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Update Final_Exam_Study_Sheet.md
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### Decimals
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- Terms:
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- Given the number `9123.456`:
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- The **`tenth`** is the `4`
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- The **`hundredth`** is `5`
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- The **`thousandths`** is `6`
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- The **`ones`* is `3`
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- The **`tens`** is `5`
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- The **`hundreds`** is `1`
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- The **`thousands`** is `9`
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- **Remember, `tens` and `tenths` may sound the same, but they are `DIFFERENT`!
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- To round to a a **`tenth`**, **`hundredth`**, and **`thousandths`**
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- The **`tenth`** is the `4`.
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- The **`hundredth`** is `5`.
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- The **`thousandths`** is `6`.
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- The **`ones`** is `3`.
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- The **`tens`** is `5`.
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- The **`hundreds`** is `1`.
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- The **`thousands`** is `9`.
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- **Remember, `tens` and `tenths` may sound the same, but they are `DIFFERENT`**!
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- To round to a **`tenth`**, **`hundredth`**, and **`thousandths`**
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- Tenths
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- If the `hundredth` is `5` or higher, round up, else, round down.
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- Hundreth
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- Example:
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- Round `12.53223` to the tenths
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- The answer is `12.5`, as the hundredths, or `3` is smaller than 5.
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- Hundredth
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- If the `thousandth` is `5` or higher, round up, else, round down.
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- Example:
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- Round `12.53521` to the hundredth
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- The answer is `12.4`, as the thousandths, or `5` is bigger or equal to 5.
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- Thousandth
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- If the number of the `thousandth` is `5` or higher, round up, else, round down.
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- If the number of the `thousandth` is `5` or higher, round up, else, round down.
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- Example:
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- Round `12.5356` to the thousandths
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- The answer is `12.536`, as the number after the thousandths, or `6` is bigger than 5.
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- To round to a **`ones`**, **`tens`**, **`hundreds`**, and **`thousands`**
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- Ones
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- If the `tenths` is `5` or higher, round up, else, round down.
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- Example:
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- Round `123.5333` to the ones
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- The answer is `124`, as the tenths, or `5` is bigger than or equal to 5.
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- Tens
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- If the `ones` is `5` or higher, round up, else, round down.
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- Example:
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- Round `123.5777` to the tens
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- The answer is `120`, as the ones, or `3` is smaller than or equal to 5.
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- Hundreds
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- If the `tens` is `5` or higher, round up, else, round down.
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- Example:
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- Round `177.34343` to the hundreds
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- The answer is `200`, as the tens, or `7` is bigger than 5.
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- Thousands
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- If the `hundreds` is `5` or higher, round up, else round down.
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- Example:
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- Round 566.777` to the thousands
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- The answer is `1000`, as the hundreds, or `5` is bigger or equal to 5.
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## Integers
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### Multiplication and Division
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- Pretend `a` and `b` are random positive numbers
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|Type|Outcome|
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|:---|:------|
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|a × b|Positive number|
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|a × (-b)|Negative number|
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|(-a) × b|Negative number|
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|(-a) × (-b)|Positive number|
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|a ÷ b|Positive number|
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|a ÷ (-b)|Negative number|
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|(-a) ÷ b|Negaitve number|
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|(-a) ÷ (-b)|Positive number|
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- Treat as normal divion and multiplacation, and just add the negative sign infront of the number according to the rules above.
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- Practice
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- 8 × -7
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- Answer: `-56`
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- 2 × 4
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- Answer: `8`
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- -7 × -7
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- Answer: `1`
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- -10 × 4
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- Answer: `-40`
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- 8 ÷ 4
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- Answer: `2`
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- -16 ÷ -8
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- Answer: `2`
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- -4 ÷ 1
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- Answer `-4`
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- 9 ÷ -3
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- Answer: `-3`
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### Addition and Division
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- Pretend `a` and `b` are random postive numbers
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|Type|Equivalent|
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|:---|:---------|
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|a+b|a+b|
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|b+a|b+a|
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|a+(-b)|a-b|
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|(-a)+b|b-a|
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|a-b|a-b|
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|b-a|b-a|
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|a-(-b)|a+b|
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|(-a)-b|a-b|-(a - b)|
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### Order Or Operation
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- BEDMAS
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- Follow ```BEDMAS``` for order of operations if there are more than one operation
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| Letter | Meaning |
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|:------:|:-------:|
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| B / P | Bracket / Parentheses |
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| E | Exponent |
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| D | Divison |
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| M | Multiplication |
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| A | Addition |
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| S | Subtraction |
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- <img src="https://ecdn.teacherspayteachers.com/thumbitem/Order-of-Operations-PEMDAS-Poster-3032619-1500876016/original-3032619-1.jpg" width="300">
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- Follow order of operation, inorder to do know which operation to do first.
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- Example: Given $`(2+4) \times 5 - 9 \div 3`$
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- First do everything in brackets: $`(6) \times 5 - 9 \div 3`$
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- Then do multiplication/division: $`30 - 3`$
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- Then finally, do subtaction/addition: $`27`$
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- The answer is `27`.
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## Fractions / Rational Numbers
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- The number on the top is called the `numerator`.
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- The number on the bottom is called the `denominator`.
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- A fraction in its most simple form is when the `numerator` and `denominator` cannot be both divided by the same number.
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### Additions / Subtractions With Fractions
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- Example: $`\frac{3}{5} + \frac{4}{3}`$
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- Find `common denominator`, which is `15`, as `5` and `3` both are factors of `15`.
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- You can do this easily with a table, just count by the number you are using, for example:
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- |Counting by 5s | Counting by 3s |
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|:--|:--|
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|5|3|
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|10|6|
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|15|9|
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|20|12|
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|25|15|
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- As you can see, both columns contain the number `15`, so `15` is the common denominator.
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- Now, after we find the denominator, we must convert the fraction so that it has the `common denominator`. To do this, we must multiply the denominator by a number, so that it equals the `common denominator`. For the first fracion $`\frac{3}{5}`$, the `denominator` is `5`, to get to `15`, we must multiply it by `3`. Now, whatever we do on the bottom, me **MUST** do it on the top too, so we also multiply the `numerator` by `3` as well, the new fraction is now $`\frac{3 \times 3}{5 \times 3} = \frac{9}{15}`$.
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- We now do the same thing to the other fraction: $`\frac{4 \times 5}{3 \times 5} = \frac{20}{15}`$
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- Now that the denominators are the same and the fractions are converted, we can just simply add the `numerators` together while keeping the `denominator` the same. The result is $`\frac{9 + 20}{15} = \frac{29}{15}`$.
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- The same steps applied to subtracion, with the only difference of subtacting the numerators rather than adding them.
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### Multiplaction With Fractions
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- To multiply a fracion, simply multiply the `numerators` together, and the `denominators` together.
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- Example: $`\frac{3}{6} \times \frac{7}{4}`$
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- Answer: $`\frac{3 \times 7}{6 \times 4} = \frac{21}{24}`$
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### Division With Fractions
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- To divide 2 fractions, flip the second fraction upside down.
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