Divison of two numbers is given by the following relation

Divison of two numbers is given by the following relation:
m/n=q
Here m is being divided by n and the result of the divison is q which is known as quotient. Let see how to do Division by 2 Digit Numbers which means dividing any dividend by 2 Digit Divisor

Step 1) Put the two digit divisor before the divison braces and put the dividend no. below the divison bar.
Step 2) Check the first digit of the dividend. If it is smaller than the divisor then take the first two digits of the dividend. Now determine how many times of divisor produces those two digits of dividend or produces a number which is just less than the dividend digits. Let x times of divisor give above result.
Step 3) Now multiply the no. x by divisor, let the result is y. Put y under the first two digits of dividend.
Step 4) Subtract no. ‘y’ from first two digits of dividend. Let the result is z. number z will be less than the divisor. So, bring down the third digit of dividend beside z. Now again follow the same steps from step 2.
Step 5) Continue following these steps till no more digits are left in the dividend and you get a remainder which is less than divisor.

Let’s use method of Long Division Two Digit Divisors through some examples:
Example 1) Divide number 7139 by 16.

Step 1) Firstly, check the first digit of the dividend. It is 7 and is smaller than divisor 16. So we will consider two digits of dividend i.e. 71. Now the largest multiple of 16 which is smaller than 71 is 64. As 16*4 = 64 so, write 4 (quotient) on right hand side of dividends and 64 below 71. Now, subtract 64 from 71.  
16)7139(4                                                                
       64                                                                                                                                                                                                                                   .       7
Step 2) Now consider number 7. As this is smaller than divisor 16, so we will bring digit 3 down with number 7. Now we will repeat the above step again. Find the largest multiple of 16 which is smaller than 73. This number is 64 which is equal to 16*4 . Write 4 at the place of quotients and subtract 64 from 73.
 16)7139(44                                                                
      64
----------
      73
-     64
----------
      99
-     96
----------
-      3
Step 3) Repeat the above steps again. This is done below:
16)7139(446
     64
--------
     73
-    64
--------
     99
-    96
--------
      3

3 is remainder.
This is how a Division 2 Digit Divisor is done. You can practice Divisibility by 2 digit divisors: 234/34, 5678/89, 7553/123etc.

What is Ratio

When we say one banana for every three apples, the relationship between the banana and the apple is shown by a term called Ratio. It is used in comparing and showing the relationship between two entities. It is denoted using the symbol colon (:) between the two values.

In the above example the proportion between banana to apple would be banana: apple read as ‘banana to apple’ the value of which would be 1:3 read as ‘one is to three’.

Hence we can say ratios tell the relationship between two values that is how one number is related to the other. It may be denoted as a fraction also, for instance the two values which are to be compared are X and Y then the proportion between them can be shown either as X:Y or X/Y or just X to Y. In the above example the proportion shows that apples are three times bananas.

One important point to remember while writing the balance is that the order should not be changed that is the respective numbers should not be interchanged.

If for instance there are 3 pencils for every 5 pens, the balance when considered as pencils to pens should also be written in the same order pencils:pens, 3:5 and not 5:3 which would mean pens to pencils

Let us now determine the value of Y, if X=6 and the balance of X to Y is 3:4. To find the value of Y first we need to determine how many times X is divisible by the corresponding part of the balance (3:4) which can be calculated by dividing 6 with 3 which gives 2.

Now we just need to multiply this 3 with the corresponding balance part of Y which gives 2x4=8. When the proportion is 3:4 and the value of X=6 then the value of Y=8.  Ratio definition can be given as comparison between two things which tells the relationship between the two. Let us now take a glance at the various ratio problems which help to understand the concept.

There are 8 children, 3 are boys and 5 girls. What is the ratio of boys to girls, girls to boys, the total children to boys and total children to girls? Given the total number of children=8, boys=3 and girls=5. So, the proportion of boys to girls is 3:5; the proportion of girls to boys is 5:3; the proportion of total number of children to boys is 8:3 and the proportion of total number of children to girls is 8:5.

Change of Base Formula for Logarithms

Logarithm is a means of expressing a number using exponents. Example log101000 is equal to 3 as 1000 is a cube of ten and can be written as log_10 10^3. Hence the value is 3.
The common base for logarithms is base ten and the other base is the natural logarithm base –e. At times while calculating logarithms we come across base other than 10 and the base e, in such cases the base change can be done using a special formula.

