Ratio and Proportion 8.1-8.2

U n i t I I A D a y 1

Do Now  Simplify each fraction  12/15   14/56 21/6  90/450

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 _____

Ex. 1: Simplifying Ratios  a.

Simplify the ratios (be careful of units!): 12 cm b. 6 ft c. 9 in.

4 m 18 in.

18 in.

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.

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.

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

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.

 If 

=

 , then ________.

Ex. 5: Solving Proportions  Solve the proportion.

a) 4

x

= 5 7 b)

y

3 + 2 = 2

y

Geometric Mean  The geometric mean of two positive numbers a and b is the positive number x such that

a

=

x x b

 If you solve this proportion for x, you find that x = _____ which is a ______ number.

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

Homework #57  The ratios of the side lengths of unknown lengths.

Δ PQR to the corresponding side lengths of ΔSTU are 1:3. Find the

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.

 Find it by __________ a and b and then _____________.

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.

Closure  What distinguishes a ratio from a quotient?