Math: 2012

Sunday, December 16, 2012

How to convert binary to decimal and decimal to binary number

Binary numbers are used in digital technology such as in telecommunications, banking, computers, gaming, and multimedia. Binary numbers have 2 as their base and have only two possible values, 0 and 1. Decimal numbers have 10 as their base and have ten possible values, (0, 1, 2, 3, 4, 5, 6, 7, 8, 9). Decimal is the default base whenever you turn on your scientific calculator from off position. Decimal numbers are the numbers that we use in ordinary day to day transactions. It is also what we teach our children in our homes to count 1 to 10. This article will demonstrate how to convert binary numbers to decimal numbers and the other way around, how to convert decimal numbers to binary numbers.

How to convert a binary number to a decimal number: (only 5 easy steps)

Example:
Convert 1001101 to decimal number.

Given:
1001101 binary number

Required:
Decimal number equivalent to 1001101 binary number

Solution:
1. Copy the binary number and separate each binary number with a space.
1 0 0 1 1 0 1

2. Count the total number of binary numbers. In this example, there are seven binary numbers. Because there are seven binary numbers we will place 0, 1, 2, 3, 4, 5, 6 at the bottom of each binary number starting from the left to the right. These are the exponents to which the bases of two will be raised to.
1 0 0 1 1 0 1
6 5 4 3 2 1 0

3. Put 2 as the base for all the binary numbers.
1 0 0 1 1 0 1
6 5 4 3 2 1 0
2 2 2 2 2 2 2

4. Looking at the given binary number, determine which binary numbers are switched ON (those with values 1). Switched OFF means those binary numbers having values of 0 (zero). Starting from the right to left, the values that are 'switched on' are the first, third, fourth, and seventh. The 'switched on' positions have 0, 2, 3, 6 as their corresponding exponents.
1 1 1 1
6 3 2 0
2 2 2 2

5. Raise the 'switched on' positions using 2 as the base to their corresponding exponents. After raising to their corresponding exponents, their invidivual results will be added.
2^0 + 2^2 + 2^3 + 2^6
1 + 4 + 8 + 64
= 77

Answer:
1001101 binary number = 77 decimal number

Example:
Convert 110001 binary number to decimal number.

Solution:
Place the exponents 0, 1, 2, 3, 4, 5
1 1 0 0 0 1
5 4 3 2 1 0

Place the base 2 in all of the numbers.
1 1 0 0 0 1
5 4 3 2 1 0
2 2 2 2 2 2

Remove all 'switched off' values.
1 1 1
5 4 0
2 2 2

Raise 2 to the exponents and add the results.
2^0 + 2^4 + 2^5
1 + 16 + 32
= 49

Answer:
110001 binary number = 49 decimal number

-------------------------------------------------------------

How to convert a decimal number to a binary number: (in 5 simple steps)

Example:
Convert 12 (decimal number) to a binary number.

Given:
12 (decimal number)

Required:
Binary representation of 12 (decimal number).

Solution:
1. Determine if the given decimal number is even or odd. In this example, 12 is an even number.

2. The result of adding two even numbers is EVEN, e.g. 4 + 8 = 12.

3. Arrange the exponents 0, 1, 2, 3. In this case, the maximum exponent is 3 because 2^4 is equal to 16 which will be greater than 12. After laying out the exponents, put 2 as bases.
3 2 1 0
2 2 2 2

4. Assuming, they are all 'switched on', the equivalent decimal will be,
2^0 + 2^1 + 2^2 + 2^3
1 + 2 + 4 + 8
= 15

5. Because we are getting the equivalent binary number for the decimal number 12, and knowing that adding two even numbers would yield an even number, and we must remove 3 from 15, therefore:
2^0 and 2^1 must be switched off --> first position is 0, second position is 0
2^2 and 2^3 must be switched on --> third position is 1, fourth position is 1
so that,
2^2 + 2^3
4 + 8
= 12

Answer:
Starting from right to left and combining the first to the fourth positions, we will get 1100.
1100 binary = 12 decimal

Example:
What is the binary representation of the decimal number 25?

Solution:
The given decimal number 25 is an odd number. Even number plus Odd number is an ODD number, e.g. 24 + 1 = 25.
In order to obtain 25, the fifth position, fourth position, and first positions must all be switched on.
The second and third positions must be switched off.
So that,
1 1 0 0 1
4 3 2 1 0
2 2 2 2 2

Adding only the switched on positions,
2^0 + 2^3 + 2^4
1 + 8 + 16
= 25

Therefore,
11001 is the binary number equivalent to the decimal number 25.

(I chose the number 25, in memory of my father's basketball shirt's number).

Tuesday, December 4, 2012

MATHEMATICS: Ratio, Proportion, Variation - Force of Gravitation, Weight, surface Illumination intensity, current, resistance

WEIGHT is inversely proportional to the SQUARE OF R

where:

WEIGHT = force of attraction between the Earth and an object

R = distance of object from the center of the Earth

example:

A satellite weighs 700 N (Newtons) on the surface of the Earth. How much will the satellite weigh at an altitude of 250 km above the Earth's surface. Take the radius of the Earth as 4000 miles.

given:

W1 = 700 N

R1 = 4000 mi

R2 = 4000 + 250

R2 = 4250

find:

W2 = weight of satellite at an altitude of 250 km above the Earth

solution:

W = k/R^2

k = W * R^2

W1 * (R1)^2 = W2 * (R2)^2

700 * (4000)^2 = W2 * (4250)^2

W2 = 700 * (4000)^2/(4250)^2

W2 = 700 * 16,000,000/18,062,500

W2 = 620 N ---> the satellite is lighter

INTENSITY OF SURFACE ILLUMINATION is inversely proportional to the SQUARE OF THE DISTANCE BETWEEN THE SOURCE AND THE SURFACE

example:

A certain light source has an illumination of 700 lux (lumens per square meter) on a surface. If the distance is doubled, find the resulting illumination.

given:

I1 = 700 lux

d1 = initial distance

d2 = 2 * d1

find:

I2 = surface illumination when the distance is doubled

solution:

I = k/d^2

k = I * d^2

I1 * (d1)^2 = I2 * (d2)^2

700 * (d1)^2 = I2 * (2 * d1)^2

700 * (d1)^2 = I2 * 4 * (d1)^2

I2 = 700/4

I2 = 175 lux ---> reduced to 1/4 or 25% of the initial illumination

ELECTRIC CURRENT is inversely proportional to CIRCUIT RESISTANCE

example:

When the resistance of a circuit is increased by 10%, by what percent will the current change?

given:

R1 = initial circuit resistance

R2 = 1.10 R1

I1 = initial current

find:

I2 = current when resistance is increased by 10%

solution:

I = k/R

k = I * R

I1 * R1 = I2 * R2

I1 * R1 = I2 * 1.10 R1

I1 = 1.10 I2 ---> equation 1

%I = [(I2 - I1)/I1] * 100%

substituting from equation 1,

%I = [(I2 - 1.10 I2)/1.10 I2] * 100%

%I = [-0.10/1.10] * 100%

%I = -9.09 % (negative value means decrease)

%I = 9.09 % decrease in Electric current

MATHEMATICS: Ratio, Proportion, Variation - Conductor wire Resistance, Reactance, Capacitance, Photograph exposure time, F stop of lens

RESISTANCE is inversely proportional to the SQUARE OF WIRE DIAMETER

example:

A wire AWG size 12 of 0.08 inches in diameter has a resistance of 15 ohms. Find the resistance of AWG 10 size conductor with a diameter of 0.10 in. Assume same length and same material.

given:

d1 = 0.08 in

R1 = 15 ohms

d2 = 0.10 in

find:

