ASVAB: MATH

two, whole, positive, negative, fractions, whole, 1

Rational numbers are numbers that can be written as the fraction of ____________ integers. Remember that integers are ____________________ numbers, both _____________________ and _____________________. For example, splitting 4 donuts among 3 people gives you 4/3, which is a rational number. Rational numbers include both ________________________ and your ___________________ numbers because you can rewrite your whole numbers as a fraction being divided by _________. Your rational expressions, then, are math statements with rational numbers in them.

common denominator, denominator, common denominator, denominator

To add fractions you must have a _________________________ __________________. When looking at your fraction, you need to make sure that the fractions you are adding have the same _______________________. Same as adding fractions, to subtract fractions you must have a __________________ ___________________. When looking at your fraction, you need to make sure that the fractions you are subtracting have the same ________________________________.

least common denominator, multiple, least common denominator

To add unlike fractions, you need to find the _____________________ ________________ _________________. The denominator is the number on the bottom. The least common denominator is the smallest shared _________________________ of the denominators. To subtract unlike fractions, we do the same thing - find the _________________ _________________ ___________________.

fraction, numerator, 98, denominator, 1, two, 98/100, 49/50, denominator, numerator

To change a decimal into a fraction, we first write the number before the decimal point down. Now we convert the number after the decimal point into a ______________________. The number after the decimal point will be our top number, our _______________________. In the case of converting the decimal 0.98 into a fraction, you'd put ____________ as our numerator. Our __________________________, our bottom number, will be a ________ followed by zeroes. The number of zeroes will equal the number of decimal spaces we have. In the case of 0.98, we'd have __________ zeroes attached to the 1. Following those steps, the decimal 0.98 converts to the fraction __________/__________ which simplifies down to _________/___________ We can turn it into an improper fraction by multiplying our whole number with the _______________________ and adding the ____________________________ part of our fraction part.

two

A binary operation is a mathematical process that uses how many numbers to accomplish something?

two, prime,

A composite number is a number that has more than _____________ factors. In other words, a composite number is the opposite of a _________________ number.

one, fractions

A decimal number is defined as a number that has a decimal point in it. A decimal point is a point or dot used to show the beginning of digits that are smaller than _________. Decimal numbers are another way to represent __________________________.

one

A non-binary operation is a mathematical process that only needs how many numbers to accomplish something?

positive, two, 2, 5

A prime number is any _____________________________ number that has just __________________ factors: 1 and itself. The first rule states that no even number (other than _______) can be a prime number. Since all these numbers are divisible by 2, they will all have a factor of 2 along with 1 and themselves. Therefore, they'll all be composite numbers. We saw an example of this earlier when we looked at the factors of 6. The second rule tells us that all numbers that end in a 5 or a 0 (other than ________ itself) can never be prime numbers. This is because they are all divisible by 5. Similar to the previous rule, these will always be composite numbers, since they will have at least a factor of 5, along with 1 and themselves. Finally, the third rule informs us that 0 and 1 are not prime numbers, even though they might seem like it at first. Remember a prime number must only have two factors, 1 and itself. Both of these numbers break this rule. For instance, 0 has an infinite number of factors. Every number is a factor of 0 because anything multiplied by 0 equals 0. So 0 is far from prime.

fraction, ratio, 2/1

A rational number is a number that can be written as a _____________________ or _____________________ of integers. Remember that a fraction is a ratio. 2 is a rational number. We could write it as a fraction: _______/________. Likewise, 7/8 is a rational number.

denominator, improper fraction

An improper fraction is a fraction where the numerator is larger than the __________________________________. When multiplying and dividing fractions, you must use an __________________ _________________ rather than mixed numbers.

length x width

Area=

multiply, 1,000, 125/1000,

Converting decimals to fractions follows the same logic as this second method. Let's try it with 0.125. To start, put the decimal in the numerator and 1 in the denominator. That looks like this: the decimal over 1. With our example of 0.125, we'd have 0.125/1. Next, ______________________ the top and bottom of the fraction by a power of ten to eliminate the decimal. In other words, count how many numbers are after the decimal. For 1, multiply by 10. For 2, multiply by 100, and so on. 0.125 has 3 numbers after the decimal, so that's ______________. You can also figure this out by reading the decimal aloud. We have 125 one-thousandths, so we want to multiply the fraction by 1000. What's 0.125/1 times 1000/1000? _________/______________. The third and final step is to simplify the fraction. Can we simplify this fraction? The first thing to do is see if 1000 is divisible by 125. Guess what? It is! This fraction simplifies to 1/8. So 0.125 is the same as 1/8. If 0.125 is our cat, in another life it's 1/8.

