  are in the extended ratio 3:4:8. Find the measures of the  s of the . 3x+4x+8x=180 15x=180 x=12 Substitute to find the angles: 3(12)=36, 4(12)=48, 8(12)= 96 Angle measures: 36 o, 48 o, 96 o

9. Proportion An equation stating 2 ratios are = b and c are the means a and d are the extremes

10. Proportions A proportion is a statement that one ratio is equal to another ratio. –Ex: a ratio of 4:8 = a ratio of 3:6 –4:8 = = and 3:6 = = –4:8 = 3:6 – = –These ratios form a proportion since they are equal to other. = 10 4 8 1 2 3 6 1 2 4 8 3 6 1 2 1 2

11. Properties of Proportions Cross product property- means=extremes 1. If then, ad=bc Reciprocal Property- (both ratios must be flipped) 2. If, then

12. Proportions In a proportion, you will notice that if you cross multiply the terms of a proportion, those cross-products are equal. 12 4 8 3 6 3 2 18 18 12 12 4 x 6 = 8 x 3 (both equal 24) 3 x 12 = 2 x 18 (both equal 36) = =

13. Proportions 13 N 12 12 3 4 = 4 x N = 12 x 3 4N = 36 4 N 36 4 4 4 4 1N = 9 N = 9 = Cross multiply the proportion Divide the terms on both sides of the equal sign by the number next to the unknown letter. (4) That will leave the N on the left side and the answer (9) on the right side

14. Proportions Solve for N 14 2 5 N 35 35 = 5 x N = 2 x 35 5 n = 70 5 N 70 5 5 5 5 1N = 14 N = 14 = 15 15 N 3 4 = 6 7 102 102 N = 4 N 6 27 27 =

15. Proportions At 2 p.m. on a sunny day, a 5 ft woman had a 2 ft shadow, while a church steeple had a 27 ft shadow. Use this information to find the height of the steeple. 2 x H = 5 x 27 2H = 135 H = 67.5 ft. 15 5 2 H 27 27 = height heightshadow = shadow You must be careful to place the same quantities in corresponding positions in the proportion

16. Ratios The ratio of freshman to sophomores in a drama club is 5:6. There are 18 sophomores in the drama club. How many freshmen are there?

17. Freshman = 5 = x Sophomore 6 18 15 freshmen

18. Example 10(s-5)=4s 10s-50=4s -50= -6s

19. The ratios of the side lengths of  QRS to the corresponding side lengths of  VTU are 3:2. Find the unknown length. Q R S X y V u T 18cm 2cm z w

20. Example cont x= 3cm a 2 +b 2 =c 2 3 2 +18 2 =y 2 9+324=y 2 333=y 2 y ≈ 18.25cm 2 2 +12 2 =z 2 4+144=z 2 148=z 2 z ≈ 12.17cm w= 12cm