Algebra 1/Unit 1: Introduction To Algebra



Algebra (from Arabic "al-jabr" (الجبر), meaning "reunion of broken parts") is quite a complicated language of maths. For someone to master the arts of Algebra, well, they are a true genius! This week, we will get into what Algebra is, and some warm ups (on arithmetic). Even though this may seem pointless, it is IMPORTANT that you review through these warm ups, and get them well and done. Your warm ups will be on mathematical problems that relate to fractions (add, subtract, multiply, divide), decimals and percentages.

Without a further to do, let's dig right into this!

What is Algebra?
A broad part of mathematics. In summary, Algebra solves for values that are not known yet. Such as □ - 4 = 6... instead of this empty box, we replace that with something known as a variable, which is a letter that is used for something we do not know yet (of it's value).

So, for the problem: □ - 4 = 6... we will have to do some math! We need to add positive 4, to the negative 4. When you add a number to another number (in which the solution becomes 0), you have cancelled out that number. But, you cannot just cancel out 4, and be done with it! You have to add 4 to the number 6, which brings out a golden rule in Algebra: 'To keep the balance, what we do to one side of the "=" we should also do to the other side!". 

Now... once you have added 6 + 4 (which is 10), you should get □ = 10.

Great! You've done what's called "solving for x", but in this case, it's an empty box... well guys, you're going to have to throw out that box, because it will be represented by a variable (a letter, from a - z). The letter that is commonly used for variables is x.

So instead of □ = 10, it's actually x = 10.

Sample problems of solving for x
 {x − 9 = 20 x={ 29_3 }
 * type="{}"}

{x − 3 = 6 x={ 9_3 }
 * type="{}"}

{4x = 12 x={ 3_3 }
 * type="{}"}

{x/2 = 0.5 x={ 1_3 }
 * type="{}"}

{x/50 = 2 x={ 100_3 }
 * type="{}"}

{x/9 = 5 x={ 45_3 }
 * type="{}"}

Seems simple, huh? Well, it will get complicated, which is why it is important for you to do some review of your arithmetic! Let's dig into that...

Arithmetic
Arithmetic has to deal with elementary/basic levels of math, such as division, multiplication, subtraction, and addition. Basically, just working with numbers.

This SHOULD be a level familiar with you. If you are not familiar with arithmetic math/rules, then PLEASE review through Arithmetic, as you won't survive even the 1st step of Algebra. Trust me, the basics are THAT important.

Fractions
Fractions (from Latin fractus, "broken") are parts of a whole. In the image of a cake, there is only $$3/4$$'s of the cake showing, the other $$1/4$$ has been eaten/taken away.

The number, 3, in $$3/4$$, is what is known as a numerator (Numerator: Number at the top, tells us of how much of the number is being talked about/being used). The number, 4, in $$3/4$$, is what is known as a denominator (Denominator: Number showing the all time total). A reduced form of a fraction is a fraction that cannot be divided by any number other than 1, and the denominator is greater than 1. So $$2/4$$ is NOT in simplest form, since we can divide 2 and 4, by 2... which results in the following number: $$1/2$$. Though, not every fraction can be divided by 2, there are fractions, such as: $$5/35$$, $$7/21$$, and $$30/5$$. The two first fractions are not divisible by 2, and $$30/5$$ can not be divided by 2 on both sides, but only on $$30$$. It's important to simplify as if you were in a test, your teacher will mark your problems as incorrect if you didn't simplify your fractions. It's also important to note that a simplified fraction and the non-simplified fraction (that, if simplified, equals that fraction) are the same, and represent the same quantity. So, $$\tfrac{2}{4}$$ and $$\tfrac{1}{2}$$ represents the same quantity, a half!
 * Simplest form/reduced form

Here, we will present a few fractions for you to simplify.