Logarithm change of base formula can be given as, log x to base a = log x to base b/log a to base b.  To understand how to arrive to this base change formula let us go through the following steps:
Consider y=log_a x, we get x = a^y
Taking log_b on both sides would result in log_b x = log_b a^y
Applying the power rule to the above equation gives, log_b x = y log_b a
Now dividing on both sides with log_b a gives, log_b x/ log_b a= y log_b a/ log_b a
So, we get, y = log_b x/ log_b a

Let us now consider a simple example, the value of log 27. This can be written as log_10 7/log_10 2

The value can be calculated as log 7=0.845 and log2= 0.3010. When these values are divided the final answer would be 2.80730…; thus using loga x= log_bx/log_b a, the change of base formula logarithms value of the given logarithmic expression can be found easily. Using Log base change formula it becomes easy to evaluate logarithms with different base. Here the logarithm is written as a fraction with the logarithm of the number as the numerator and the logarithm of the base as the denominator, such as log_a x = log x/log a.

Then each of the logarithms is evaluated using the log table or a scientific calculator, the final value is got by dividing these values. The evaluation of other logarithms with base different from natural logarithm base or the common logarithm base can be done using the base change formula, log_a x = log_b x/log_b a. Let us now evaluate the logarithm log_5 9. This problem can be solved by either using natural logarithm or the common logarithm. Using the natural logarithm that is base-e it would be, log_5 9 = ln9/ln5 = 2.1972/1.6094 which would be approximately equal to 1.3652… Now using the common logarithm that is the base ten it would be, log_5 9 = log 9/log5= 0.9542/0.6989 = 1.3652… Using either of the logarithms we arrive at the same result.

Simple interest

Definition:
Consider a house that one would have rented. The tenant has to pay some amount of money to the owner of the house as rent for using the property. Similarly if a person borrows money from another person, he has to pay some amount of money as rent for using the borrowed money. This charge paid for use of funds is called interest. Therefore the amount charged on a fixed amount of principal, that is lent by a lender for a specific period of time is called simple interest. In simple interest the principle amount over the period of loan remains constant and is not reduced or increased.
Formula for simple interest:
Some important terms related to simple interest:
(1) Principal (P): The money borrowed or lent.
(2) Interest (I): The additional amount paid to the lender, for the use of the money borrowed.
(3) Rate( R ): Interest for one year per 100 units of currency.
(4) Time (T): The time period for which the money is borrowed.
(5) Simple interest or (S.I.): When the interest is paid to the lender regularly every year or every half year, we call the interest simple interest.
(6) Amount (A): Principal + Interest = amount at the end of the term of T years.

Formula used for calculating simple interest is like this:
S.I. = P x R x T
100
A = P + S.I.

When we calculate simple interest, the following points need to be noted:
(1) Rate of 4% per annum means $ 4 for every $ 100 per year. Similarly a rate of 1.5% per month means $ 1.5 for every $ 100 per month = $ 1.5 * 12 = $ 18 for every $ 100 per  year = 18% per annum.
(2) When time is given in days, we convert it to years by dividing by 365. When time is given in  months, we convert it to years by dividing by 12. When dates are given, the day on which the sum is borrowed is not included but the day on which the money is returned is included, while counting the number of days.

Decimal to Hexadecimal

We are now going to look at Decimal to Hexadecimal converter. So let us understand what exactly a hexadecimal number and what its digits mean. So we are going to look at three digits of hexadecimal number the first unit represents units, which is 16 to the power of zero that is one. That represents units. The second digit represents tenths, which is 16 to the power of one. And the third digit represents hundredths, which is 16 to the power of two. That is nothing but 256. So important thing to do when one is working on how to convert a decimal to hexadecimal, is the start of working out how many hexadecimal number is going to have?

Let us understand it with an example, convert decimal to hexadecimal. Say number 74, here we need to decide, what we are going to and how many digits this hexadecimal number is going to have. Now because 256 is less than 74, there is any going to be two digits. So we now going to see, that in 16 to the power of one column, here we divide 74 by 16 and the result is 4. This means 4 times 16 is 64 and we have the remainder as 10. Now 10 is a single digit in a hexadecimal, simply represents a A , that tells us 74  = 4 A.

 Let us understand with a complicated example. This time it represents 680 as a hexadecimal number. Here that we see 680 is greater than 256 so we are going to have three digits in a hexadecimal number. What we going to do first is divide 680 by 256. And the result of that is 2. Two times 256 is 512, so our remainder is 168. Next we go back as what we did in our first example, we are going to divide 168 by 16. The result of this is 10. 10 times 16 equals to 160, as we are left with the remainder 8. Now we have three digits in hexadecimal number, thus we notice that we have 10, which is represented by ( A ) . This tells us that 680 when written as a hexadecimal number as 2A8. That is, 680 = 2A8. This is how we do a decimal to hexadecimal conversion. The method to do this is to keep on dividing the decimal number by 16 till it gets the most significant remainder.