threes, twos, ones, 5*4, 20, common, 1, 1/20

Dividing Factorials 5!/3! First, we write out what each factorial means. So, we have 5*4*3*2*1/3*2*1. Well, now we see that there are some things we can cancel out since we have the same number in the numerator AND the denominator. We can cancel out the _____________ and the _________________ and the ______________. So, we are left with _______*________. We know that 5*4 equals _____________, and we are done. So, what happens when our denominator is larger than our numerator? What if we divided three factorial by five factorial? Well, we would have 3!/5!. Written out we have 3*2*1/5*4*3*2*1. Canceling out the numbers that are _________________________to both the numerator and denominator, we have 1/5*4. I left the one in the numerator because we know from algebra that if everything cancels out, there is always a _____ there, as in x/x = 1. Now, to finish evaluating our factorial, we multiply out the 5*4 to get 20. We keep our answer in fraction form. So, our answer is ______/________.

index, evenly, 6, radical,

Division by a Radical We are going to go about dividing our radicals in two different ways depending on what kind of division problem we see. Recall that each radical has an ___________________ number which is the little number written in the little dip of the radical symbol. If our division problem has the same index for both the numerator and the denominator, and if the denominator divides ____________________ into the numerator, then we will go ahead and divide the numerator and denominator, combining them under the same radical symbol. So, for example, the third root of 24 divided by the third root of 4 becomes the third root of 24 divided by 4, which is the third root of __________. Now, if we had the third root of 4 divided by the third root of 8, we would actually go ahead and evaluate the third root of 8, because that would eliminate one of the radicals, thus simplifying our expression. So, we would have the third root of 4 divided by 2. So, a rule here is if we can go ahead and evaluate a radical thus removing the __________________________, then we should by all means do so. Now, if we have any other case where we still have a radical in the denominator, then we'll have to simplify our radical so that we don't have a radical in the denominator.

whole, 2, right, 72.4

Division is also similar to dividing whole numbers. The only difference here is that if the number we are dividing by is a decimal, then we will want to convert it to a ________________ number before we divide. If we are dividing 7.24 by 0.2, we would first change the 0.2 to a _______. To do that, we move the decimal place over one space to the ________________. Because we are doing this to one number, we also need to move the decimal place one place to the right in the other number. So my 7.24 becomes _____________. Now we can go ahead with our long division of 72.4 divided by 2. Once we get our answer, we write in the decimal point so that it matches the position of the decimal point in the ____________

multiplication

Division of factorials just gives us shortened chains of _______________________________. As a quick note, while division of factorials has this neat little trick, there is no such trick with multiplication.

multiply

Factors of a number are the different numbers that you can ____________________ together to get that original number.

rate = gain or loss/ time gain or loss= rate x time

The Rate Formula There are two ways you can write this mathematically. You can write it using division as - ________________________________ - or, you can write it using multiplication like this - _____________________________________

improper fractions, top, bottom, simplify

Let's talk about how we multiply mixed numbers. This is a four-step process. Step one: convert to _______________________ ____________________. Step two - well, really, we just follow the steps from earlier: multiply the _____________, multiply the ________________, then _____________________.

natural, counting

On the numbers side of things, the most basic number is the number 1, and the most specific descriptor of where it lives is called the ______________________ numbers. The natural numbers are all the numbers that you learn when you're a baby, like 1, 2, 3, 4, 5, 6 and on and on. The natural numbers are also sometimes called the _________________________ numbers because they're the first numbers you learn how to count.

% increase or decrease= (increase or decrease / first total) x 100

Percent Increase and Decrease Formula

2l + 2w

Perimeter=

(n + 2) and (n + 1).

Simplify (n + 2)! / n! These types of problems can be intimidating because the lack of numbers makes it seem entirely different, but that's not the case. We can still use the exact same division of factorial skills we just learned to simplify this factorial expression out. We want to think, how much will cancel out and what will be left over? Well the n! in the denominator can be rewritten as n * (n - 1) * (n - 2) * (n - 3) *... forever or as long as we need to go until we get to 1. While the (n + 2)! on the top is (n + 2) * (n + 1) * n * (n - 1) * (n - 2) and again, on and on as long as it would actually go. Now, we simply have to cancel anything out that's in the numerator and the denominator. There are two ns, there are two (n - 1)s, there are two (n - 2)s, and the rest of these chains in multiplication would be exactly the same, so everything would cancel out. The only two things we would be left with are the two things in the numerator: (_______________) and (______________).

