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