diff --git a/Grade 9/Math/MFM1P1/Final_Exam_Study_Sheet.md b/Grade 9/Math/MFM1P1/Final_Exam_Study_Sheet.md index d161869..756db5f 100644 --- a/Grade 9/Math/MFM1P1/Final_Exam_Study_Sheet.md +++ b/Grade 9/Math/MFM1P1/Final_Exam_Study_Sheet.md @@ -5,19 +5,155 @@ ### 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 a **`tenth`**, **`hundredth`**, and **`thousandths`** + - 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. - - Hundreth + - 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. + - 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. + \ No newline at end of file