Understanding Proportions

Understanding Proportions

Proportions is a way of comparing 2 similar things. You do them every day. Have you ever figured out how many miles your car can get per gallons of gasoline? Or how much something is per ounce? Or How how long it will take you to finish a project? Or how much money you can make per day? If you have you have just used the concept of proportions.

Before we look at problems and solutions using proportions, let’s look at and example together.

Question: Al painted 500 square feet in 6 hours. At that rate how many hours will it take him to paint 1500 square feet?

What is the question? At that rate how many hours will it take him to paint 1500 square feet?

What is the necessary information? 500 square feet, 4 hours, and 1500 square feet. The words at this rate are a clue that we will be using proportions as a setup to solve this problem.

What is the operation? They have given us three pieces of information and we will need to solve for the fourth. To do this we are going to put like things across from each other, cross multiply, and then divide.

So we have our

500 sq ft   =   1500 square feet

6 hours                      x

Remember we are going to place like things across from each other. And we are solving for x, the number of hours. And we need to cross multiply and then divide.

Solve it. So we are going to multiply 1500 x 6 = 9000 and then we need to divide 9000 by 500.

That is going to give us 18.

Does this answer make sense? Do you see that 1500 is 3 times larger than 500? So is 18 three times larger than 6? Yes, so this makes sense and is the correct choice. If you see patterns that double or triple, you can just multiply to get the answer.

 

500 x 3 = 1500 and

   6 x 3 = 18

So the answer is it will take 18 hours.

 

Understanding Fractions

Understanding Fractions

Don’t be afraid of fractions.  Fractions are nothing more than a piece of a whole number. For example, if you baked a birthday cake and you were expecting 10 people to come, but an 11th person showed up would you back an entirely new cake? No, you would take that cake and cut it into smaller pieces. A fraction is a piece of the whole number, each piece of the cake when added together creates the whole cake, the whole number.

Fractions are a piece of cake!

Let’s look at an example.

So now I want to show you how to use a dollar to turn fractions into decimals to make it easier. So let’s look. A dollar is represented as 1.00. Do we need the zeros? No, the zeros just it means there is no change. How much of a percentage is one dollar? Well if we move the spaces over by two places to the right we get 100%.

Now, let me show you how I am going to break this dollar into factions and how I am going to turn those fractions into percents.

I am going to break the dollar into 4ths. Because we know that there are four 25-cent pieces in a dollar or in other words, 4 quarters make up a dollar. Right? so,  we have…

Each one is also represented as a fourth because there are four pieces.

If you had just one piece, you’d have one-fourth or .25. If you have 2-pieces, you’d have .25 and/plus .25 or .5 which is half a dollar, or 50%.

No, if you have three-fourths, then you would have .25 added three times, which is .75 which as a percent moved over three times that would be 75%. And the same with the whole dollar.

Now let’s say you had three friends and you wanted to divide the dollar evenly among you and your friends. So, each would be .33 cents. Also written as ⅓. What happened to that extra penny?  I will show you where that extra penny goes.

Shows you division of 1 divided by 3. The three is repeated… Remember the rule for rounding. Fi it is less that 5 we round down. This is why we get .33.

Next, 2 divided by 3.

Now let’s take all of your money and add them all up.

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ASVAB Practice Exams

Congruence

Today we are going to go over Congruency.

This lesson will be broken up into four sections. We will start off by defining congruency and experiment with transforming shapes in any plane, understand congruence and how they relate to rigid motions, prove geometric theorems, and how to make geometric constructions with ease.

At the end of today’s lesson you will be able to make sense of geometric theorem problems and be able to solve them.

Defining Congruence

Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc.

 

geometry_congruence_definition

Now, let’s take a look at planes, perpendicular lines, parallel lines, circles and arcs.

geometry_congruence_definition 2

 

As you can see above, in geometry we use notation to show congruency among shapes. Now lets represents these transformations and compare their distance and angles.