rational, fractions

That trend continues with the numbers. As we take another step back, we come to what are called the ______________________ numbers. The new additions to the club are _______________________. This means we could have things like -1/2 or 1/3 or 3/4 or maybe 11/7.

radicals, denominator, denominator, 24, 24, 2, 6, 12

Simplifying Radicals One of the rules of simplifying radicals is that we have to remove the _____________________ from the _______________________________. We can't have radicals in the denominator. So, if either our reciprocal of a radical or our division by a radical gives us a radical in the denominator that we can't evaluate, then we would have to use the method that I'm going to show you right now to remove it. This method involves multiplying the numerator and denominator by the radical in the _______________________________. So, for 1 divided by the square root of 24, I would multiply the 1 with a square root of 24, and we would multiply the square root of 24 with the square root of 24. What happens when we multiply a radical by itself? We get the number inside the radical. So the square root of 24 multiplied by the square root of 24 gives us __________. So 1 divided by the square root of 24 simplifies into the square root of 24 divided by __________. Now, we aren't quite done yet. We still need to check our radicals to see if we can simplify them even further. The square root we can actually simplify further, because we can split it into the square root of 4 times the square root of 6 which becomes ___________ times the square root of 6. Now we see that we have a 2 over a 24 so we can actually simplify this even further by dividing the 2 and the 24. So, our final answer is the square root of ________ over __________.

addition or multiplication

The Associative Property states that terms in an ______________________ or _________________________ problem can be grouped in different ways, and the answer remains the same.

integers, negatives

Taking a step out in terms of the numbers brings us to what are called the __________________. Again, not all of the integers are whole numbers and natural numbers. But all of the whole numbers and natural numbers are integers. The integers now also add in the ________________________: -1, -2, -3 and on and on in the negative direction as well.

numerator, least common denominator, 20, denominator, 6, Divide, reciprocal, multiply,

The first way involves handling the top and bottom separately. Step 1: Simplify the ___________________________. Here's an example: (3/4 + 2/5)/(1/2 - 1/6) The top is 3/4 + 2/5. We need to make the denominators the same by finding the __________________ _____________________ ____________________, or the smallest shared multiple of the denominators. Here, that's ___________. To make the denominator of 3/4 20, we multiply it by 5/5, which gets us 15/20. We multiply 2/5 by 4/4 to get 8/20. 15/20 + 8/20 is 23/20. Now for Step 2: Simplify the _______________________. That's 1/2 - 1/6. What's the least common denominator? ___________. So, 1/6 is fine. We multiply 1/2 by 3/3 to get 3/6. 3/6 - 1/6 is 2/6, or 1/3. Now we have (23/20)/(1/3), and we're ready for the final step: Step 3: ___________________ the Fractions. (23/20)/(1/3) is really 23/20 divided by 1/3. That's all the line in the middle means. To divide fractions, we first flip the second one, giving us its ___________________________. So, 1/3 becomes 3/1. Then, we __________________________ them together. So, 23/20 * 3/1 is 69/20. That can be simplified to 3 and 9/20.

real numbers

The irrational and the rational come together, and with them combined, they form the _____________________ ________________________.

smallest, divided

The least common multiple of two numbers is the ______________________ number that can be _____________________ evenly by your two original numbers.

1, 1/3, -1, -1, flipping, 3/2, original, 1/23, 23/1, 23, multiply, 1, 1, 0, 1

The reciprocal of a number is _______ divided by that number. So, for example, the reciprocal of 3 is 1 divided by 3, which is ______/_______. A reciprocal is also a number taken to the power of ________. So, 1/8 is the same as 8 to the power of ________. You can also take the reciprocal of a fraction by ______________________ the fraction. For instance, the reciprocal of 2/3 is ________/________. This operation allows us to come to an interesting conclusion, one that's especially important for using reciprocals in algebra. The reciprocal of a reciprocal is the ________________________ number. To understand this principle more fully, let's take the number 23. The reciprocal of 23 is ______/_______. But if we take the reciprocal again and flip that fraction, 1/23 becomes _______/_______. And 23 divided by 1 is just __________. To reverse a reciprocal, you take the reciprocal all over again. Additionally, if you _________________________ a number by its reciprocal, you always get ___________. For example, 16 multiplied by 1/16 is just __________. Last of all, every number has a reciprocal except for __________. If you take the reciprocal of 0, you get _________ over 0. When you divide by 0, you get an undefined number; or, in physics, you might argue that you get infinity. But the key point is that reciprocals of 0 are not especially helpful.

decimal places, two, one, three, three

The trick to multiplying decimals is to count the number of __________________ __________________we have in total. We have ____________ from 1.25 and we have ______________ from 3.5. We have a total of ____________________ decimal places, so that tells us that we need to count ____________________ decimal places, and that is where we put our decimal point.

reciprocal, division

There are actually two scenarios where we are dividing by a radical. The first is if we take the _________________________ of a radical. The second is when we are performing straight ________________________ with radicals.

