1. Ratio and Proportion
Unit IIA Day 1
8.1 and 8.2

2. Do Now
Simplify each fraction
12/15
14/56
21/6
90/450
4/5; ¼; 7/2; 1/5

3. Computing Ratios
If a and b are two quantities that are measured in the same units, then the ratio of a to b is a/b.
Can also be written as _____.
Because a ratio is a quotient, its denominator cannot be ________.
Ratios are usually expressed in simplified form. For instance, the ratio of 6:8 is usually simplified to _____
a:b
zero
3:4

4. Ex. 1: Simplifying Ratios
Simplify the ratios (be careful of units!):
12 cm b. 6 ft c. 9 in.
4 m in in.
12 cm/ 400 cm = 3/100
72 in/ 18 in = 4/1 = 4
9 in/ 18 in = 1/2

5. Ex. 2: Using Ratios
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.
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.
2L + 2w = P
2(3x) + 2(2x) = 60
6x + 4x = 60
10x = 60
x = 6
So the dimensions are 18 x 12 cm.

6. Ex. 3: Using Extended Ratios
The measures of the angles in ∆JKL are in the extended ratio 1:2:3.
Find the measures of the angles.
Begin by sketching a triangle. Then use the extended ratio of 1:2:3 to label the measures of the angles as x°, 2x°, and 3x°.
X = 30 so the measures are 30°, 2(30°) = 60°, and 3(30°) = 90°.
x°+ 2x°+ 3x° = 180°
6x = 180
x = 30

7. Properties of proportions
An equation that equates (sets equal) two ratios is called a __________________
The numbers a and d are called the extremes of the proportions.
The numbers b and c are called the means of the proportion.
Means
Extremes
 = 

8. Properties of proportions
Cross product property: The product of the “extremes” equals the product of the “means”.
If  = , then ________.
Reciprocal property: If two ratios are equal, then their reciprocals are also equal.

9. Ex. 5: Solving Proportions
Solve the proportion.
x = 28/5
y = 4

10. Geometric Mean
The geometric mean of two positive numbers a and b is the positive number x such that
If you solve this proportion for x, you find that x = _____ which is a ______ number. a x = b
x = sqrt(a*b); positive
compare to average aka arithmetic mean, where a – x = x – b
geometric mean — common ratio; arithmetic mean — common difference

11. Homework #46
The ratio of the width to the length for each rectangle is given. Solve for the variable.

12. Homework #57
The ratios of the side lengths of ΔPQR to the corresponding side lengths of ΔSTU are 1:3. Find the unknown lengths.

13. Comparing Arithmetic and Geometric Means
If x is the arithmetic mean of a and b, then x is the same ___________ from a and b.
Find it by __________ a and b and then _____________.
If x is the geometric mean of a and b, then x is the same ___________ from a and b.
distance
adding; dividing by 2
ratio
multiply; take the square root

14. Ex. 7: Using a Geometric Mean
International standard paper sizes all have the same width-to-length ratios.
Two sizes of paper are shown. The distance labeled x is the geometric mean of 210 mm and 420 mm. Find the value of x.
x = sqrt(210*420) = sqrt(88200) ≈ 297 mm

15. Closure What distinguishes a ratio from a quotient?
The numerator and denominator of a ratio must be measured in the same units.