Sample problems for simplifying fractions (use / as the fraction line)
 { $$\tfrac{6}{8}=$${ 3/4_7 }
 * type="{}"}

{ $$\tfrac{4}{60}=$${ 1/15_7 }
 * type="{}"}

{ $$\tfrac{30}{90}=$${ 1/3_7 }
 * type="{}"}

{ $$\tfrac{8}{18}=$${ 4/9_7 }
 * type="{}"}

{ $$\tfrac{9}{72}=$${ 1/8_7 }
 * type="{}"}

{ $$\tfrac{64}{46}=$${ 32/23|1 9/23_7 }
 * type="{}"}

{ $$\tfrac{206}{340}=$${ 103/170_7 }
 * type="{}"}

Adding Fractions
To simply add fractions, make sure the denominators of the fractions you are adding are the same. If they are not, find the least common denominator (LCD). So, if you have $$\tfrac{4}{2}$$ and $$\tfrac{4}{6}$$, you have to multiply the 2 in $$\tfrac{4}{2}$$ by 3, which equals 6... BUT you cannot just multiply 2 only, you also have to multiply 4 by 3, since that's what you did to 2, the denominator. If you change the denominator, you have to change the numerator.

Alright, we got that out of the way, so once we have $$\tfrac{12}{6}$$ + $$\tfrac{4}{6}$$, we can simply add. So $$12$$ + $$4$$ = $$16$$, but don't add the denominators, they stay the same. So the answer is $$\tfrac{16}{6}$$, and then we simplify down to $$\tfrac{8}{3}$$ dividing by 2.

But... did you notice something? $$\tfrac{8}{3}$$? That doesn't seem right, does it? The denominator is smaller than the numerator. When you have a fraction like this, you have to convert it to a mixed fraction (skip to section 2.1.1.4).

Multiplying Fractions
To multiply fractions, simplify your fraction to simplest terms. Once you have done that, you can simply multiply the numerators and the denominators. And obviously, simplify your final product, if you can. So, we have $$\tfrac{6}{8}$$ and $$\tfrac{2}{6}$$. We can't multiply yet, as we need to simplify those fractions. We simplify 6 and 8 by dividing both by 2, we also divide 2 and 6 by 2. So the fractions are now $$\tfrac{3}{4}$$ and $$\tfrac{1}{3}$$. You simply multiply those two fractions by multiplying the numerator by the numerator, and doing the same for the denominators. After completing this process, you will get a solution (in fraction form). $$\tfrac{3}{4}$$ × $$\tfrac{1}{3}$$ $$=$$ $$\tfrac{3}{12}$$.

$$\tfrac{3}{12}$$ is not going to be our final product, though, since we can simplify the fraction by dividing the fraction by 3, which results in $$\tfrac{1}{4}$$.

Dividing Fractions
There is an interesting twist when it comes to dividing fractions. You have to turn the fraction you want to divide by (second fraction) upside-down, also known as "Keep, Change, Flip" where you keep the first fraction the same, change the operation to multiplication, and replace the second fractions numerator with the denominator and the denominator with the numerator. Not only that, you have to turn the division symbol (÷) into a multiplication symbol (× or •). After that, you use your skills you learned in multiplying a fraction, and you multiply both of the fractions. Simplify if you need to.

So, $$\tfrac{6}{8}$$ ÷ $$\tfrac{7}{12}$$. Change the division symbol to a multiplication symbol, and turn upside-down the fraction you want to divide by (the upside-down fraction is known as a reciprocal). So $$\tfrac{6}{8}$$ × (or •) $$\tfrac{12}{7}$$. Multiply the numerators and denominators. The answer is $$\tfrac{72}{56}$$, simplified down to $$\tfrac{9}{7}$$.

Improper Fraction --> Mixed Fraction
Divide the numerator by the denominator. The quotient (result of the division taking place/number above the division line) will be the whole number of the mixed fraction, while the numerator will be the remainder. The denominator remains unchanged, so don't change the denominator at all!

See also: https://www.tes.com/lessons/bJieZ4sFPJbSTw/fractions-4-mixed-numbers-and-improper-fractions

Decimals
Ever wondered how to write 8$$\tfrac{47}{100}$$? Well, you've got the answer: 8.47! How did we get that answer? Let's look at a few more and maybe you'll see the pattern: You see? We simply put the mixed number in front of the dot, and with the numerator, we slap that behind the dot! Throw out the 100, it's not important when building your decimal.
 * 1) 6$$\tfrac{98}{100}$$ = 6.98
 * 2) 2$$\tfrac{56}{100}$$ = 2.56
 * 3) 9$$\tfrac{27}{100}$$ = 9.27
 * 4) 5$$\tfrac{83}{100}$$ = 5.83

Decimals are all about place value, the value of a number in a specific place in a number. So, when we have $$6.72$$, the $$6$$ is in the Ones place. Now, let's throw $$9$$ in the tens place, which is 10 times bigger than the Ones place: $$96.72$$. But... that's doesn't seem enough, does it? Let's throw in a $$6, 2, 8$$ and a $$3$$ in there! And now, we have: $$628,396.72$$.