0, whole

There is one whole number that is not a natural number. That is the number __________. When we're talking about the ___________________ numbers, we're talking about the numbers that start with 0 and starting going up 1, 2, 3 and on and on.

irrational, Irrational, pi

There's a similar thing going on with the numbers. There's a separate group of numbers that don't fit with the rest of these. Those are called the _____________________ numbers. _________________________ numbers are all the numbers that can't be described as fractions. These irrational numbers are their own group. They are not part of the rationals, or naturals or whole numbers; they are not related, but separate. Examples of irrational numbers are pi

second, multiply, numerators, denominators, simplify

To divide fractions, we follow three steps. Step one: flip the ______________________ fraction. This gives you its reciprocal. Step two: _________________________ the fractions. This means you multiply the ______________________, then the ___________________________. Finally, step three: ___________________________ as needed.

1, 1 / (4/3), divided, bottom, 3/4, 3/4, flipped

To get the reciprocal of (4/3), we use the definition of reciprocal and we do _____ ___________________by our number. We get ______ / (____/______). Now, we flip the ________________________ rational number so that we can turn our problem into a multiplication problem. We get 1 * (3/4). Our answer, then, is ________/__________. The reciprocal of 4/3 is _______/_______. To make it easy on yourself, just remember that the reciprocal of any rational number is simply the _______________________ version.

numerators, denominators, simplify

To multiply fractions, we follow three steps. Step one: multiply the ________________________; those are the top numbers. Step two: multiply the __________________________; those are the bottom numbers. Step three: ________________________ the fraction, if necessary.

look at Ch 5/ Lesson 8

Using the Rate Formula, make a table to solve a rate formula problem

length x width x height

Volume=

first, greater

When alphabetizing, you look at the first letter, then the second, and so on. If the first letter is bigger, then it goes behind the other folder no matter how large of a word the other folder has. So, just like with alphabetizing and looking at the first letter, then the next, and so on, you do the same with decimal numbers. You look at the _________________ digit after the decimal point, then the next, and so on. If the first digit after the decimal point is bigger, then that number is automatically _____________________ no matter how long the other decimal is.

common denominator, numerators, denominators

When comparing and ordering fractions, you must have a __________________ _____________________. After you find a common denominator, you can simply compare the ________________________ of the fractions. The easiest method to find a common denominator is to multiply your _________________________________ together. After you have multiplied the denominators together, the product will become your new denominator.

division, 12, 8, quotient, numerator, denominator

When converting an improper fraction to a mixed number, we will think of the fraction bar as ______________________. For example, using the improper fraction 12/8, we would divide the numerator _________ by the denominator __________. To convert an improper fraction to a mixed number, we will start by dividing the numerator by the denominator. Once you are finished dividing, your ___________________ will become your whole number. Your remainder will also become your ______________________, and you will keep the same _________________________.

4/1 , 11/1, 100, 75, 75/100, 2, 0.75

Whole integers, like 4 or 11, are simplified fractions. Those two are 4/1 and 11/1. To use the second method, you need to find the number that converts the denominator to 10, 100, 1000, or any subsequent power of ten. Why does this work? Think about 0.2. We call that 2 tenths. What about 0.02? That's 2 one-hundredths. 0.002 is 2 one-thousandths. And, well, it goes on from there. Decimals are numbers where, as a fraction, the denominator is a power of ten. Let's say we have 3/4. How can we make that 4 into a power of ten? 4 * 25 is _____________, which is a power of ten. That gets us to step two: multiply the numerator and denominator by this number. We know 4 * 25 is 100. What's 3 * 25? _________. That gets us ________/__________. And then there's step three: write the numerator as a decimal, moving the decimal point left one place for each 0 in the denominator. Here, we write 75 and move the decimal how many places? There are _______ zeros in 100, so two places. That gives us _____________, which is 75 one-hundredths. So 3/4 is the same as 0.75. And we didn't have to harm any cats to figure it out.

1/ t1 + 1/ t2 = 1/ tt

Work Rate Formula

1

You can't use the LCM shortcut for numbers that share a factor that besides what number?