March 26, 2016

Proportion

Proportion
The equality of two ratio is called Proportion.
If a/b = c/d or a : b = c : d then we say a, b, c and d are proportional. It is denoted by : : symbol. In simple words, a : b : : c : d denotes the four numbers are proportional.
Types of Proportion

Mean Proportion:- Mean proportion between a and b is √ab
Example: - Find the mean proportion of 4 and 9
Solution:- Mean Proportion = √4 x 9 = √36 = 9

Third Proportion: - If a : b : : b : c then c is called the third proportion to a and b.
Formula to find Third Proportion: - c = b^2 / a
Example: - Find the third proportion of 4 : 6
Solution: - Third proportion = 6^2 / 4 = 9

Fourth Proportion:- If a : b : : c : d then d is called the fourth proportion to a, b , c and d. Fourth proportion = b c /a
Example: Find the fourth proportion to 3, 7 and 9
Solution: - Here, 3 : 7 : : 9 : x then the fourth proportion = bc / a = 7 x 9 /3 = 21

Basic Rule involved in Proportion

 Invertendo:- For a : b :: c : d the invertendo = b : a : : d : c

 Alternando : - For a : b :: c : d the alterando = a : c :: b : d

 Componendo :- For a : b :: c : d the componendo = a + b : b : : c +d : d

 Dividendo : For a : b :: c : d the dividendo = a – b : b : : c – d : d

 Componendo and Dividendo :- For a : b :: c : d the componendo and dividendo = a + b : a = b = c + d : c – d

Continue...........


Dr Rajesh Kumar Thakur

Ratio

Ratio and Proportion
Ratio is a relationship between two numbers that indicates how many times the first number contains the second. Suppose I bought 2 orange and 8 apples then the ratio of oranges to apple is two to eight.

This ratio can be therefore written as 2: 8 which is equivalent to 1: 4.
A ratio is a comparison of two numbers. For example, if 25 students -- 10 boys and 15 girls -- are in your class, then the ratio of boys to girls is 10 to 15.
The numbers compared in ratio can be any quantities of a comparable kind such as lengths, persons, any objects etc. The ratio of numbers a and b can be expressed as
1. a to b
2. a is to b
3. a : b
In ratio the numbers a and b are called antecedent and consequent respectively.
Ratios are sometimes used with three or more terms. Have you ever gone to a construction sites? A mason mixes sand, cement and water in a proper ratio. If for every 2 kg of sand and 4 kg of cement the mason mixes 9 kg of water then the ratio in this circumstance is 2: 4: 9.

According to the Oxford English Dictionary online, ratio is 'the relation between two similar magnitudes in respect of quantity, determined by the number of times one contains the other (integrally or fractionally)'.
It means the ratio can be expressed in fraction. The ratio of oranges to that of apple as taken above can be written in fraction as 2/8.

There can be a ratio between `100 aand `500 but there can’t be the ratio between `500 and 1000 apples.
Types of Ratio

 Duplicate Ratio: - If two numbers are in ratio, then the ratio of their squares is called duplicate ratio. The duplicate ratio of two numbers a and b is a^2: b^2.
Example: - Duplicate ratio of 5: 6 is 5^2: 6^2 = 25: 36

 Sub- duplicate Ratio: - If two numbers are in ratio then the ratio of their square roots is called sub- duplicate ratio. The sub-duplicate ratio of a and b is a^1/2 : b^1/2
Example: - Sub duplicate of 25: 36 = √25: √36 = 5 : 6

 Triplicate ratio: - If two numbers are in ratio, then the ratio of their cubes is called triplicate ratio. The triplicate ratio of two numbers a and b is a^3: b^3.
Example: Triplicate ratio of 2: 5 is 2^3: 5^3 = 8: 125

 Sub- triplicate ratio:- If two numbers are in ratio, then the ratio of their cube- root is called sub-triplicate ratio. The sub-triplicate ratio of two numbers a and b is a^1/3: b^1/3.
Example:- Sub- triplicate ratio of 8 : 125 is 2: 3

 Inverse Ratio: - If two numbers are in ratio such that when the antecedent and consequent of ratio is interchanged a new ratio is obtained which is called Inverse ratio. The inverse ratio of a: b is b : a.
Example:- Inverse ratio of 4 : 7 is 7 : 4

 Compound Ratio:- If two or more ratio are given, then the antecedent of one is multiplied with antecedent of other and respective consequents are also multiplied then we get compound ratio. If a : b , c : d, e : f are three ratio then their compound ratio will be ace : bdf
Example: The compound ratio of 2 : 5 , 5 : 9 and 7 : 18 is 2 x 5 x 7 / 5 x 9 x 18 = 7 : 81


Dr Rajesh Kumar Thakur

March 16, 2016

Why should I read Mathematics???

Why should I read mathematics?

Mathematics is omnipresent. It is in your life, in your body, in your kitchen, in your garden, in vegetables, fruits, music, art, painting, medicine, buildings, etc. etc. in one word mathematics is in every particle of the world. If it is so then why there does so much rumor about mathematics. Mathematics is a fun loving subject based on truth. Yes, you have the choice to hate mathematics but can’t think to disassociate yourself with mathematics.
Mathematics is present in everyday life; it surrounds a person wherever a person goes. It is involved in our day to day life. People sometimes don’t realize the amount of math that surround them and the amount of math that they use every day. In order to live happily and be competent to live in a society you should be able to do simple arithmetic like addition, subtraction, multiplication and division. You need to be able to handle money, prepare food do your job, purchase the necessary thing for household and all this require the basic knowledge of mathematics.
The study of mathematics is extremely important for many reasons. Most importantly, math surrounds us in many aspects of our everyday life. Our monetary system, system of measurement, mechanical objects such as automobiles and many other aspects of life we encounter daily are highly dependent upon math. If an individual is unable to perform simple mathematical processes, he or she finds himself in a highly disadvantaged situation that lacks a certain understanding of how things operate. These individuals lack beneficial abilities such as pattern recognition and logical reasoning that are developed in mathematical instruction.