Woah! That's a pretty big number, but we can easily break this number down to it's place value. Let's do it!

So, our number, $$628,396.72$$, is the number we need to break down. Let's start from the decimal point, and move left:
 * The number $$6$$ is in the Ones place. x10
 * The number $$9$$ is in the Tens place. x10
 * The number $$3$$ is in the Hundreds place. x10
 * The number $$8$$ is in the Thousands place. x10
 * The number $$2$$ is in the Ten thousands place. x10
 * The number $$6$$ is in the Hundred Thousands place.

Now we have broken up the numbers left of the decimal--What about the numbers on the right? Let's throw in a $$5, 2, 4$$ and a $$7$$. Now, we have $$628,396.725,247$$. Let's break this number up like we did above.

So, our number, $$628,396.725,247$$, is the number we need to break down. This time, we need to start on the the decimal point, and move right:
 * The number $$7$$ is in the Tenths place. x-10
 * The number $$2$$ is in the Hundredths place. x-10
 * The number $$5$$ is in the Thousandths place. x-10
 * The number $$2$$ is in the Ten Thousandths place. x-10
 * The number $$4$$ is in the Hundred Thousandths place. x-10
 * The number $$7$$ is in the Millionths place.

We have just now gone over the importance of Place Value in the Decimal World. Now, we will go into how to work with decimals, in the Decimal World!

See also: http://www.shmoop.com/fractions-decimals/place-value-naming-decimals.html

Adding/Subtracting Decimals
To add decimals, in addition column-style, put the decimals in its place with the decimals lined up. Then simply add on. So, for $$1.5$$ + $$2.5$$. We'd line up the decimal points. But, if we had a problem like $$1.15$$ + $$2.0$$, we'd add a $$0$$ after the $$0$$ that is behind the decimal. Adding a zero to a place in a decimal means "no value". So $$10$$ basically means no ones, and $$100$$, means no ones or hundreds. Same things goes for subtracting as well folks.

Sample problems for adding/subtracting decimals
 { 6.8 - 2.5={ 4.3_6 }
 * type="{}"}

{ 3.4 + 5.6={ 9_6 }
 * type="{}"}

{ 9 + 4.50={ 13.5_6 }
 * type="{}"}

{ 41.89 + 25.00={ 66.89_6 }
 * type="{}"}

{ 9.01 + 3.089={ 12.099_6 }
 * type="{}"}

{ 10.90 + 11.1={ 22_6 }
 * type="{}"}

{ 9.5 + 3.44={ 12.94_6 }
 * type="{}"}

{ 9.00 x 2.00={ 18_6 } (BONUS!)
 * type="{}" coef="2.5"}

Multiplying Decimals
Multiplying decimals isn't as hard as it really seems to be. So, we have $$9.83$$ × $$5.73$$. For me, and for most people, column multiplication is a lot easier than side-by-side multiplication. That being mentioned, let us column these numbers:

$$9.83$$ × $$5.73$$ ---

Now that we have our problem, we should simply ignore the decimal points and just multiply as usual, so you should get this answer once you are done with that (remember to add a zero (and grow with zeros in each line) to each and every line of addition):

$$9.83$$ × $$5.73$$ --- $$2949$$ $$+$$$$68810$$ $$+$$$$491500$$ ---

With the simple usage of addition, we should get:

$$9.83$$ × $$5.73$$ --- $$2949$$ $$+$$$$68810$$ $$+$$$$491500$$ --- $$563259$$

Now, we need to bring back our handy dandy decimal point, but where? In $$9.83$$ and $$5.73$$, there are FOUR numbers in these 2 numbers overall that are behind the decimal point (in each number, there are two numbers behind the decimal points). So, we have 9.$$83$$ and 5.$$73$$. Now, that totals up to four numbers overall behind the decimal point. So in $$563259$$, we need to move the decimal point four times (beginning from the right). So watch as follows:

$$563259.$$ $$56325.9$$ $$5632.59$$ $$563.259$$ $$56.3259$$

That simple. Now, review your work, your whole work should look like this:

$$9.83$$ × $$5.73$$ --- $$2949$$ $$+$$$$68810$$ $$+$$$$491500$$ --- $$56.3259$$

Dividing Decimals
If you want to divide a decimal by a whole number, you should divide the 2 numbers, omitting the decimal point. After you are done dividing, add the decimal point to the quotient (final product/answer at the top of the long division symbol). The decimal should be right above the decimal point in the dividend (number in the box/number that is being divided). It's quite easy and simple, as long as you know how to do long division and if you are still familiar with long division.
 * Dividing a decimal by a whole number