R2 = resistance of AWG 10 (diameter = 0.10 in)

solution:

R = k * 1/d^2

k = R * d^2

R1 * (d1)^2 = R2 * (d2)^2

15 * (0.08)^2 = R2 * (0.10)^2

R2 = 15 * (0.0064)/(0.10)^2

R2 = 15 * (0.0064)/(0.0100)

R2 = 9.6 ohms

CAPACITIVE REACTANCE is inversely proportional to CAPACITANCE

example:

If the capacitance of a circuit is increased by 50%, what percent will the capacitive reactance change?

given:

c1 = initial capacitance

c2 = 1.5 c1

R1 = initial capacitive reactance

R2 = final capacitive reactance

find:

%R = percent change in capacitive reactance corresponding to a 50% increase in capacitance

solution:

R = k/c

k = R * c

k = R1 * c1 = R2 * c2

R1 * c1 = R2 * c2

R1 * C1 = R2 * 1.5 C1

R1 = 1.5 R2 ---> equation 1

%R = [(R2 - R1)/R1] * 100%

substituting from equation 1,

%R = [(R2 - 1.5 R2)/1.5 R2] * 100%

%R = [-0.5/1.5] * 100%

%R = -33.33 % (negative value means decrease)

%R = 33.33 % decrease in capacitive reactance

TIME OF EXPOSURE is directly proportional to the SQUARE OF THE F STOP OF A LENS

example:

A photograph is exposed at a shutter speed of 0.02 seconds with a lens opening of f6. What shutter speed is required if lens opening is changed to f8.

given:

t1 = 0.02 sec

f1 = 6

f2 = 8

find:

t2 = shutter speed for a lens opening of f8

solution:

f stop of a lens = L/D

where:

L = focal length

D = diameter

t = k * f^2

k = t/f^2

t1/(f1)^2 = t2/(f2)^2

(0.02)/(6)^2 = t2/(8)^2

t2 = (0.02) * (8)^2/(6)^2

t2 = 0.02 * 64/36

t2 = 0.036 seconds

MATHEMATICS: Ratio, Proportion, Variation - FREEFALL, DISPLACEMENT, RESISTOR POWER and CURRENT, PENDULUM OSCILLATIONS and LENGTH

1. DISTANCE TRAVELLED is directly proportional to the SQUARE OF THE TIME OF TRAVEL (freefall, freely falling bodies)

example:

If a body falls 200 meters in 8 seconds, How far will it fall in 10 seconds?

given:

S1 = 200 m

t1 = 8 sec

t2 = 10 sec

solution:

S = kt^2

k = S/t^2

S1/(t1)^2 = S2/(t2)^2

200/(8)^2 = S2/(10)^2

200/64 = S2/100

S2 = 200 * 100/64

S2 = 312.5 m

2. POWER IN A RESISTOR is directly proportional to the SQUARE OF THE CURRENT

example:

How many times the current in an electric heating coil be increased such that the power consumed will be 4 times.

given:

P1 = x

P2 = 4x

solution:

P = kI^2

k = P/I^2

P1/(I1)^2 = P2/(I2)^2

x/(I1)^2 = 4x/(I2)^2

(I2)^2 = (I1)^2 * 4x/x

(I2)^2/(I1)^2 = 4

if I1 = 7 amp

(I2)^2/(7)^2 = 4

(I2)^2/49 = 4

(I2)^2 = 4 * 49

(I2)^2 = 196

(I2) = sqrt (196)

I2 = 14 amp ---> twice of I1

therefore, if the POWER consumed is 4 times, the CURRENT will be doubled

3. PENDULUM OSCILLATIONS inversely proportional to SQUARE ROOT OF LENGTH OF PENDULUM

example:

A pendulum of 12 in length oscillates at 2 oscillations per second. If the length is increased twice, that is, 24 in., how many oscillations does the pendulum produces?

given:

n1 = 2 oscillations/sec

L1 = 12 in

L2 = 24 in

find:

n2 = oscillations if the length is doubled

solution:

n = k/sqrt(L)

k = n * sqrt(L)

n1 * sqrt(L1) = n2 * sqrt(L2)

2 * sqrt(12) = n2 * sqrt(24)

n2 = 2 * sqrt(12)/sqrt(24)

n2 = 2 * 3.464/4.899

n2 = 2 * 0.707

n2 = 1.4 oscillations/second

MATHEMATICS: Ratio, Proportion, Variation - Power, Resistance of Conductor, Volume, Turbine Flowrate

1. POWER is directly proportional to DISPLACEMENT (total volume swept out by the pistons)

example:

A Honda engine delivers 240 horsepower and has a displacement of 3 liters. If displacement is changed to 3.8 liters, how much horsepower does the Honda engine deliver?

given:

P1 = 240 horsepower

D1 = 3 liters

D2 = 3.8 liters

find:

P2 = horsepower corresponding to a displacement of 3.8 liters

solution:

P = kD

k = P/D

k = P1/D1 = P2/D2

P1/D1 = P2/D2

240/3 = P2/3.8

P2 = 240 * 3.8/3

P2 = 304 horsepower

2. RESISTANCE OF CONDUCTOR is directly proportional to LENGTH OF CONDUCTOR

example:

If the resistance of 3 miles of transmission line is 170 ohms, what is the resistance of a 30-mile transmission line?

given:

R1 = 170 ohms

L1 = 3 miles

L2 = 30 miles

find:

R2 = resistance of a 30-mile transmission line

solution:

R
= kL

k = R/L

k = R1/L1 = R2/L2

R1/L1 = R2/L2

170/3 = R2/30

R2 = 170 * 30/3

R2 = 1700 ohms

3. POWER is directly proportional to the FLOWRATE THROUGH THE TURBINES

example:

A turbine flowrate of 6000 gpm produces 45 MW. If the flowrate through the turbines is reduced to 3000 gpm (half), how much power would be produced?

given:

Q1 = 6000 gpm (gallons per minute)

P1 = 45 MW

Q2 = 3000 gpm

find:

P2 = power produced with 3000 gpm (half the flowrate through the turbines)

solution:

P = kQ

k = P/Q

k = P1/Q1 = P2/Q2

P1/Q1 = P2/Q2

45/6000 = P2/3000

P2 = 45 * 3000/6000

P2 = 22.5 MW --------> also half of the POWER would be produced, thus confirming a direct proportion

Thursday, November 15, 2012

TRIGONOMETRY: Sine law, Law of cosines, pythagorean identities, trigonometric identities, double angle formulas, reduction formulas, power-reducing formulas, common right triangle combination sides

Below are the most important and most useful formulas in Trigonometry which have so many applications in various fields of study, discipline and professions:

1. Sine Law (Law of Sines)
2. Cosine Law (Law of Cosines)
3. Pythagorean Theorem
4. Pythagorean Identities, Twice angle formulas
5. Double angle formulas
6. Power - reducing theorem
7. Radian measure, Reduction formulas

let:

t, theta = angle measure

a = shortest side of triangle

b = medium side of triangle

c = longest side of triangle

A = angle opposite a

B = angle opposite b

C = angle opposite c

Right Triangles

45 x 45:

a = b = 1

c = sqrt 2 = 1.4

30 x 60:

a = 1

b = sqrt 3 = 1.7

c = 2

Common Right Triangle combinations

3,4,5 ---> a=3, b=4, c=5

5,12,13 ---> a=5, b=12, c=13

Sine Law (Law of Sines):

MATHEMATICS: Statistics, Motion, Fencing

STATISTICS:

Mode, Median, Mean

Mode:

Rearrange the data in ascending order, the mode is occuring MOST FREQUENT

example: 5, 7, 8, 8, 8, 9 ---> mode is 8

Mean:

Mean (arithmetic mean) is the Average

Median:

odd median ---> middle

example: 5, 6, 7, 8, 9 ---> median is 7

even median ---> average of 2 medians

example: 5, 6, 8, 9

n = 4 (even number of terms)

there are two medians ---> 6 and 8

to find the median, get the average of 6 and 8

median = (6 + 8)/2 = 14/2 = 7

MOTION:

A train travels at constant speed 70 mph from Station1 to Station2. The distance between the stations is 1000 miles. Find the expression for the distance as a function of time.

solution:

Distance diminishes as it travels from station1 to station2

S = v*t

S = 70t ---> distance travelled for a certain time t

Expression:

S(t) = 1000 - 70t ---> this is the distance as a funtion of time (the distance for a specific value of time)

where:

S = distance travelled

S(t) = distance as a funtion of time as the train travels from station1 to station2

v = constant speed or velocity

t = time of travel

FENCING:

A farmer wants to fence a length of 100 ft. Determine the number of posts required if the posts are spaced 25 ft apart.

find:

P = L/s + 1 ---> number of posts required

given:

L = 100 ft, length of fence

s = 25 ft, spacing between posts

solution:

P = L/s + 1

P = 100/25 + 1

P = 4 + 1

P = 5

GEOMETRY: Perimeter, Circumference, Largest Area, Square, Circle, Fencing, Differential Calculus, Maxima, minima

1. If 400 ft of fence is to be used, which shape generates the larger area, a circle or a square?

SQUARE:

A = s^2

using 400 ft of fence, each side of the square should be

s = 100 ft

A = s^2

A = (100)^2

A = 10,000

CIRCLE:

C = 2 * pi * r

400 = 2 * pi * r

r = 400/(2 * pi )

r = 63.7

A = pi * r^2

A = pi * (63.7)^2

A = 12,740

Comparing the areas of square and circle,

CIRCLE ---> larger area

2. What are the dimensions of a triangle of maximum area that can be inscribed in a circle such that one side of the triangle passes through the center of the circle.

solution:

let

b = base of triangle

h = height of triangle

t = angle between the hypotenuse C and the height h

C = hypotenuse = 2r

A = 1/2 b * h ---> equation1

h = C * cos t

b = C * sin t

substituting b and h in equation1

A = (1/2) C^2 cos t sin t ---> equation2

from double angle formulas:

sin(2t) = 2 sin t cos t

cos t sin t = (1/2)sin(2t) ---> equation3

substituting equation3 in equation2

A = (1/4) C^2 sin (2t)

using the double angle form,

A = (1/4) C^2 (2 cos t sin t)

derivative of A with respect to t ---> derivative of a product

dA/dt = (1/4) C^2 (2 cos t sin t)

dA/dt = (1/4) C^2 [ 2 (cos t cos - sin t sint) ]

dA/dt = (1/4) C^2 [ 2 (cos^2 t - sin^2 t) ]

from double angle formulas:

cos^2 t - sin^2 t = cos 2t

substituting

dA/dt = (1/4) C^2 [ 2 (cos 2t) ]

dA/dt = (1/4) C^2 * 2 cos 2t

to find the maximum area, equate dA/dt = 0

dA/dt = (1/4) C^2 * 2 cos (2t) = 0

as the angle between the hypotenuse and the adjacent side(h) approaches 90, cos t approaches zero because the adjacent side (h) gets shorter and shorter while the opposite side (b = base of triangle) gets longer and longer

sin 90 = 1

cos 90 = 0

for dA/dt = 0

2 * t = 90

t = 45 degrees

substituting the values for C = 2r and t = 45

h = C * cos t

h = 2r * cos 45

h = 2r * sqrt(2)

b = C * sin t

b = 2r * sin 45

b = 2r sqrt(2)

The area is maximum when t = 45 degrees

and

b = h = 2r sqrt(2)

which is an ISOSCELES right triangle.

Wednesday, November 14, 2012

GEOMETRY: Interior, Exterior angles of any polygon, ratio of the angles

Interior angles of any polygon

Ia = 180(s - 2)

where:

Ia = sum of all interior angles of any polygon

s = number of sides of the polygon

for triangles: Ia = 180

for rectangles, trapezoids, quadrilaterals, parallelograms and any four-sided polygons: Ia = 360

for a pentagon: Ia = 540

for a hexagon (hex-bolt): Ia = 720

Exterior angles of any polygon

Ea = 360

where:

Ea = sum of all exterior angles of any polygon

1. Determine the largest angle of a Triangle if the ratio of the angles is 1:3:5

find:

x = shortest angle

5x = largest angle of triangle

solution:

1x + 3x + 5x = 180

9x = 180

x = 180/9

x = 20

3x = 3 * 20 = 60

5x = 5 * 20 = 100

IQ TEST: Math, Percentages, Markdowns, Discounts, Original, discounted Price, Ratio, Proportion

1. The ratio of two numbers is 1:4, if the smaller number is increased by 7, the ratio becomes 1:2. Find the two numbers.

find:

x = smaller number (based on 1:4 ratio)

4x = bigger number (based on 1:4 ratio)

solution:

(x + 7)/4x = 1/2

4x = 2x + 14

2x = 14

x = 7

4x = 4 * 7 = 28

2. A certain product has been marked down twice. The first was 20% and the second 25%. If the original price was $100, what is the final price after the two markdowns?

find:

x = final price

solution:

after first mark down of 20%

Price = 100 * 0.80 = 80

after second mark down of 25%

x = 80 * 0.75 = 60

3. The original price was $100. A discount of 10% has been offered, then the price went back to the original price. What is the percentage of increase with respect to the discounted price?

find:

x = percent of increase with respect to the discounted price

solution:

after 10% discount

discounted price = 100 * 0.90 = 90

to go back to original price

increase = 100 - 90 = 10

x = (increase/discounted price) * 100%

x = 10/90 * 100%

x = 1/9 * 100%

x = 11.11% ---> [ 1/9 is repeating 0.1111111111111111111111111111 ]

IQ TEST: Math, Pie, Cake, Divisions, Total, Sum, Parts, Age Problems

1. A birthday cake is to be divided and distributed such that the birthday celebrant will have a quarter share and the rest will have half as big as the celebrant's share. How many pieces was the cake divided?

find:

x = number of equal shares

solution:

TOTAL = sum of parts

1 = 1/4 + x(1/2 * 1/4)

x(1/8) = 3/4

x = 6

Cake divisions = 6 + 1 = 7

2. A school has 55 students for each teacher. If the school has a total of 2240 students and teachers, how many teachers are there?

find:

x = number of teachers

solution:

TOTAL = sum of students and teachers

2240 = 55x + x

56x = 2240

x = 40

3. Roland is twice as old as Adrienne. Twenty years ago, Roland was four times as old as Adrienne. How old is Adrienne now?

find:

A = Adrienne's age now

given:

now

A = Adrienne's age

R = Roland's age

R = 2A ---> equation 1

twenty years ago

A - 20 = Adrienne's age

R - 20 = Roland's age

R - 20 = 4(A - 20) ---> equation 2

substituting equation 1 in 2

2A - 20 = 4(A - 20)

2A - 20 = 4A - 80

2A = 60

A = 30

Saturday, November 3, 2012

IQ TEST: Math, Working together, Job, Work problems at the same rate, inverse proportions

1. John can finish the job in 3 hours while Amy can finish the same job in 4 hours. If both work on the same job, how long will they finish?

find:

T = total time if both work together

t1 = time for John to finish the job alone

t2 = time for Amy to finish the job alone

solution:

1/T = 1/t1 + 1/t2

1/T = 1/3 + 1/4

1/T = 7/12

T = 12/7 or 1.7 hours

2. Two carpenters can finish 4 cabinets in 4 hours. If the working rate is the same, how many cabinets can 7 carpenters finish in 24 hours?

