1. a(b+c) = ab + ac

2. (a+b)² = a² + 2ab + b²

3. (a-b)² = a²-2ab + b²

4. (a+b)² = (a-b)² + 4ab

5. (a-b)² = (a+b)² – 4ab

6. (a-b) (a+b) = a² – b²

7. (a+b)³ = a³ + 3ab(a+b) + b³

8. (a-b)³ = a³ – 3ab(a-b) – b³

9. a³ + b ³ = (a+b) (a² – ab + b²)

10. a³ – b ³ = (a-b) (a² + ab + b²)

11. (a³ – b³) / (a² + ab + b²) = a-b

12. (a³ + b³) / (a² – ab + b²) = a+b

 Adding Rules:

Positive + Positive = Positive: 5 + 4 = 9
Negative + Negative = Negative: (- 7) + (- 2) = – 9

Sum of a negative and a positive number: Use the sign of the larger number and subtract

(- 7) + 4 = -3
6 + (-9) = – 3
(- 3) + 7 = 4
5 + ( -3) = 2

  Subtracting Rules:

Negative – Positive = Negative: (- 5) – 3 = -5 + (-3) = -8
Positive – Negative = Positive + Positive = Positive: 5 – (-3) = 5 + 3 = 8
Negative – Negative = Negative + Positive = Use the sign of the larger number and subtract (Change double negatives to a positive)
(-5) – (-3) = ( -5) + 3 = -2
(-3) – ( -5) = (-3) + 5 = 2

Multiplying Rules:

Positive x Positive = Positive: 3 x 2 = 6
Negative x Negative = Positive: (-2) x (-8) = 16
Negative x Positive = Negative: (-3) x 4 = -12
Positive x Negative = Negative: 3 x (-4) = -12

Dividing Rules:

Positive ÷ Positive = Positive: 12 ÷ 3 = 4
Negative ÷ Negative = Positive: (-12) ÷ (-3) = 4
Negative ÷ Positive = Negative: (-12) ÷ 3 = -4
Positive ÷ Negative = Negative: 12 ÷ (-3) = -4

Tips:

When working with rules for positive and negative numbers, try and think of weight loss or poker games to help solidify ‘what this works’.
Using a number line showing both sides of 0 is very helpful to help develop the understanding of working with positive and negative numbers/integers.

Midpoint Formula

The Midpoint forumla is used when you need the point that is exactly between two other points. The midpoint formula is applied when you need to find a line that bisects a specific line segment. Essentially, the ‘middle point’ is called the “midpoint”.

The Slope Formula

Sometimes called ‘Rise over Run’.

The formula for the slope of the straight line going through the points (x1, y1) and (x 2, y 2) is given by:

The subscripts refer to the two points.

(m=rise/run)

Note:
Parallel lines have equal slope.
Perpendicular lines have negative reciprocal slopes.

Common Formulas

CIRCUMFERENCE

Circle:

C = πd, in which π is 3.1416 and d the diameter.

AREA

Triangle:

A = (ab)/2 , in which a is the base and b the height.

Square:

A = a², in which a is one of the sides.

Rectangle:

A = ab, in which a is the base and b the height.

Trapezoid:

A = (h(a + b))/2, in which h is the height, a the longer parallel side, and b the shorter.

Regular pentagon:

A = 1.720a², in which a is one of the sides.

Regular hexagon:

A = 2.598a², in which a is one of the sides.

Regular octagon:

A = 4.828a², in which a is one of the sides.

Circle:

A = πr², in which π is 3.1416 and r the radius.

VOLUME

Cube:

V = a³, in which a is one of the edges.

Rectangular prism:

V = abc, in which a is the length, b is the width, and c the depth.

Pyramid:

V = (Ah)/3, in which A is the area of the base and h the height.

Cylinder:

V = πr²h, in which π is 3.1416, r the radius of the base, and h the height.

Cone:

V = (πr²h)/3, in which π is 3.1416, r theradius of the base, and h the height.

Sphere:

V = (4πr³)/3, in which π is 3.1416 and r the radius.

TEMPERATURE SCALES

Degrees Fahrenheit to Degrees Celsius:

T_C = ⁵/9 (T_F – 32)

Degrees Celsius to Degrees Fahrenheit:

T_F = ⁹/5 T_C + 32

Degrees Celsius to Kelvins:

T_K = T_C + 273.15

Mean and Median

The arithmetic mean, also called the average, of a series of quantities is obtained by finding the sum of the quantities and dividing it by the number of quantities. In the series 1, 3, 5, 18, 19, 20, 25, the mean or average is 13—in other words, 91 divided by 7.

The median of a series is that point which so divides it that half the quantities are on one side, half on the other. In the above series, the median is 18.

The median often better expresses the common-run, since it is not, as is the mean, affected by an excessively high or low figure. In the series 1, 3, 4, 7, 55, the median of 4 is a truer expression of the common-run than is the mean of 14.

MISCELLANEOUS

Distance in feet traveled by falling body:

d = 16t², in which t is the time in seconds.

Speed of sound in feet per second through any given temperature of air:

take the square root of (273 + t), in which t is the temperature Centigrade, multiply it by 1087, and divide the result by 16.52.

Cost in cents of operation of electrical device:

C = (Wtc)/1000, in which W is the number of watts,t the time in hours, and c the cost in cents per kilowatt-hour.

Conversion of matter into energy (Einstein’s Theorem):

E = mc², in which E is the energy in ergs, m the mass of the matter in grams, and c the speed of light in centimeters per second: (c² = 9 × 10²⁰)

Number Sequences- find the missing number in a sequence of numbers

1. Find the next number in the series

4     8     16     32     —
A) 48     B) 64     C) 40      D) 46

2. Find the next number in the series

4     8     12     20     —
A) 32     B) 34     C) 36      D) 38

3. Find the missing number in the series

54     49     —     39     34
A) 47     B) 44     C) 45      D) 46

4. Find the first number in the series

—     19     23     29     31
A) 12     B) 15     C) 16      D) 17

5. Find the next number in the series

3     6     11     18     —
A) 30     B) 22     C) 27      D) 29

6. Find the next number in the series

48     46     42     38     —
A) 32     B) 30     C) 33      D) 34

These number sequences usually consist of four visible numbers plus one missing number.

7. Find the missing number in the series

.
4     3     5     9     12     17     —
A) 32     B) 30     C) 24      D) 26

8. Find the missing numbers in the series

5     6     7     8     10     11     14     —     —-
A) 19     B) 17     C) 15      D) 16

9. Find the missing numbers in the series

1     –     4     7     7     8     10     9     —-
A) 6     B) 3     C) 11      D) 13

Answers
1. B  The numbers double each time
2. A  Each number is the sum of the previous two numbers
3. B  The numbers decrease by 5 each time
4. D  The numbers are primes (divisible only by 1 and themselves)
5. C  The interval, beginning with 3, increases by 2 each time
6. B  The interval, beginning with 2, increases by 2 and is subtracted each time
7. D  Each number is the sum of the previous and the number 3 places to the left
8. C A  There are 2 simple interleaved sequences 5,7,10,14,19 and 6,8,11,15
9. A D  There are 2 simple interleaved sequences 1,4,7,10,13 and 6,7,8,9