Multiplying mixed fractions

Mixed fractions are just the mixture or a combination of two things of whole number and a fraction. Mixed fractions are specially used when we have to show how much a whole thing we have and how much of part of something we have. The form for mixed fraction looks like this, a and b/c, where ‘a’ is our whole number and b over c is b/c is our fraction.

Multiplying Mixed Number Fractions - For Multiplication of Mixed Fractions, we first convert the mixed fractions into improper fractions and the multiply them and then convert the solution back into mixed fraction. How do I Multiply Mixed Fractions – let us understand by taking an example, if we have to multiply 1 times 3/4 and 5 times 7/9 then we can convert 1 times 3/4 to improper fraction by multiplying 4 with 1 and adding 3 to it. We will get 7, which will be our numerator for the improper fraction. The denominator will remain same that is 4. So our improper fraction will be 7/4. Similarly we can convert 5 times 7/9 as (9*5) +7/9 which will be 52/9.

Hence we got two fractions as 7/4 and 52/9. We can multiply both the numerators and the denominators. We will get 7*52/4*9 which will be 364/36. We can reduce this fraction by dividing both the numerator and the denominator by 4 which makes the fraction as 91/9.

Now we can convert it into a mixed fraction by dividing. So 91/9 can be written as 10 times 1/9. Multiplying Mixed Fraction always gives a mixed fraction as the solution. The same we witnessed in the above example. We multiplied two mixed fractions and we got a mixed fraction as an answer. So to multiply mixed fractions we need to follow the following steps: -
1. Convert the mixed fractions that we have to multiply into improper fractions.
2. Multiply the improper fractions by multiplying the numerators and the denominators separately that is if we have two fractions, the numerator of both the fractions should be multiplied together and the denominator of both the fractions together. For example if we have 11/2 times 8/5. Then we can multiply 11 with 8 which will be 88 and 2 with 5 which will be 10. Therefore 11/2 times 8/5 gives 88/10
3. Reduce the fraction as much as we can.
4. Convert the answer back into the mixed fraction form.

Discovery of algebra

The discovery of the algebra starts from the place Egypt and Babylon. In Egypt and Babylon the people were very interested to learning the linear equations such as cx=d and the quadratic equations such as c2+dx = e and in intermediate equations. The Babylonians basic steps are used to solve the quadratic equations. Nowadays we also use these basic steps only. Now in this article we just simply see about the discovery of algebra.

Explanation for Discovery of Algebra:

The Diaphanous who is called the father of algebra. The book named Arithmetic book of Diaphanous gives an advanced level and many unexpected solutions to the intermediate equations for discovery of algebra. The first Arabic algebras that is a systematic expose of the basic theory of equations, is written by AL-Khwarizmi. The basic laws and identities of algebra are stated and solved by the Egyptian mathematician Abu Kamil in nineteenth century.

Discovery of Algebra in Persian Mathematics:

The Persian mathematician, who called omar kayyam, mentioned in his book how to express roots of cubic equations by line segment which is obtained by intersecting conic equations. But he did not find the formula for the roots. But in 13th century, the great Italian mathematician named Leonardo Fibonacci achieved the close approximation to the solution of the cubic equation.

After the introduction of the symbols for unknown and for algebraic powers and operations there was a development in algebra. It was happened in 16th centuries.

Algebra entered in to the modern phase in the gauss time. After the discovery the Hamilton, the German mathematician Hermann Grassmann examining the vector. Because of its abstract character, American physicist named Gibbs acknowledged the vector algebra for system of great utility for physicists. In that time period, algebra of modern that means abstract algebra has continued to develop. After that it is used in all branches of mathematics and in other science.

Mathematical Sentence

In Algebra, Mathematical sentence is a term which is also known as an expression. An expression is usually defined as a sentence that has a number, an operation and a letter in it.

When a mathematical sentence is not in an algebraic form, it will just have two numbers and an operation. In other words, an expression is the mathematical analogue of an English noun. It is a correct arrangement of mathematical symbols which is used to represent a mathematical object of interest.

Example for Mathematical Sentence:

In general, a mathematical sentence is a formula that is right or wrong, true or false. If a mathematical sentence has an equal sign, it is referred as an equation.

Let us consider the following simple examples to show what a Mathematical Sentence means:

3 + 2 = 5. We know this to be true, it is a mathematical sentence.

3 + 4 = 5

we know this to be false, however, since we know definitely that it is false, But still it is a mathematical sentence.

5x + 8 = 13, for all values of x

This statement is neither true nor false: For some values of x, the statement is true and for some other values, it is false. Hence as the statement is neither true nor false, this is not a mathematical sentence.