find:

J2 = number of cabinets finished by 7 carpenters in 24 hours

given:

m1 = 2 carpenters

t1 = 4 hours

J1 = 4 cabinets

m2 = 7 carpenters

t2 = 24 hours

solution:

m1*t1/J1 = m2*t2/J2

2(4)/4 = 7(24)/J2

2 = 168/J2

J2 = 84 cabinets

3. A factory has an output of 100 units. An additional 20% of output has required an increase of 4 units per worker. If they are working at the same rate, determine the number of workers.

find:

x = number of workers

solution:

OUTPUT = rate * workers

additional output = additional rate * workers

100(0.2) = 4x

20 = 4x

x = 20/4

x = 5

check:

normal capacity:

output = 100

rate = 100/5 = 20 units per worker

at 20% over normal capacity (120 units):

rate = 20 + 4 = 24

output = 24 * 5 = 120 units

Wednesday, October 24, 2012

IQ TEST: Math, Mixture, Solutions, Concentration, Coin Problems

1. It takes 3 parts of cement and 4 parts of sand to make a specific mixture. How many containers of cement is required to make a mixture of 28 containers?

find:

x = number of containers of cement

solution:

TOTAL = sum of parts

7/7 = 3/7 + 4/7 ---> denominator (7) = 3 + 4

mixture = cement + sand

7/7(28) = 3/7(28) + 4/7(28)

x = 3/7(28)

x = 12

2. How many liters of a 20% solution should be added to 40 liters of 80% solution to make a mixture containing 50% of concentration?

find:

v1 = number of liters of 20% solution

solution:

v1*concentration1 + v2*concentration2 = Vtotal*finalconcentration

v1(0.2) + 40(0.8) = (v1 + 40)(0.5)

0.2v1 + 32 = 0.5v1 + 20

0.3v1 = 12

v1 = 40

3. How many quarters, dimes, and nickels are there if their total is $3.60 and there are a total of 21 coins and the number of nickels is twice the number of dimes.

find:

n = number of nickels

d = number of dimes

q = number of quarters

given:

cents = 360

n = 2d ---> equation1

n + d + q = 21 ---> equation2

solution:

TOTAL = sum of parts

360 = 5n + 10d + 25q ---> equation3

substituting n = 2d in equation2

n + d + q = 21

2d + d + q = 21

3d + q = 21

q = 21 - 3d ---> equation4

substituting n = 2d in equation3

360 = 5n + 10d + 25q

360 = 5(2d) + 10d + 25q

360 = 10d + 10d + 25q

360 = 20d + 25q ---> equation5

substituting equation4 in equation5

360 = 20d + 25q

360 = 20d + 25(21 - 3d)

360 = 20d + 525 - 75d

55d = 525 - 360

55d = 165

d = 3

n = 2d = 2(3) = 6

q = 21 - 3d = 21 - 3(3) = 12

checking:

d cents = 3 * 10 = 30 cents

n cents = 6 * 5 = 30 cents

q cents = 12 * 25 = 300 cents

total = 30 + 30 + 300 = 360

Tuesday, October 23, 2012

IQ TEST: Math, Fractions, Series, Sequence, Military Time, Clock, Days of the week

1. What letter is three to the left of the letter that is immediately
to the right of the letter that is two to the left of the letter L?

A B C D E F G H I J K L M N O P Q R

solution:

start from the "inside" just like starting from the parentheses in math

two to the left of L ----> J

right of letter J ----> K

three to the left of K ----> H

2. What is the number that is one half of one quarter of one eighth of 448?

solution:

start from the "inside"

1/8 of 448 = 56

1/4 of 56 = 14

1/2 of 14 = 7

3. How many minutes is it before midnight if six tenth of an hour ago it
was twice as many minutes past 10 pm?

find:

x = minutes before midnight

given:

Total time from 10 pm to midnight ---> 120 minutes

t1 = 6/10 hr

unit analysis: hr * 60 min/hr

t1 = 6/10 * 60 = 36 minutes

solution:

TOTAL = sum of parts ---> working equation

substituting:

120 = x + 2x + 36

3x = 84

x = 84/3

x = 28 ---> answer

check:

28 minutes before midnight is 11:32 pm

36 minutes ago, the time is 10:56 pm

10:56 pm is 56 minutes after 10 pm

56 = twice of x

4. What time is it now if 2 hours later it would be half as long until 5 pm as it would be if it were an hour from now?

find:

x = time now

given:

5 pm ---> 17:00

solution:

17 - (x + 2) = 1/2 [17 - (x + 1)]

17 - x - 2 = 1/2 (17 - x - 1)

15 - x = 1/2 (16 - x)

15 - x = 8 - x/2

x/2 = 15 - 8

x = 14 ---> or 2 pm

5. a. What is the day today if the day before yesterday is 2 days after Friday? b. What is the day today if the day after tomorrow is 2 days before Saturday?

solution:

a.

put Friday first on the sequence, then start from the "inside"

FRIDAY sat SUNDAY mon TUESDAY wed THURSDAY fri SATURDAY

2 days after friday ---> sunday

sunday is the day before yesterday

monday is yesterday

TUESDAY = today ---> answer

b.

2 days before saturday ---> thursday

thursday is the day after tomorrow

wednesday is tomorrow

TUESDAY = today ---> answer

Sunday, October 21, 2012

Trigonometry - Law of sines, Bearing, Ship, Motorboat, Navy, Marines, Sailor

1. A ship takes a sighting of two motorboats. Motorboat A has a bearing of N 44 W and Motorboat B has a bearing of N 59 E. The distance between the two motorboats is 4 km and the bearing of Motorboat B from A is N 88 E. Compute the distance of the ship from each motorboat.
find:

SA = distance of the ship from Motorboat A

SB = distance of the ship from Motorboat B

given:

Bearing of Motorboat A = N 44 W

Bearing of Motorboat B = N 59 E

Bearing of Motorboat B from A = N 88 E

AB = 4 km, distance between Motorboat A and Motorboat B

solution:

A_B
S\/

angle ASB = 44 + 59 = 103

angle BAS = 180 - 88 - 44 = 48

angle ABS = 180 - 103 - 48 = 29

using Sine Law:

SA/sin ABS = SB/sin BAS = AB/sin ASB

SA/sin 29 = AB/sin 103

SA/0.485 = 4/0.974

SA = 0.485 * 4/0.974

SA = 1.99 km

SB/sin 48 = AB/sin 103

SB/0.743 = 4/0.974

SB = 0.743 * 4/0.974

SB = 3.05 km

IQ TEST: Math, Probability, Permutations, Repetitions, Ordered Combinations, Exclusive events, Number of digits

1. If there are 10 people in a room and they would make handshakes with each one in the room, how many unique handshakes would be made?

find:

H = number of unique handshakes

solution:

n = number of people = 10

H = (n - 1) + (n - 2) + (n - 3) .....until LAST ADDEND is 1

first term = PEOPLE - 1

H = 9 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1

H = 45

if there are only 5 people in the room

H = 4 + 3 + 2 + 1

H = 10

2. If a dice and a coin are tossed, what is the probability that you will get a 4 on the dice and heads on the coin?

solution:

P = Pdice * Pcoin

P = 1/6 * 1/2

P = 1/12

3. A circular spinner is marked W, X, Y, Z. what is the probability of NOT getting an X if spinned thrice?

solution:

if spinned once ---> P1 = 3/4

if spinned twice ---> P2 = 3/4 * 3/4 = 9/16

if spinned thrice ---> P1 = 3/4 * 3/4 * 3/4 = 27/64

4. There are 5 choices each question. What is the probability of correctly guessing 3 questions?

solution:

P = 1/5 * 1/5 * 1/5

P = 1/125

5. If you have 5 pants and 8 shirts, how many ways can you wear them?

solution:

W = 5 * 8

W = 40

6. In a restaurant, there are 3 choices of soup, 5 choices of entrees, and 2 choices of desert. How many ways can you order them if each order consists of one soup, one entree, and one desert?

solution:

W = 3 * 5 * 2

w = 30

7. If 4 warriors line up so that the order matters, how many ordered combinations (permutations) are possible?

P = 4 * 3 * 2 * 1

P = 24

8. If you count from 1 to 100, How many zeroes are there? How many sevens are there?

solution:

let

Zt = total number of 0's

St = total number of 7's

1 to 10 : z = 1, s = 1

11 to 20: z = 1, s = 1

21 to 30: z = 1, s = 1

31 to 40: z = 1, s = 1

41 to 50: z = 1, s = 1

51 to 60: z = 1, s = 1

61 to 70: z = 1, s = 2 (67, 70)

71 to 80: z = 1, s = 10 ( 71, 72, 73, 74, 75, 76, 77, 78, 79 ---> there are two 7's on 77 )

81 to 90: z = 1, s = 1

91 to 100: z = 2, s = 1 ( 100 ---> two zeroes on 100 )

Zt = 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 2 = 11

St = 1 + 1 + 1 + 1 + 1 + 1 + 2 + 10 + 1 + 1 = 20

Greek and Latin Prefixes, Numerals used in English numerical system, counting, ordering, arrangement, measurement

SI, Metric Prefixes for Large and Small Numbers:

Prefix Symbol Value and their Meaning

Yotta--- Y --- 10^24 --- septillion
Zetta--- Z --- 10^21 --- sextillion
Exa----- E --- 10^18 --- quintillion
Peta---- P --- 10^15 --- quadrillion
Tera---- T --- 10^12 --- trillion
Giga---- G --- 10^9 ---- billion
Mega---- M --- 10^6 ---- million
myria--- my -- 10^4 ---- ten thousand
kilo---- k --- 10^3 ---- thousand
hecto--- h --- 10^2 ---- hundred
deka---- da--- 10 ------ ten
-------------------------------------
deci---- d --- 10^-1 ---- tenth
centi--- c --- 10^-2 ---- hundredth
milli--- m --- 10^-3 ---- thousandth
micro--- u --- 10^-6 ---- millionth
nano---- n --- 10^-9 ---- billionth
pico---- p --- 10^-12 --- trillionth
femto--- f --- 10^-15 --- quadrillionth
atto---- a --- 10^-18 --- quintillionth
zepto--- z --- 10^-21 --- sextillionth
yocto--- y --- 10^-24 --- septillionth

Latin Numbers and Prefix [Examples in brackets]:

1 = unus -------- uni [unary, unitary, unilateral]

2 = duo --------- bi, duo [binary, duple, biennial]

3 = tres, tria -- tri, ter [tertial, trinary, ternary]

4 = quattuor ---- quadri, quart [quartal, quaternary, quadriennial, quadrillion]

5 = quinque ----- quinque, quint [quintal, quinary, quinquenary, quinquennial]

6 = sex --------- sex(t), se [sextal, senary, sexenary]

7 = septem ------ sept [septimal, septenary, septet, septuple]

8 = octo -------- oct [octal, octaval, octonary, octet]

9 = novem ------- nonus, novem [nonal, nonary, nonet, nonuple]

10 = decem ------ dec(a), de [decimal, denary, dectet, decuple]

11 = undecim ---- undec, unde [undecimal, undenary, undecillion]

12 = duodecim --- duodec, duode [duodecimal, duodenary, duodecennial]

13 = tredecim -------- tredec, tridec [tridecimal, tridenary]

14 = quattuordecim --- quatuordec [quatuordecimal, quatuordenary]

15 = quindecim ------ quinde(c) [quindecimal, quindenary]

16 = sedecim --------- sede(c) [hexadecimal, sedenary]

17 = septendecim ----- septende(c) [septendecimal, septendenary]

18 = duodeviginti ---- decennoct [decennoctal, decenoctonary]

19 = undeviginti ----- decennov [decennoval, decennonary]

20 = viginti --------- vige, vice [vigesimal, vicesimal, vigenary, vicenary]

30 = triginta -------- trige, trice [trigesimal, tricenary]

40 = quadraginta ----- quadrage [quadragesimal, quadragenary]

50 = quinquaginta ---- quinquage [quinquagesimal, quinquagenary]

60 = sexaginta ------- sexage [sexagesimal, sexagenary]

70 = septuaginta ----- septuage [septuagesimal, septuagenary]

80 = octoginta ------- octage [octagesimal, octogenary]

90 = nonaginta ------- nonage [nonagesimal, nonagenary]

100 = centum --------- cente [centesimal, centenary, century, centennial]

1000 = mille --------- mille [millesimal, millenary, millennium, millennial]

Greek Numbers and their Prefixes [Examples in brackets]:

1 = en -------------------- eis, mia, ev, mono [monarch, monad, monogamy]

2 = dyo, duo, di ---------- duo, dis, di, dy [diarch, dyarch, dimeter, dyad, biathlon, bicycle]

3 = treis, tria ----------- treis, tria, tris, tri [triad, tripod, triangle]

4 = tessera --------------- tettares, tettara, terakis, tetra, quad [tetrahedron, tetragramaton, tetrapack, quadrangle, quadrilateral]

5 = pente ----------------- pente, pentakis, penta [pentagon, pentangle, pentameter, pentecost]

6 = hexa ------------------ hex, hexa [hexagon, hexahedron, hexangle, hexameter, hex bolt]

7 = hepta ----------------- hepta, hepta [heptagon, heptameter, heptangle]

8 = okto ------------------ octô, octa [octagon, octave, octopus, octohedron]

9 = ennea ----------------- ennea, nona [nonagon, nano technology, ennead,enneahedron]

10 = deka ----------------- deca [decameter, decathlon, decagon, decade, decapolis]

11 = hendeka -------------- endeca, undeca, hendeca [hendecagon, undecagon, hendecarch, hendecahedron]

12 = dodeka --------------- dodeca [dodecagon, dodecahedron, dodecathlon]

13 = treiskaideka --------- triskaideca, trideca [triskaidecagon, trisdecagon, triskaidecahedron]

14 = tettares kai deka ---- tetrakaideca, tetradeca [tetradecagon, tetrakaidecagon, tetrakaidecahedron]

15 = pentekaideka --------- pendeca [pendecagon, pendecahedron]

16 = hekkaideka ----------- hexadeca [hexadecagon, hexadecahedron, hexadecimal]

17 = heptakaideka --------- heptadeca [heptadecagon, heptadecahedron]

18 = oktôkaideka ---------- octodeca [octodecagon, octodecahedron]

19 = enneakaideka --------- enneadeca [enneadecagon, enneadecahedron]

20 = eikosi (icosa-) ------ icos(a) [icosagon, icosahedron]

100 = hekaton ------------- hecato [hecatogon, hecatohedron]

1000 = chilioi, chiliai --- chilia [chiliagon, chiliahedron]

Conversion of commonly used Fractions into Decimals and Percentage equivalents

Applications: These fractions and decimal equivalents are often used in (by)

