Math 6A ( Periods 1 & 2)

Proportions 7-7

Let’s continue our discussion of mythical middle schools

The 6^th grade class at Madison Middle School has 160 students and 10 teachers.

The 6^th grade class at Jefferson Middle School has 144 students and 9 teachers.

Let’s compare the two teacher to student ratios!

Thus the two ratios are equal

An equation that states that two ratios are equal is called a proportion.

The proportion above may be read as

10 is to 160   AS    9 is to 144

The numbers 10, 160, 9, and 144 are called the TERMS of the proportion.

Sometimes (especially in this textbook!) one of the terms of the proportion is missing—or is a variable.

For example, 

Let’s say we know that next year the student population  at Madison will be at 192 students. How many teachers will be needed if the teacher to student ratio is to remain the same?

First, write a “let statement” to identify your variable

Let n = the number of teachers needed next year

Then, if the teacher to student ratio is to be the same, we must have

To solve this proportion, we find the value of the variable that makes this equation true.

This can be done by finding equivalent fractions with a common denominator…

 Since the denominators are equal the numerators must also be equal so we have 160n = 10(192)

What do you do NOW?

divide carefully

n = 12

Notice that this results could also have been obtained by cross-multiplying in the original proportions. That is

to get

160n = 10(192)

n = 12

There for the school will need 12 teachers next year.

Property of Proportions                        

 with b≠  0 and d ≠ 0       Then ad = bc

3n = 8(12)

3n = 96

divide both sides by 3

n =  32

But WAIT—could you have done this another way?

Sure

What do you do to 3 to get 12? ( multiply by 4)

… so what must you do to 8?  ( multiply by 4)

that’s a great check.

So what happens if you have

from the textbook we learned

m² = 3(27)

m² = 81

Now, we have worked with square roots before—so you should be able to solve this problem.

Chapter 5 covered square roots.

Technically, you perform the following

m= 9