Hey, this seems too easy--Let's figure out how to divide a decimal by a decimal! The trick to dividing a decimal by a decimal is to shift the decimal point as many times as it gets to a whole number, so follow along: $$69.45$$ ÷ $$5.78$$. Now, we simply move the decimal point as many times as we need to make the number we are going to use to divide 69.45 a whole number, so watch as followed: $$69.45$$ ÷ $$5.78$$ → $$694.5$$ ÷ $$57.8$$ → $$6945$$. ÷ $$578$$.
 * Dividing a decimal by a decimal

Now that we have finally got our dividend a whole number (and now our first number that we are going to divide), we can go ahead and divide normally (using long division). In the end, $$69.45$$ divided by $$5.78$$ should get you $$12.0155709$$!

A pretty simple one we could go is $$6.4$$ ÷ $$0.4$$, here, we simply move our dots like so: $$6.4$$ ÷ $$0.4$$ $$64$$ ÷ $$04.$$ $$64$$ ÷ $$4$$ Then, we can simply divide, heck... we don't even need to do long division! The answer should pop in your head, which is $$16$$.

Percentages
A good definition of "percent" is a fraction in which the denominator is the number $$100$$. So for example, the numbers $$59%$$, $$63%$$, $$91%$$, and $$85%$$, you are simply just saying $$\tfrac{59}{100}$$, $$\tfrac{63}{100}$$, $$\tfrac{91}{100}$$, and $$\tfrac{85}{100}$$. You could also say 59 out of 100 parts, 63 out of 100 parts, 91 out of 100 parts, and 85 out of 100 parts.

Converting Percentages
Now that we got the basis of percentages and how they operate, we should now look into changing percentages.

Percentage → Decimal
Let's look in turning a percentage into a decimal point first. Very simple. Let's say you have $$\tfrac{9}{100}$$, which, in percentage form, is $$9%$$. So, we have 9%. Now, we want to change it to a decimal (I don't know, think of a reason). We simply convert the percentage symbol into a decimal point, so like this: $$9.$$. Now, we have $$9.$$, so then we move the decimal number two places to the left, like so: $$9.$$ → $$.9$$ → $$.09$$. So now, we have $$0.09$$. We added the 2 zeros in because there is no value in the tenths place, and because $$.09$$ does not look quite right. Looks a bit off.

Samples problems for converting percentages to decimals
 { 59%={ 0.59_5 }
 * type="{}"}

{ 63%={ 0.63_5 }
 * type="{}"}

{ 91%={ 0.91_5 }
 * type="{}"}

{ 85%={ 0.85_5 }
 * type="{}"}

{ 9%={ 0.09_5 }
 * type="{}"}

{ 9834%={ 98.34_5 }
 * type="{}"}

{ 20%={ 0.2_5 }
 * type="{}"}

{ 4%={ 0.04_5 }
 * type="{}"}

{ 7.6%={ 0.076_5 }
 * type="{}"}

{ 6%={ 0.06_5 }
 * type="{}"}

Decimal → Percentage
Vice versa. Complete opposite. We have $$98.34$$. We need this to be a percentage (easier to read). Move the decimal point two places to the right. So, watch: $$98.34$$ → $$983.4$$ → $$9834.$$ --Now, we have $$9834.$$, but the decimal point, since it's now a percentage, should not be there, but instead, a percentage should talk the decimal point's place. Now, we have our final product of $$9834%$$.

Finding percent of a number
So, 95 students applied to a university (the fictional Mustafa Einhoonansebadoi University, for example), and only 20% of the students made it. 20%? What? With this in mind, we want to find $$20%$$ of $$95$$. We take the percentage, $$20%$$, and divide it by $$100$$. So we get $$20/100$$ = $$.2$$. Then, we multiply $$.2$$ by $$95$$, in which we get $$19$$. So $$20%$$ of $$95$$ is $$19$$. Therefore, only 19 students out of 95 students made it into the fictional Mustafa Einhoonansebadoi University.