-- Aptitude tests
-- IQ tests
-- College entrance examinations
-- Assessment tests
-- Achievement tests
-- Admission tests
-- Math wizard award tests
-- Military job selection tests
-- Psychometric entrance tests
-- Competitions & contests
-- Employment applications
-- Job promotion exams
-- Trade eligibility exams
-- Everyday accounting
-- Inventory listing
-- Machine shops
-- Car mechanic and repair shops
-- Sailors and Marine navigators
-- Engineers
-- Architects
-- Draftsmen
-- Physicists
-- Astronomers
-- Doctors
-- Economists
-- Merchandisers
-- Presidents in their speeches full of numbers and figures
-- Ordinary homes and offices
.. and other numerous applications in various trades, disciplines, crafts, commerce, businesses, professions & occupations

fifty percent (half):
1/2 = 0.5
1/2 = 2/4 = 3/6 = 4/8 = 0.5 (50%)

the third part:
1/3 = 0.333
1/3 = 2/6 = 3/9 = 4/12 = 5/15 = 0.333 (33.3%)

"the devil's number", "number of the beast":
2/3 = 0.666
2/3 = 4/6 = 6/9 = 8/12 = 10/15 = 0.666 (66.6%)

twenty-five percent:
1/4 = 0.25
1/4 = 2/8 = 3/12 = 5/20 = 0.25 (25%)

seventy-five percent:
3/4 = 0.75
3/4 = 6/8 = 9/12 = 12/16 = 15/20 = 0.75 (75%)

twenty-forty-sixty-eighty percent: (magic of the fifth part)
1/5 = 0.2 (20%) = 3/15 = 4/20 = 5/25
2/5 = 0.4 (40%) = 6/15 = 8/20 = 10/25
3/5 = 0.6 (60%) = 9/15 = 12/20 = 15/25
4/5 = 0.8 (80%) = 12/15 = 16/20 = 20/25

common divisions of six:
1/6 = 0.166
1/6 = 2/12 = 0.166 (16.6%)

5/6 = 0.833
5/6 = 10/12 = 0.833

divisions of seven:
1/7 = 0.143
2/7 = 0.286
3/7 = 0.429
4/7 = 0.571
5/7 = 0.714
6/7 = 0.857

common divisions of eight: (x/8 * 12.5) sangngapulo + "benteng"
1/8 = 0.125
1/8 = 2/16 = 0.125 (12.5%)

3/8 = 0.375
3/8 = 6/16 = 0.375 (37.5%)

5/8 = 0.625
5/8 = 10/16 = 0.625 (62.5%)

7/8 = 0.875
7/8 = 14/16 = 0.875 (87.5%)

repeating sequence:
1/9 = 0.111
2/9 = 0.222
4/9 = 0.444
5/9 = 0.555
7/9 = 0.777
8/9 = 0.888

ten percent to ninety percent:
1/10 = 0.1 (10%)
2/10 = 0.2 (20%) = 1/5
3/10 = 0.3 (30%)
4/10 = 0.4 (40%) = 2/5
5/10 = 0.5 (50%)
6/10 = 0.6 (60%) = 3/5
7/10 = 0.7 (70%)
8/10 = 0.8 (80%) = 4/5
9/10 = 0.9 (90%)

first two digits (repeating sequence):
1/11 = 0.0909
2/11 = 0.1818
3/11 = 0.2727
4/11 = 0.3636
5/11 = 0.4545
6/11 = 0.5454
7/11 = 0.6363
8/11 = 0.7272
9/11 = 0.8181
10/11 = 0.9090

divisions of the dozen:
1/12 = 0.083
5/12 = 0.416
7/12 = 0.583
11/12 = 0.916

common divisions of sixteen: (diff = 0.125); (+12 +13, 25 75 alt)
1/16 = 0.0625
3/16 = 0.1875
5/16 = 0.3125
7/16 = 0.4375
9/16 = 0.5625
11/16 = 0.6875
13/16 = 0.8125
15/16 = 0.9375

common division of twenty: (magic 5 --> multiply by 5 and get percent)
1/20 = 0.05 (5%)
2/20 = 0.1 (10%)
3/20 = 0.15 (15%)
4/20 = 0.2 ( 20%)
5/20 = 0.25 (25%)
6/20 = 0.3 (30%)
7/20 = 0.35 (35%)
8/20 = 0.4 (40%)
9/20 = 0.45 (45%)
10/20 = 0.5 (50%)
11/20 = 0.55 (55%)
12/20 = 0.6 (60%)
13/20 = 0.65 (65%)
14/20 = 0.7 (70%)
15/20 = 0.75 (75%)
16/20 = 0.8 (80%)
17/20 = 0.85 (85%)
18/20 = 0.9 (90%)
19/20 = 0.95 (95%)

common division of twenty five: (magic 4 --> multiply by 4 to get percent)
1/25 = 0.04 = 4%
2/25 = 0.08 = 8%
3/25 = 0.12 = 12%
4/25 = 0.16 = 16%
5/25 = 0.2 = 20%
6/25 = 0.24 = 24%
7/25 = 0.28 = 28%
8/25 = 0.32 = 32%
9/25 = 0.36 = 36%
10/25 = 0.4 = 40%
11/25 = 0.44 = 44%
12/25 = 0.48 = 48%
13/25 = 0.52 = 52%
14/25 = 0.56 = 56%
15/25 = 0.6 = 60%
16/25 = 0.64 = 64%
17/25 = 0.68 = 68%
18/25 = 0.72 = 72%
19/25 = 0.76 = 76%
20/25 = 0.8 = 80%
21/25 = 0.84 = 84%
22/25 = 0.88 = 88%
23/25 = 0.92 = 92%
24/25 = 0.96 = 96%

common divisions of thirty-two: (sequence difference = 1/16 = 0.0625)
1/32 = 0.03125
3/32 = 0.09375
5/32 = 0.15625
7/32 = 0.21875
9/32 = 0.28125 (*)
11/32 = 0.34375
13/32 = 0.40625
15/32 = 0.46875
17/32 = 0.53125 (*)
19/32 = 0.59375
21/32 = 0.65625
23/32 = 0.71875
25/32 = 0.78125 (*)
27/32 = 0.84375
29/32 = 0.90625
31/32 = 0.96875

one percent to twenty percent: (x/25 * 4); (x/50 * 2)
1/100 == 0.01 = 1%
1/50 === 0.02 = 2%
3/100 == 0.03 = 3%
1/25 === 0.04 = 4%
1/20 === 0.05 = 5%
3/50 === 0.06 = 6%
7/100 == 0.07 = 7%
2/25 === 0.08 = 8%
9/100 == 0.09 = 9%
1/10 === 0.1 == 10%
11/100 = 0.11 = 11%
3/25 === 0.12 = 12%
13/100 = 0.13 = 13%
7/50 === 0.14 = 14%
3/20 === 0.15 = 15%
4/25 === 0.16 = 16%
17/100 = 0.17 = 17%
9/50 === 0.18 = 18%
19/100 = 0.19 = 19%
1/5 ==== 0.2 == 20%

divisions of forty: (increments --> quarter of ten percent) (x/40 * 2.5)
1/40 = 0.025 == 2.5% == 1/40
2/40 = 0.05 === 5% ==== 1/20
3/40 = 0.75 === 7.5% == 3/40
4/40 = 0.1 ==== 10% === 1/10
5/40 = 0.125 == 12.5% = 1/8

6/40 = 0.15 === 15% === 3/20
7/40 = 0.175 == 17.5% = 7/40
8/40 = 0.2 ==== 20% === 1/5
9/40 = 0.225 == 22.5% = 9/40
10/40 = 0.25 == 25% === 1/4

11/40 = 0.275 = 27.5% = 11/40
12/40 = 0.3 === 30% === 3/10
13/40 = 0.325 = 32.5% = 13/40
14/40 = 0.35 == 35% === 7/20
15/40 = 0.375 = 37.5% = 3/8

