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Algebra

Multiple Fraction

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Multiplying Fraction

Fractions are numbers which are used to denote a whole number that has been divided in equal parts. We can say fraction is the part of everything or collection of objects. There are two parts of fractions, the top and the bottom (Numerator or Denominator).The number on the line is called Numerator and the number below the line is called Denominator. We use fractions everyday. Fractions are important because they show you the parts of the whole. Fractions are significant for u they tell us which part of the collection we want.

In multiple fractions, first multiplication of numerators, second multiplication of denominators then simplify the resultanting fraction. In this chapter we discuss the explanation of the numerator and the denominator of the fraction. In this, we discuss rules of multiplying fraction, multiplying fraction with same denominator and multiplying fraction with same numerators. Also we learn multiplying fraction with mixed numbers, multiply of the improper fractions, multiply fraction on a number line, and multiply fraction to whole number

Explanation of the numerator and the denominator of a fraction

In fraction, Number on the line is called numerator and number below the line is called denominator. Denominator describes the whole part and the numerator describes the part, which we want.

Numerator and Denominator represented by the figure given below, shape denote 1 out of 2 parts is shaded, therefore numerator is 2 and denominator is 2. 

Rules of Multiplying Fractions

In Fractions, Multiplying Fractions is described as the multiple of a fraction with one or more fractions. The multiple of fractions are multiple numerator to numerator. Multiple denominator to denominator.

Multiplying fractions needs to be simplified. This is useful for us.it makes the fraction multiplication much easier because the numbers to be multiplied are smaller after the simplification.

First method

There are following steps for multiplying fractions.

  • Multiple the numerators from each fraction by each other. The outcome is the numerator of the answer.
  • Multiple the denominator from each fraction by each other. The outcome is the denominator of the answer.
  • Simplify or reduce the fraction. If needed.

Multiplying fraction

    \[\dfrac{{20}}{{15}}and\dfrac{6}{{12}}\]

 

    \[\begin{array}{l}\dfrac{{20}}{{15}} \times \dfrac{6}{{12}}\\\dfrac{{20 \times 6}}{{15 \times 12}} = \dfrac{{120}}{{180}} = \dfrac{2}{3}\end{array}\]

 

Second method 

There are following steps for multiplying fractions.

  • Simplify or reduce the fractions themselves.
  • Multiple the numerators from each fraction by each other. 
  • Multiple the denominators from each fraction by each other.

    \[\begin{array}{l}\dfrac{{20}}{{15}} = \dfrac{4}{3}\\\dfrac{6}{{12}} = \dfrac{1}{2}\\\dfrac{4}{3} \times \dfrac{1}{2} = \dfrac{2}{3}\end{array}\]

 

Multiplying fraction with same denominator

Multiplying a fraction with the same denominator is the same as multiplying other fractions. 

Let us explain with example:

Multiplying fraction  68 and 78

By the first method multiply the numerator from each fraction by each other 67=42, then we multiply the denominator from each fraction by each other, 88=64 this will give us the product as 4264  Since this can be reduce then the  2132  will be the answer.

The second method simplifies the given fraction. If needed. 

    \[\dfrac{6}{8} = \dfrac{3}{4}\]

,

    \[\dfrac{7}{8}\]

 

Now multiply the numerator to numerator and denominator to denominator.

    \[\dfrac{{3 \times 7}}{{8 \times 4}} = \dfrac{{21}}{{32}}\]

 

Multiplying fraction with different denominator

Multiplying fraction with different denominator is same as multiplying same denominator.

Example:

Multiplying 26 or 34  first we multiply the numerator that is,23=6, then we multiply the denominator that is, 64=24, this will give us the product 624 this can be reduce then the answer will be 14.

Multiplying with mixed numbers

Multiplying mixed fractions first changes the mixed fraction into improper fractions then multiplies the numerators from each fraction with each other, multiplying the denominators from each fraction with each other. Then simplify or reduce the fractions.

For example:

Multiplying fraction 712 or 315. By the rule first convert the mixed fraction into improper fraction 

    \[\begin{array}{l}7\dfrac{1}{2} = \dfrac{{2 \times 7 + 1}}{2} = \dfrac{{15}}{2}\\3\dfrac{1}{5} = \dfrac{{5 \times 3 + 1}}{5} = \dfrac{{16}}{5}\end{array}\]

 

Then multiplying the numerator to numerator and multiplying the denominator to denominator that is

    \[\dfrac{{15 \times 16}}{{5 \times 2}} = \dfrac{{240}}{{10}} = \dfrac{{24}}{1}\]

 

 Answer will be the 24

Multiply of the improper fractions

An improper fraction is where the numerator is bigger than or one and the same to the denominator. Improper fractions multiply the numerator of the fraction and multiply the denominators of the fraction then reduce the fractions.