It is necessary for people to study Mathematics for many reasons. The universal language of the world is math, and people have been using it for thousands of years across the world. Today’s society would not be in existence without the application of mathematics. The application of math can be seen everywhere throughout the world, and without it a majority of things would not be possible. Complicated things such as building a bridge or buildings, flying an airplane, setting up an industrial units or mass producing anything would not be possible. Even simple things such as buying vegetables or groceries, paying a mobile, electric, or water bill, or cooking would not be possible without math.

Even if someone has no interest in how to build a bridge, or how things are mass produced, to live in today’s society it is necessary for someone to be able to support themselves financially, and money involves math. In order to cook numerous things exact measurements and temperatures, as well as exact times are needed, all involving math. As much as the “normal person” hates math, no one can escape the need for mathematics in their everyday life. Galileo said, "The great book of nature can be read only by those who know the language in which it was written. And that language is mathematics.” Math is simply logic and reasoning. Math is present in everyday life and is being used by every layman. It is the universal language which is understands by every person regardless of what spoken language one use. The product of 2 and 4 is 8 in India, US, UK and any other part of the world
Let me figure out few more reasons to boost your moral if you think mathematics reading is not important.

• Learning math involves a different type of thinking that is not addressed in other subjects. It helps us to think logically and rationally.

• It helps us keep score- not just in sports, but in everything that we measure such as time, distance, speed, money, cooking quantities, etc.

• It helps us to make our choice better. Suppose you are in a shop to buy a motorcycle and your budget is between ` 50,000-60,000. After discussing much you freeze out on two bikes that fit your budget but the problem is which one to buy. Now you will certainly look out for other option that is mileage. With the increasing cost of patrol you would love to buy that bike which is fuel efficient. Keeping this on mind your next query would be which one is fuel efficient. The first one shown here gives 54 Kmpl and the second one gives 87 Kmpl then certainly your choice would be the second one. This clearly shows that mathematical knowledge is essential in making a better choice.


• Let us see the ingredient of matar- paneer recipe. Matar Paneer Recipe
Prep & Cooking: 40 mts Serves 4-5 persons
Ingredients:
1/4 kg paneer 1small cup fresh green peas
1 large onion, finely chop ginger-green chili paste (1″ ginger piece+3 green chillis)
2 tomatoes, finely chopped 3/4 tsp red chili powder
1 pinch of turmeric powder 1 1/4 tsps coriander powder
1/2 tsp kasuri methi (dry fenugreek leaves) 1pinch of garam masala pwd
1 tsp masala powder 1 tsp malai salt to taste 1 tbsp oil

The above ingredient shows that until you are aware of the mathematical terms like half, quarter, etc or understand the basic measurement you can’t be a good cook. Cooking uses fractions and ratios. If you are baking a cake in microwave you need to understand the timing. In order to make cake you have to understand the measurement. If your measurement is incorrect the recipe will not turn out right.

• The most obvious place where mathematics is used in daily life is balancing the household budget. Whether you are a student, daily wage worker or employee, managing house budget, saving, other expense all depends on mathematical calculation. If you are getting a salary of `25000 pm and as per the requirement you need to save `5000 pm to avoid tax deduction. Besides that you pay `2000 for utility bills like electricity, water, gas and `10000 for your household budget such as grocery, vegetables, milk etc. Now you are left with `8000 and you have other expenses to meet such as tuition fee for your children, medical expenses if any. In order to run your house smoothly you have to decide how to expense the remaining amount judiciously so that your daily requirement is met. This all depends upon the mathematical calculation.

• Many of us use credit card. An understanding of mathematical logic helps you determine whether a credit card is really as great a deal as the bank wants you to think. The bank charges some interest and some banks provide you a grace period and after the end of that grace period interest starts to accrue. The interest rate is called APR that stands for Annual Percentage Rate which accrues on unpaid balances. Suppose you have a credit card with an 18% APR with 25 days grace period. If you purchased goods worth `20000 on your credit card and pays back `5000 within the grace period. Now once the grace period is over you will be charged with 1.5% monthly interest on the balance plus the monthly periodic rate. So after a month you will receive the bill of 20000 x 1.5% + 20000 =23000. Since you paid the sum of `5000, so you will be charged the interest of remaining `18000. The best method of using credit card is to pay the balance in full so that you can avoid paying any interest.

• In order to decorate your home you need to understand measurement, area, perimeter, and simple arithmetical calculation. If you want to plant flowers in your garden, you need to calculate the area in order to determine the number of plants that will fit within the space. To calculate how much soil or stone you will need to fill an area or the amount of water you need to water the plant you must have the knowledge of volume. In order to put wire fencing around the garden you need to understand its perimeter.

The above examples are sufficient to understand that mathematics should be read by all and it should be enjoyed because mathematics is the language of truth. It makes us rational in our approach. Mathematics teaches us judgment, truth, honesty and many other virtues which are important for our life. So be ready to embrace mathematics.



Dr Rajesh Kumar Thakur
rkthakur1974@gmail.com