16/40 = 0.4 === 40% === 2/5
17/40 = 0.425 = 42.5% = 17/40
18/40 = 0.45 == 45% === 9/20
19/40 = 0.475 = 47.5% = 19/40
20/40 = 0.5 === 50% === 1/2

21/40 = 0.525 = 52.5% = 21/40
22/40 = 0.55 == 55% === 11/20
23/40 = 0.575 = 57.5% = 23/40
24/40 = 0.6 === 60% === 3/5
25/40 = 0.625 = 62.5% = 5/8

26/40 = 0.65 == 65% === 13/20
27/40 = 0.675 = 67.5% = 27/40
28/40 = 0.7 === 70% === 7/10
29/40 = 0.725 = 72.5% = 29/40
30/40 = 0.75 == 75% === 3/4

31/40 = 0.775 = 77.5% = 31/40
32/40 = 0.8 === 80% === 4/5
33/40 = 0.825 = 82.5% = 33/40
34/40 = 0.85 == 85% === 17/20
35/40 = 0.875 = 87.5% = 7/8

36/40 = 0.9 === 90% === 9/10
37/40 = 0.925 = 92.5% = 37/40
38/40 = 0.95 == 95% === 19/20
39/40 = 0.975 = 97.5% = 39/40

increments of quarter percent:

1/400 = 0.0025 = 0.25% = quarter of a percent ====== 1/4%
1/200 = 0.005 == 0.5% == half of a percent ========= 1/2%
3/400 = 0.0075 = 0.75% = three quarters of a percent = 3/4%

1/100 = 0.01 === 1%
1/80 == 0.0125 = 1.25%
3/200 = 0.015 == 1.5%
7/400 = 0.0175 = 1.75%

1/50 === 0.02 === 2%
9/400 == 0.0225 = 2.25%
1/40 === 0.025 == 2.5%
11/400 = 0.0275 = 2.75%

3/100 == 0.03 === 3%
13/400 = 0.0325 = 3.25%
7/200 == 0.035 == 3.5%
3/80 === 0.0375 = 3.75%

1/25 === 0.04 === 4%
17/400 = 0.0425 = 4.25%
9/200 == 0.045 == 4.5%
19/400 = 0.0475 = 4.75%

1/20 === 0.05 === 5%
21/400 = 0.0525 = 5.25%
11/200 = 0.055 == 5.5%
23/400 = 0.0575 = 5.75%

3/50 === 0.06 === 6%
1/16 === 0.0625 = 6.25% --> 100/16 = 6.25
13/200 = 0.065 == 6.5%
27/400 = 0.0675 = 6.75%

7/100 == 0.07 === 7%
29/400 = 0.0725 = 7.25%
3/40 === 0.075 == 7.5%
31/400 = 0.0775 = 7.75%

2/25 === 0.08 === 8%
33/400 = 0.0825 = 8.25%
17/200 = 0.085 == 8.5%
7/80 === 0.0875 = 8.75%

9/100 == 0.09 === 9%
37/400 = 0.0925 = 9.25%
19/200 = 0.095 == 9.5%
39/400 = 0.0975 = 9.75%

1/10 === 0.1 ==== 10%
41/400 = 0.1025 = 10.25%
21/200 = 0.105 == 10.5%
43/400 = 0.1075 = 10.75%

11/100 = 0.11 === 11%
9/80 === 0.1125 = 11.25%
23/200 = 0.115 == 11.5%
47/400 = 0.1175 = 11.75%

3/25 === 0.12 === 12%
49/400 = 0.1225 = 12.25%
1/8 ==== 0.125 == 12.5% --> 100/8 = 12.5
51/400 = 0.1275 = 12.75%

other interesting fractions with definite patterns:
1/14 = 0.0714
1/15 = 0.0666 (6.66%)
1/20 = 0.05 (5%)
1/22 = 0.04545
1/25 = 0.04 (4%)
10/25 = 0.4 (40%)
24/25 = 0.96 (96%)
1/30 = 0.03333 (3.33%)
1/40 = 0.025 (2.5%)
1/45 = 0.02222
1/49 = 0.020408
1/50 = 0.02 (2%)
49/50 = 0.98 (98%)
1/60 = 0.016666
1/64 = 0.015625
1/66 = 0.0151515 (*)
1/75 = 0.013333
1/90 = 0.011111
1/99 = 0.010101
75/99 = 0.7575
1/100 = 0.01 (1%)
1/120 = 0.0083333
1/128 = 0.0078125
1/256 = 0.004 (0.4%)
1/512 = 0.002 (0.2%)
123/999 = 0.123123
1/1000 = 0.001 (0.1%)

Monday, October 15, 2012

Shortcuts in Multiplication, Division, Addition & Subtraction

These are some of the practical secrets in mental math calculation speed. Knowing these can help you figure out answers to most of life's daily activities involving numbers. There are many useful Math techniques, tricks, and secrets that can be valuable in your day to day work or study. Knowing these shortcuts are key to computation speed and accuracy in your Mental Mathematical Aptitude. This can be useful in applications such as IQ tests, aptitude tests, job application tests, military tests, college entrance tests and so many more uses in the office, workplace, or in your own home.

SHORTCUTS IN MULTIPLICATION:

Multiplication using multiples
12 x 15
= 12 x 5 x 3
= 60 x 3
= 180

Multiplication by distribution
12 x 17
= (12 x 10) + (12 x 7) ---> 12 is multiplied to both 10 & 7
= 120 + 84
= 204

Multiplication by "giving and taking"
12 x 47
= 12 x (50 - 3)
= (12 x 50) - (12 x 3)
= 600 - 36
= 564

Multiplication by 5 --> take the half(0.5) then multiply by 10
428 x 5
= (428 x 1/2) x 10 = 428 x 0.5 x 10
= 214 x 10
= 2140

Multiplication by 10 ---> just move the decimal point one place to the right
14 x 10
= 140   ---> added one zero

Multiplication by 50 ---> take the half(0.5) then multiply by 100
18 x 50
= (18/2) x 100 = 18 x 0.5 x 100
= 9 x 100
= 900

Multiplication by 100 ---> move the decimal point two places to the right
42 x 100
= 4200 ---> added two zeroes

Multiplication by 500 ---> take the half(0.5) then multiply by 1000
21 x 500
= 21/2 x 1000
= 10.5 x 1000
= 10500

Multiplication by 25 ---> use the analogy $1 = 4 x 25 cents
25 x 14
= (25 x 10) + (25 x 4) ---> 250 + 100 ---> $2.50 + $1
= 350

Multiplication by 25 ---> divide by 4 then multiply by 100
36 x 25
= (36/4) x 100
= 9 x 100
= 900

Multiplication by 11 if sum of digits is less than 10
72 x 11
= 7_2 ---> the middle term = 7 + 2 = 9
= place the middle term 9 between 7 & 2
= 792

Multiplication by 11 if sum of digits is greater than 10
87 x 11
= 8_7 ---> the middle term = 8 + 7 = 15
because the middle term is greater than 10, use 5 then
add 1 to the first term 8, which leads to the answer of
= 957

Multiplication of 37 by the 3, 6, 9 until 27 series of numbers --> the "triple effect"
solve 37 x 3
multiply 7 by 3 = 21, knowing the last digit (1), just combine two more 1's giving the triple digit answer 111

solve 37 x 9
multiply 7 by 9 = 63, knowing the last digit (3), just combine two more 3's giving the triple digit answer 333
solve 37 x 21
multiply 7 by 21 = 147, knowing the last digit (7), just combine two more 7's giving the triple digit answer 777

Multiplication of the "dozen teens" group of numbers --
(i.e. 12, 13, 14, 15, 16, 17, 18, 19) by ANY of the numbers within the group:
solve 14 x 17
4 x 7 = 28; remember 8, carry 2
14 + 7 = 21
add 21 to whats is carried (2)
giving the result 23
form the answer by combinig 23 to what is remembered (8)
giving the answer 238

