- 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.
- You can cut any normal polygon into standard polygons you know, for example, a polygon shaped as a house can be split into a recntalge and a triangle. All normal polygon, can be simplified into triangles.
## Perimeter of Composite Figures
- Same as an area of composite figures, cut them down into much simplified shapes so you can easily calculate its area. Alternatively, if its possible, you can even measure the shape using a tool such as ruler.
## Optimization (For Maximimizing Area/Volume, or Minimizing Perimeter/Surface Area)
### 2D Shapes
|Shape|Maximum Area|Minimum Perimeter|
|:----|:-----------|:----------------|
|4-sided rectangle|A rectangle must be a square to maximaze the area for a given perimeter. The length is equal to the width<br>$`A = lw`$<br>$`A_{max} = (w)(w)`$<br>$`A_{max} = w^2`$|A rectangle must be a square to minimaze the perimeter for a given area. The length is equal to the width.<br>$`P = 2(l+w)`$<br>$`P_{min} = 2(w)(w)`$<br>$`P_{min} = 2(2w)`$<br>$`P_{min} = 4w`$|
- An easy way to see if ratios are equivalent is to see if the ratios are the same in its simpliest form. A ratio is a fraction in essence, the first number is the `numerator`, and the second number is the `demominator`.
- An easy way to find an equivalent ratio is just to multiply the everything by 2, for example, given a ratio `2:5`, multiply everything by 2, so your new ratio will be `2x2:5x2 = 4:10`.
## Ratio and Proportions
- Ratios link directly with proportions, as a ratio of one thing can be applied to another ratio of another thing.
- Example
- Given one ratio of one triangle's height to base and another ratio of another triangle's heigth to base, we can find one missing side length given that we have the other 3.
- Lets say the 2 ratios are `1:5` and `3:x`, given that `x` is a random number.
- To solve `x`, we can cross-multiply, or multipling the `numerators` with the `denominators` and setting them equal to each other. Thus, we can first make the ratios into fractions and then do: $`\frac{1}{5} \text{ cross-multiply } \frac{3}{x} \rightarrow 1x = 3 \times 5`$. Which we can then solve for `x`, which is `15`.
## Solving Algebraic Equations
- To solve equations, just remove the extra bits on both sides by doing the opposite. So if its addition, do subraction to get rid of the added numbers, and vice versa:
- Simply divide the `numerator` by the `denominator`. So $`\frac{3}{4} = 3 \div 4, \text{or } 0.75`$.
## Fraction to Percent
- The percentage symbol `%`, means the number has been multiplied by 100, so given a fraction $`\frac{3}{4}`$, convert the fraction into a decimal. Then multiply that decimal by `100`.
- Simply divide the first number by the second number, and multiply by one `100`. It is the same as making the ratio into a fraction, then converting the fraction into a percentage.