Mathematical Sentence as an Open Sentence:

A mathematical sentence that contains one or more variables is referred as an open sentence

Some examples for an open sentence are listed below:

2a = 5+ b, 4x = b + 2a

Solved example for an open sentence:

To show ‘3(2b) = 3' an open sentence?

Solution:

Step 1: In the equation given, the number of variable is One

Step 2: Open Sentence is a mathematical sentence with one or more variables

Step 3: So, '3(2b) = 3' is an open sentence.

Subtracting Fractions

Subtraction is one of the basic operations in math. It is a process of finding the difference of two numbers. In case of whole numbers the process is simple but in case of fractions, the process may involve a number of steps.

How to Subtract Fractions
As mentioned earlier, subtraction of fractions require a number of steps. Of course, in the simplest cases, where the denominators of the fractions are same, the process is just like subtracting whole numbers. All that need to be done is just subtract the numerators over the given common denominator.

But in case of fractions with different denominators, first task is to find the equivalents of the given fractions with same denominator.  Such a equivalent common denominator is also called as Lowest Common Divisor. It is same as the lowest common multiple of the denominator.

How do you Subtract Fractions
Let us concentrate on the case of subtractions of fractions with different denominators, as the case of subtraction of fractions with same denominators is as simple as subtraction of whole numbers.
The easiest method for students to understand is the method of equivalent fractions. The first step here is, determine the lowest common multiple of the given denominators.

Then find ‘equivalent fractions’ of the given ones with the lowest common multiple as the denominator. That is rewrite the given fractions as if their denominators are changed to the lowest common multiple found. For example, 1/2 can be rewritten in the equivalent form as 2/4, 4/8,8/16 etc. depending on the need.
The subtraction of given fractions is same as the subtraction of their equivalent fractions and now the process is simple because the denominators are made same.

Subtract Fractions
Let us discuss a specific example to illustrate how to subtract fractions with different denominators. Let is consider that 2/5 is subtracted from 3/7.

The denominators are 5 and 7 and the lowest common multiple is 35. The equivalent fraction of 2/5 with denominator is 14/35 and that of 3/7 with the same denominator is 15/35.

Now subtraction of 2/5 from 3/7 is same as subtraction of 14/35 from 15/35 which is (15 – 14)/35 = 1/35. Thus the answer is 1/35.

Subtracting Fractions with Whole Numbers
For subtracting fractions with whole numbers, first subtract the fraction parts and then the whole number parts and combine them For example, 5(1/2) – 3 (1/4) is done by (1/2) – (1/4) 1/4 and 5 – 3 = 2, which gives the final answer as 2(1/4).

However trouble arises when the fraction part of the first mixed number is less than that of the second mixed number. In such cases, convert the first fraction part to an improper fraction by borrowing 1 from the whole number. The rest of the procedure is same as explained but remember the whole number part of the first one is reduced by 1!

How to Multiply Two Digit Numbers

If we have practiced and hopefully memorized the multiplication tables and Multiplication Rules, then we can solve any multiplication problem. We just have to understand the system to how to do it. We know that multiplication is just repeated addition but if we have large numbers, we cannot add them repeatedly to get the solution. To multiply one digit number and two digit number with a one digit number is an easy process but multiplying two double digits numbers uses a different process that needs to be followed.

How to Multiply Two Digit Numbers - We will learn here how Double Digit Multiplication works. Let us start with an example of the multiplication problem of two double digits numbers. Let us do 16 times 19; we can break down the Two Digit Multiplication into series of steps which are given as follows: -

• In 2 Digit Multiplication, firstly we take the numbers present in ones place and multiply them together; in this case we multiply 9 times 6 which equal 54.
• With the product of the digits at ones that is 54, we only write the 4 down while multiplication and 5 is carried forward on the tenth place just like when we add two numbers.
• Then we multiply 9 times 1 which is 9 and then add 5, and which is 14 so the solution to 9 times 16 is 144. Thus 16 times 9 is 144.
• Then we take the digit on the tens place on one number that is one in the number 19 and multiply it with 16, which gives us 16.
Now we have two solutions, one is 144 and other is 16, to find the final answer we add a zero to the second solution and then add the solutions together. That means 16 will become 160 and then we add it to 144 which equals 304. So the solution of 16 times 19 is 304.

 Only by using these breaking down method we have solved this big problem.  To multiply two digit numbers we can either break the one double digit number into two digits. For example, in the previous example, we can break 19 into 1 and 9. 9 is at ones place so we can write it as 9 but 1 is at tens place so we can use its face value which is 10. Then we can multiply 9 by 16 which give us 144 and then we multiply 10 by 16 which equals 160. We can add both the solutions to get the solution 304.