Multiplication of numbers ending in 5 with difference of 10
45 x 35
first term = [(4 + 1) x 3] = 15; (4 is the first digit of 45 and 3 is the first digit of 35 --> add 1 to the higher first digit which is 4 in this case, then multiply the result by 3, giving 15)
last term = 75
combining the first term and last term,
= 1575

75 x 85
first term = (8 + 1) x 7 = 63
last term = 75
combining first and last terms,
= 6375

15 x 25
= 375

Multiplication of numbers ending in 5 with the same first terms (square of a number)
25 x 25
first term = (2 + 1) x 2 = 6
last term = 25
answer = 625 ---> square of 25

75 x 75
first term = (7 + 1) x 7 = 56
last term = 25
answer = 5625 ---> 75 squared

SHORTCUTS IN DIVISION:

Division by parts ---> imagine dividing $874 between two persons
874/2
= 800/2 + 74/2
= 400 + 37
= 437

Division using the factors of the divisor: "double division"
70/14
= (70/7)/2 ---> 7 and 2 are the factors of 14
= 10/2
= 5

Division by using fractions:
132/2
= (100/2 + 32/2) ---> break down into two fractions
= (50 + 16)
= 66

Division by 5 ---> divide by 100 then multiply by 20
1400/5
= (1400/100) x 20
= 14 x 20
= 280

Division by 10 ---> move the decimal point one place to the left
0.5/10
= 0.05 ---> 5% is 50% divided by ten

Division by 50 ---> divide by 100 then multiply by 2
2100/50
= (2100/100) x 2
= 21 x 2
= 42

700/50
= (700/100) x 2
= 7 x 2
= 14

Division by 100 ---> move the decimal point two places to the left
25/100
= 0.25

Division by 500 ---> divide by 100 then multiply by 0.2
17/500
= (17/100) x 0.2
= 0.17 x 0.2
= 0.034

Division by 25 ---> divide by 100 then multiply by 4
500/25
= (500/100) x 4
= 5 x 4
= 20

750/25
= (750/100) x 4
= 7.5 x 2 x 2
= 30

SHORTCUTS IN ADDITION:

Addition of numbers close to multiples of ten (e.g. 19, 29, 89, 99 etc.)
116 + 39
= 116 + (40 - 1)
= 116 + 40 - 1 ---> add 40 then subtract 1
= 156 - 1
= 155

116 + 97
= 116 + (100 - 3)
= 116 + 100 - 3   ---> add 100 then subtract 3
= 216 - 3
= 213

Addition of decimals
12.5 + 6.25
= (12 + 0.5) + (6 + 0.25)
= 12 + 6 + 0.5 + 0.25   ---> add the integers then the decimals
= 18 + 0.5 + 0.25
= 18.75

SHORTCUTS IN SUBTRACTION:

Subtraction by numbers close to 100, 200, 300, 400, etc.
250 - 96
= 250 - (100 - 4)
= 250 - 100 + 4    ---> subtract 100 then add 4
= 150 + 4
= 154

250 - 196
= 250 - (200 - 4)
= 250 - 200 + 4    ---> subtract 200 then add 4
= 50 + 4
= 54

216 - 61
= 216 - (100 - 39)
= 216 - 100 + 39
= 116 + (40 - 1) ---> now the operation is addition, which is much easier
= 156 - 1
= 155

Subtraction of decimals
47 - 9.9
= 47 - (9 + 0.9) ---> "double subtraction"
= 47 - 9 - 0.9   ---> subtract the integer first then the decimal
= 38 - 0.9
= 37.1

18.3 - 0.8
= 18 + 0.3 - 0.8
= (18 - 0.8) + 0.3 ---> subtract 0.8 from 18 first
= 17.2 + 0.3
= 17.5

WORKING ON PERCENTAGES:

30% of 120
= 10% x 3 x 120 ---> it is much easier working with tens (10%)
= 10% x 120 x 3
= 12 x 3
= 36

five percent of a number: 5%
360 x 5%
= 360 x 10%/2   ---> take the 10% and divide by 2
= 36/2
= 18

360 x 5%
= 360 x 50%/10   ---> take the half(0.5) and divide by 10
= (360/2)/10
= 180/10
= 18

ninety percent of a number: 90%
90% of 700
= (100% - 10%) x 700
= (100% x 700) - (10% x 700) ---> 100% minus 10% of the number
= 700 - 70
= 630

What is 18 as a percentage of 50?
= 18/50
= (18/100) x 2 ---> method: division by 50 (explained above)
= 0.18 x 2
= 0.36
= 36%

What is 132 as a percentage of 200?
= 132/200
= (132/2)/100
= [100/2 + 32/2]/100 ---> solution by "double division"
= (50 + 16)/100
= 66/100
= 0.66
= 66%

What is 270 as a percentage of 300?
= 270/300
= [(270/3)/100] ---> "double division" (using the factors of 300)
= 90/100
= 90%

What is 17 as percentage of 500?
= 17/500
= (17/50)/10
= (17/100) x 2/10   ---> solution using the procedure: division by 50
= (0.17 x 2)/10
= 0.34/10
= 0.034
= 3.4 %

percentages close to 100:
95% of 700
= (100% - 5%) x 700
= (100% x 700) - (5% x 700)
= 700 - (10% x 700/2) -------> 5% is 10%/2
= 700 - 70/2
= 700 - 35
= 665

percentages less than 10 percent:
3% of 70
= (3/100) x 70
= (70/100) x 3 ---> divide by 100 then multiply the percent value
= 0.7 x 3
= 2.1

DECIMALS:

To convert or express percentages as decimals, divide by 100 or simply just move the decimal point by two places to the left of the given number, thus:

1% = 1/100 = 0.01
2% = 2/100 = 0.02 = 1/50
3% = 3/100 = 0.03
4% = 4/100 = 0.04 = 1/25
5% = 5/100 = 0.05 = 1/20
6.25% = 6.25/100 = 0.0625 = 1/16
7% = 7/100 = 0.07
7.5% = 7.5/100 = 0.075
10% = 10/100 = 0.1 = 1/10
12.5% = 12.5/100 = 0.125 = 1/8
20% = 0.2 = 1/5
21% = 0.21
25% = 0.25 = 1/4
30% = 0.3 = 3/10
33.33% = 33.33/100 = 0.3333 = 1/3
37.5% = 0.375 = 3/8
40% = 0.4 = 2/5
50% = 0.5 = 1/2
60% = 0.6 = 3/5
62.5% = 0.625 = 5/8
66.66% = 66.66/100 = 2/3
75% = 0.75 = 3/4
80% = 0.8 = 4/5
87.5% = 0.875 = 7/8
100% = 1
125% = 1.25 = 1 1/4
150% = 1.5 = 1 1/2
200% = 2

FRACTIONS:

What is three quarters of 80?
= 3/4 x 80
= (80/4) x 3 ---> divide by 4 then multiply by 3
= 20 x 3
= 60

How many quarters in two and a half?
2.5/.25
= 10 ---> there are 10 quarters in $2.50

Improper fractions:

3/2 = 1 1/2 = 1.5 = 150%

4/3 = 1 1/3 = 1.3333 = 133.33% ---> useful number for volume of sphere, etc.

9/5 = 1 4/5 = 1.8 = 180% ---> conversion factor for Celsius/Fahrenheit temperatures

V = 4/3 pi * r^3

where:

V = volume of sphere
r = radius of sphere

F = (1.8 C) + 32

where:

F = temperature in Fahrenheit
C = temperature in Celsius