Warm Up Let’s Review Classroom Rules!

True or False A pass is not needed to go to the bathroom.

True or False After sitting in your assigned seat, prior night’s homework should be placed on the right side of your desk.

True or False If I am absent, it is my responsibility to find out what I missed.

True or False When the bell rings, it is ok to get up and leave.

True or False It is ok to be disrespectful to my teachers and peers. This includes using words like shut up.

8.1 Ratio and Proportions

Ratio: a ratio is a quotient of two numbers.

a:b a to b a÷b

ab

Always given in lowest terms.

Slope of a line is a ratio between two points. (rise over run)

Proportions: two or more ratios set equal to each other.

ab

cd= a:b = c:d

a is the first termb is the second termc is the third termd is the fourth term

Product and Ratio TheoremsIn a product containing four terms:First and fourth terms are the extremes.Second and third terms are the means.T59: In a proportion, the product of the means is equal to the product of the extremes. (means-extremes product theorem.)

ab

cd= ad = bc

If they aren’t equal, then the ratios aren’t in proportion.

T60: If the product of a pair of non-zero numbers is equal to the product of another pair of non-zero numbers, then either pair of numbers may be made the extremes, and the other pair the means, of a proportion. (means-extremes ratio theorem.)

This theorem is harder to state than to use!Given: pq = rsThen:

pr

sq

ps

rq

rp

qs

= = =

pq = rs pq = rs pq = rs

These proportions are all equivalent since their cross products are equivalent equations.

In a mean proportion, the means are the same.

416

14 =

ax

xr=

4 is the geometric mean

x is the geometric mean

Definition: If the means in a proportion are equal, either mean is called a geometric mean or mean proportional between the extremes.

Arithmetic mean:

2273

= 15

Geometric mean:

3x

x27=

x2 = 81 x = 9

Solve:

714

3x

= You might want to reduce the fraction first.

7x = 42 x = 6

23

4x=

2x = 12 x = 6

Find the mean proportional(s) between 4 and 16.

4x

x16=

x2 = 64

x = 8If we are looking for the length of

a segment, then only the positive number works.

If 3x = 4y, find the ratio of x to y.

xy

Make x and 3 the extremes and y and 4 the means.

3x = 4y

xy

43

=

Is

ab

xy = equal to

a 2bb

x 2yy

= ?

Cross multiply and simplify both sets.

ay = bx

b(x-2y) = y(a-2b)bx-2by = ay-2by bx = ay

Yes, they are equal.