Bell Work:

The ratio of apples to oranges was 7 to 2. if there were 714 apples, how many oranges were there?

Answer:

7/2 = 714/O

204 Oranges

Lesson 39:Trichotomy Axiom,

Negated Inequalities,

Advanced Ratio Problems

Trichotomy Axiom:

Johnny wrote a number on a piece of paper. Then he turned the paper over and wrote a number on the other side. There are exactly three possibilities.

1. The second number is less than the first

2. The second number is the same as the first

3. The second number is greater than the first number

Since this property has three parts, we give it the name trichotomy.

Trichotomy Axiom*: for any two real numbers a and b, exactly one of the following is true:

a < b a = b a > b

Negated Inequalities:

Symbols of inequality can be negated by drawing a slash through the symbol.

x > 10

“x is not greater than 10”

There are only three possibilities under the trichotomy axiom, and if x is not greater than 10, then it must be less than or equal to 10.

x ≤ 10

Practice:

What is another way we could say this statement?

x ≥ 6

Answer:

x < 6

Example:

Draw a number line and graph the solution of x < 2.

Answer:

Practice:

Write both an inequality and a negated inequality that describe this graph.

Answer:

x < 3 x ≥ 3

Some ratio problems are difficult because key information is hidden by the way the problem is worded. If we are told that the ratio of red marbles to blue marbles is 5 to 7, we would write

R = 5

B 7

Now if we are told that we have a total of 156 marbles and are asked for the number of marbles that are red, we would have difficulty because there is no place for the total in the equation we have written. If we use the following four-step procedure, we can work any ratio problem with ease because this method will produce three useful equations.

1. Write the information given to include the total

2. Use the cover-up method to write three equations

3. Reread the problem to determine which equation to use

4. Substitute in the selected equation and solve the problem

Example:

The ratio of red marbles to blue marbles is 5 to 7. if there are 156 marbles in the bag, how many marbles are red?

Step 1

Write all information to include the total. If 5 are red and 7 are blue, the total is 12.

R = 5

B = 7

T = 12

Step 2

Write all the implied equations. In this problem there are three implied equation. We can recognize the equations if we cover part of the information with a finger.

R = 5

B = 7 a) B/T = 7/12

T = 12

R = 5

B = 7 b) R/T = 5/12

T = 12

R = 5

B = 7 c) R/B = 5/7

T = 12

Step 3

Now we reread the questions. It says we have 156 total and asks for the number that are red. This tells us to use equation (b) because the variables in this equation are R and T.

R + 5

T 12

Step 4:

Now we substitute 156 for T and solve for R.

R = 5 12R = 5 156 R = 65

156 12 12 12

Practice:

The ratio of fish to crabs in the sea cave was 13 to 4. if there were 119 fish and crabs in the cave, how many were fish?

Answer:

F = 13

C = 4

T = 17

a) C/T = 4/17 b) F/T = 13/17 c) F/C = 13/4

Equation (b) fits.

F/119 = 13/17

F = 91

HW: Lesson 39 #1-30