Rhombus

Properties of a Rhombus

• Opposite angles of the rhombus have the equal measure.
• The diagonals of the rhombus intersect each other at right angles.
• Rhombus has two diagonals that connects the opposite pair of vertices.
• Rhombus is symmetric along the diagonals.
• Every rhombus is a parallelogram.
• All the parallelograms are not a Rhombus.
• Dual polygon of a rhombus is called as a Rectangle.
• Sum of the adjacent sides of any rhombus is equal to 180 degree. That is the adjacent angles of a rhombus are supplementary.

Area of a Rhombus

Area of a Rhombus is same as that of the area of a parallelogram. Area is the product of the base and the height.
The Area of a Rhombus Formula is,
A=b * h
The base of the rhombus is the length of one of its sides and the height is the perpendicular distance between the opposite sides.
Perimeter of a Rhombus

The perimeter of a rhombus is the total sum of all the side lengths. All the sides are having equal length in a Rhombus. And so, Perimeter of a Rhombus Formula is,
Perimeter = 4s, where s is length of the sides.

Factorial upto 10

n	n!
0	1
1	1
2	2
3	6
4	24
5	120
6	720
7	5,040
8	40,320
9	362,880
10	3,628,800

DATA SUFFICIENCY questions

Mark A) If the question can be answered with statement I alone but not statement II alone, or can be answered with statement II alone but not statement I alone
Mark B) If the question cannot be answered with statement I alone or with statement II alone, but can be answered if both statements are used together
Mark C) If the question can be answered with either statement alone
Mark D) If the question cannot be answered with the information provided 

1. A point P is identified as P(m,n). What is the ratio AP:BP given that the points A and B are identified as A(5,-4) and B(1,6)?

Stmt 1. m = 3. Not sufficient.
Stmt 2. n = 2.5 Not sufficient.

Either statement alone is not sufficient. Both put together, we can answer the questions. Ans B

2. A point P is identified as P(m,n). What is the ratio AP:BP given that the points A and B are identified as A(5,-4) and B(5,6)?

Stmt 1. m = 5. Not sufficient. It tells us that all points lie on the line x = 5, but this is not enough
Stmt 2. n = 3 Not sufficient.

Both put together, this is enough.

In the above question, would n = 1 have been sufficient? Think about that.

3. A survey of 100 people tried to find the number of people who can write with both their left and right hands. What is the maximum number of people who could write left-handed and right-handed?

Stmt 1. 50 people can write only with their left hand. 40 people can write only with their right hand. Sufficient: Maximum of 10 people could write left-handed and right-handed.
Stmt 2. 50 people can write with their left hand. 40 people can write with their right hand. Sufficient: Maximum of 40 people could write left-handed and right-handed.

Answer Choice C

4. What is the slope of a line?

Stmt 1: The line makes 135 degrees with the negative direction of x - axis. Sufficient: The line makes 45 degrees with positive x axis. This should be enough.
Stmt 2: The line makes an isosceles right triangle with the coodinate axes and the product of the intercepts is negative. Sufficient: Either both intercepts are positive and equal or negative and equal. Slope = -1
Answer Choice C 

DATA SUFFICIENCY

“The ultimate goal of mathematics is to eliminate any need for intelligent thought.”-A. N. Whitehead
Well, for starters try to think what’s the meaning of the above quote, it has a nice and beautiful meaning. To make things so simple that an average mind is able to see it. So lets put this into practice, with another concept lesson.
Today, we will discuss DATA SUFFICIENCY, one of the very scoring problems in cat and other mba entrance exams. The good thing about DS is we get DS both in quant and DI, and it makes for 6-8 problems in almost every paper. As there are no theorems in DS, we will take things to note
Things To Note:
1) DS problems, we need to answer if it is sufficient information to answer, means, there should be one conclusive answer.  We do not need to find the answer, just if it can be found or not?
2) Some questions ask , is this true? So if we can find that the information available is enough to prove that it is not, we are still able to answer the question, that it is not true. Hence we are able to answer the question.
3) Check for all possibilties, that is using one statement, using second, then only combine the two.

Permutation and Combination

Conversion : Distance and weight

1 mile = 1760 yards
1 yard = 3 feet
1 mile² = 640 acres
I gallon = 4 quarts
1 quart = 2 pints
1 pint = 2 cups
1 cup = 8 ounces
1 pound = 16 ounces
1 ounce = 16 drams
1 kg = 2.2 pounds

Time and Work - Examples

Q
If A can do a work in 10 days, B can do it in 20 days and C in 30 days in how many days will the three together do it?

Soln:
The efficiencies are A = 1/10, B = 1/20 and C = 1/30
So work done per day by the three = 1/10 + 1/20 + 1/30 = 11/60 => No of days = 60/11 = 5.45 days.


Q
If A and B can do a work in 10 days , B and C can do it in 20 days and C and A can do it in 40 days in what time all the three can do it?

Soln:

A+B = 1/10
B+C = 1/20
C+A = 1/40
Adding all the three we get 2(A+B+C) = 7/40 => A+B+C = 7/80 => No of days = 80/7 days.

If A can do a work in 12 days, B can do it in 18 days and C in 24 days. All the three started the work. A left after two days and C left three days before the completion of the work. How many days are required to complete the work?

Soln:

Let the total no of days be x.

A worked only for 2 days, B worked for x days and C worked for x-3 days.

So, mA + nB + oC = 1
ð      2(1/12) + x(1/18) + (x-3)(1/24) = 1
ð      12 + 4x + 3(x-3) = 72
ð      x = 69 / 7 days.

Note:

The ratio of dividing wages = ratio of efficiencies = ratio of parts of work done

Q:

A can do a work in 10 days and B can do it in 30 days and C in 60 days. If the total wages for the work is Rs. 1800 what is the share of A?

Soln:

Ratio of wages = 1/10 : 1/30 : 1/60 = 6 : 2 : 1  (Multiplying each term by LCM 60)

So total 9 equal parts in Rs. 1800 => each part = Rs. 200 => share of A = 6 parts = Rs. 1200.
Applying the same logics to pipes and cistern

Q:

A pipe can fill a tank in 5 hrs but because of a leak a the bottom it takes 1 hr extra. In what time can the leak alone empty the tank?

Soln:

Let the filling pipe be A.
A = 1 / 5.

But with the leak L,  A – L = 1 / 6   ( A-L because leak is outlet)

So, 1/L = 1 / 5 – 1/ 6 = 1/30 => Leak can empty the tank in 30 hrs.

Q:

A pipe A can fill the tank in 10 hrs, B can fill it in 20 hrs and C can empty in 40 hrs. All are opened at the same time. After how many hours shall the pipe B be closed such that the tank can be filled in 10 hrs?

Soln:

Let the pipe B be closed after x hrs.

Then A worked for 10 hrs, B worked for x hrs and C worked for 10 hrs.

mA + nB – oC = 1    (since C is outlet)

10(1/10) + x(1/20) – 10(1/40) = 1

x = 5 hrs.