# Math tricks

## Arithmetic

There are several tricks that can make arithmetic *much* easier to calculate.

You can do *any* arithmetic you want, as long as you do the same thing to the other side to keep it equal.

While using the same operator on the opposite side of the formula creates an equivalent result, using the *opposite* operator on the same side does it as well:

1. x = 50 * 6
2. x = 25 * 12
3. x = 5 * 60
4. x = 300

Anything multiplied or divided by a multiple of 10 simply requires moving a decimal point or adding a zero:

- 65.1 * 10 = 651
- 8749 / 10 = 874.9
- Generally, removing decimals is less complicated than adding them.

Multiplying by 9 is simply the number times 10, minus one instance:

- 11 * 9 = 110 - 11 = 99
- 95 * 9 = 950 - 95 = 855
- 856 * 9 = 8,560 - 856 = 7,704

Multiplying 2-digit numbers by 11 involves adding the 2 digits together, then placing the sum in between those 2 digits:

- 12 * 11, add 1 + 2 to make 3, convert 12 to 132
- 72 * 11, add 7 + 2 to make 9, convert 72 to 792
- 57 * 11, add 5 + 7 to make 12, convert 57 to 627 (because the 1 carries leftward)

Multiplying and dividing decimals is easy:

1. Move the decimals all the way to the right and count the number of places, which is effectively multiplying the entire operation by increments of 10 (e.g., 5.8 * 7.21 becomes 58 * 721 with 3 decimal places).
2. Perform standard multiplication or division (e.g., 58 * 721 = 41,818).
3. Put the decimal places back in, which is effectively dividing the entire operation by the same increments of 10 (e.g., 41,818 becomes 41.818).

Fractions, like equations, can be multiplied and divided by arithmetic as long as the numerator and denominator are treated the same:

- 7/10 = 14/20
- 8.25/11 = 17/22
- 360/400 = 9/10

Fractions can be easily divided if you flip the numerator and denominator and multiply them:

- (4/5) / (8/3) = 4/5 * 3/8 = 12/40 = 3/10

Percentages can be converted to to fractions by making it the percentage, divided by 100:

- 59% = 59/100
- 63.8% = 63.8/100 = 638/1000

A number can be divided in no particular order, so you can factor a large number step-by-step by dividing it piece-by-piece:

- It can be divided by 2 if the last digit is even.
- It can be divided by 3 if the digits added together are divisible by 3 (e.g., 732).
- It can be divided by 4 if the last 2 digits together are divisible by 4.
- It can be divided by 5 if the last number is 0 or 5.
- It can be divided by 6 if the rules for 2 and 3 above apply.
- It can be divided by 9 if the sum of the digits can be divided by 9 (e.g., 360).
- It can be divided by 10 if the number ends in a 0.
- It can be divided by 12 if the rules for 3 and 4 apply.

## Geometry

Since area and volume are commutative, you can factor the shapes into smaller shapes and move them around to more easily calculate them:

- A parallelogram can be vertically split, then moved to create a square.
- Opposite vertices of a parallelogram can *always* slice the shape into 2 triangles.

The area of a triangle is *always* 1/2 the area of a rectangle, so you can multiply the base by the height of that triangle, then divide it by 2 (i.e., a = bh/2)

A triangle's 3 sides always add up to 180°, and each side adds another 180°, which creates some tricks:

- If you know 2 sides of a triangle, you can deduce the 3rd.