Let us explain with the example:

Multiple the fraction 34  and  15

    \[\dfrac{3}{4} \times \dfrac{1}{5} = \dfrac{3}{{20}}\]

 

Answer will be the

    \[\dfrac{3}{{20}}\]

 

Multiply fraction on a number line

Multiplying on a number line is the application of multiplication operation on a number line.

Example:

Find the product of the fraction 35and12

Step1. Draw out a number line from 0 to 1 divide into two one and the same parts, the value of 2 is the denominator of second fraction 12

Step2. first part on number a line. 1 is the numerator of the second fraction.

Step3.  Subdivide each of the 2 parts into 5 equal parts. The value 5 is the denominator of the first fractions35. Note that there are now a total of 10 equal parts.


Step4. 3 of 5 subdivided parts. 3 is the numerator of the first fraction.

Step5. There are 10 equal parts.

Answer will be 310

Multiply fraction to whole number

Multiply fractions to whole numbers are defined as repeated addition where the fraction is added the same number of times as the whole number.

There are three steps to multiply a fraction to the whole number.

  1. Write the whole number with a denominator of 1.
  2. Product the numerators.
  3. Product the denominators.
  4. Simplify, if needed.

For example:

Multiplying

    \[5\]

and

    \[\dfrac{{17}}{{51}}\]

 

By the rule write whole number as a fraction

    \[\dfrac{5}{1}\]

 

 Multiply the numerators and denominators 

    \[\dfrac{{5 \times 17}}{{1 \times 51}} = \dfrac{{85}}{{51}} = \dfrac{5}{3}\]

 

Answer will be the

    \[\dfrac{5}{3}\]

 

Conclusion
In this article we discuss the fractions and multiplying fractions. In this, learn about multiplying fractions with the same numerator and same denominator. Also discuss multiple with mixed numbers, multiple of proper fraction, multiple fraction on number line, and multiple fraction to whole number.

Solved Example

Example 1: Find the multiplication of the fractions. 67,17

    \[\dfrac{6}{7} \times \dfrac{1}{7} = \dfrac{6}{{49}}\]

 

Example: 2. Solve this

    \[\dfrac{6}{7} \times \dfrac{2}{{30}}\]

 

    \[\dfrac{6}{7} \times \dfrac{2}{{30}} = \dfrac{{6 \times 2}}{{7 \times 30}} = \dfrac{2}{{35}}\]

 


Example: 3. Multiple the following fractions.

    \[\dfrac{{16}}{{30}}and\dfrac{5}{{35}}\]

 

    \[\begin{array}{l}\dfrac{{16}}{{30}} = \dfrac{8}{{15}}\\\dfrac{5}{{35}} = \dfrac{1}{7}\\\dfrac{{8 \times 1}}{{15 \times 7}} = \dfrac{8}{{105}}\end{array}\]

 

    \[{\raise0.7ex\hbox{$8$} \!\mathord{\left/ <!-- /wp:paragraph --> <!-- wp:paragraph -->  {\vphantom {8 {105}}}\right.\kern-\nulldelimiterspace} <!-- /wp:paragraph --> <!-- wp:paragraph --> \!\lower0.7ex\hbox{${105}$}}\]

will be the answer

Example 4: Multiple the following fractions.

    \[4\dfrac{3}{6}\]

    \[3\dfrac{3}{5}\]

 

    \[\dfrac{{6 \times 4 + 3}}{{5 \times 3 + 3}} = \dfrac{{27}}{{18}} = \dfrac{3}{2}\]


Example: 5 solve this 

    \[8 \times \dfrac{3}{{10}} \times 4\dfrac{1}{2}\]

 

    \[\dfrac{8}{1} \times \dfrac{3}{{10}} = \dfrac{{4 \times 3}}{5} = \dfrac{{12}}{5}\]

 

FAQs

1. What is a fraction?

Ans: A fraction is a whole that has been divided into equal parts.

2. In how many parts fractions are divided?
Ans: Fractions are divided into two parts.

3. What does the numerator show?

Ans: The number of equal parts the full is divided into.

4. What does the denominator tell?

Ans: The denominator is the parts we are counting.

5. How many kinds of fractions are there?

Ans: There are 3 kinds of fractions:

Proper fraction

    \[\dfrac{2}{4}\]

 

Improper fraction

    \[\dfrac{8}{3}\]

 

Mixed fraction

    \[4\dfrac{1}{2}\]

 

References

https://www.cuemath.com/numbers/multiplying-fractions/

https://byjus.com/maths/multiplying-fractions/

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