Fractions – Their Types, Comparison & Sum

fraction
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A fraction is a number representing parts of a whole.”

The Whole may be a single object or group of objects but the Parts have to be equal.

A fraction is written as parts above the whole  and horizontal line or slash in between.

\frac{x}{y} or x/y

e.g,  \frac{5}{12} (five-twelfths) i.e., 5 parts out of 12 equal parts.

Denominator

It is the number of equal parts into which the whole has been divided. (12 in 5⁄12).

Numerator

It is the number of equal parts which have been taken out. (5 in 5⁄12)

Proper fraction

In proper fraction numerator is always less than denominator.

e.g., 1⁄2, 3⁄4, 5⁄12.

Improper fraction

how can we share 5 apples among four person?

Divide each apple into four equal part, then each person will get a equal share. i.e., (1⁄4 + 1⁄4 + 1⁄4 + 1⁄4 + 1⁄4 = 5⁄4).

“The fractions, where the numerator is greater than the denominator are called improper fractions.” e.g.,3⁄2, 7⁄4, 19⁄12 .

Improper fraction can be also be expressed as mixed fraction. e.g., in above example, each share is made up of one whole and one quarter (1⁄4 + 1⁄4 + 1⁄4 + 1⁄4 + 1⁄4 = 4⁄4 + 1⁄4 = 1 + 1⁄4).

Mixed fraction

A mixed fraction has combination of whole and part.

17⁄4 = 16⁄4 + 1⁄4
= 4 + 1⁄4
= 4\frac{1}{4}

We can express an improper as a mixed fraction by dividing the numerator by denominator to obtain the quotient and the remainder. Then, the mixed fraction will be written as:  Quotient \frac{Remainder}{Divisor}

Proper fraction on the Number line

Fractions can be placed on a number line just like the whole numbers.

→ Proper fraction lie between 0 and 1.

→ Draw a line and mark a point 0 near the left end and 1 near right end.

→ Divide the line segment (gap) between these number into same number of equal parts as denominator.

→ Then the first part will be equal to 1⁄D, the second part will be equal to 2⁄D and so on.

Equivalent Fractions

Fractions which represent the same part of a whole are called equivalent fractions. e.g., 1⁄2, 2⁄4, 2⁄6, 4⁄8, 5⁄10, … are all equivalent fractions.

↪ Equivalent fractions of a fraction can be obtained by –

multiplying the numerator & the denominator of the fraction with same numbers.

↪ Cross product of any two equivalent fractions are always equal.

Simplest form of a Fraction

A fraction is said to be in simplest (or lowest) form when its denominator and numerator have no common factor except 1.

e.g., 1⁄2, 2⁄3, 3⁄5 etc.

↬ How to find equivalent fraction in the simplest form/ How to simplify the fraction?

Find HCF of the numerator & the denominator, and then divide both of them with the HCF.

e.g., Let’s find the simplest form of 45⁄75

45 = 3×3×5, 75 = 3×5×5

HCF = 3×5 = 15  ⇥| How to Find HCF of Natural Numbers |⇤

∴ Simplest form of \frac{45}{75}

= \frac {45\div15}{75\div15} = \frac {3}{5}

Like & Unlike Fractions

Like Fractions –Fractions with same denominator are called like fractions.

e.g., 1⁄5, 2⁄5, 3⁄5 etc.

Unlike Fractions –Fractions with different denominators are called unlike fractions.

e.g., 1⁄3, 2⁄5, 3⁄7 etc.

Comparing Fractions

Comparing Like Fractions:

↬ The denominators are same.

Fraction with greater numerator will be greater.

1⁄5 < 2⁄5 < 3⁄5

Comparing Unlike Fractions

↬ When the numerators are same:

Fraction with smaller denominator will be greater.

2⁄5 < 2⁄4 < 2⁄3

↬ When both the numerators and denominators are different.

We find the LCM of the denominators and then find equivalent fraction of the each given fractions with denominator equal to the LCM  and then we can compare them as like fractions.⇥| How to Find LCM of Natural Numbers |⇤

Compare : 3⁄4 , 1⁄5 & 5⁄6

→ LCM of 5, 4 and 6 is 60

→ 3⁄4 = 3×15⁄4×15 = 45⁄60,

1⁄5 = 1×12⁄5×12 = 12⁄60,

5⁄6 = 5×10⁄6×10 = 50⁄60,

→ Now we can compare the numerators, 12<45<50

→ This implies that, 1⁄5 < 3⁄4 < 5⁄6

Addition and Subtraction of Fractions

Addition and Subtraction of Like fractions

→ Add/Subtract the numerators.

→ Retain the common denominator.

→ Write the resultant fraction as : Result of Step 1 / Result of Step 2

→ Add 3⁄5 & 4⁄5 :3⁄5 + 4⁄5 = 3+4⁄5 = 7⁄5

→ Subtract 4⁄5 from 3⁄5 :3⁄5 – 4⁄5 = 3-4⁄5 = -1⁄5

Addition and Subtraction of Unlike fractions

→ Find the LCM of denominators.

→ Write equivalent fractions with common denominator (LCM).

→ They are like fractions now, so proceed as with like fractions.

→ Add 3⁄5 & 4⁄7 :

 LCM of 5 & 7 = 35

    3⁄5  = (3×7)⁄(5×7) = 21⁄35

   4⁄7  = (4×5)⁄(7×5) = 20⁄35

  ∴3⁄5 + 4⁄7 = 21⁄35 + 20⁄35 = (21+20)⁄35  = 41⁄35

→ Subtract   4⁄7 from 3⁄5 :

 LCM of 5 & 7 = 35

    3⁄5  = 3×7⁄5×7 = 21⁄35

 4⁄7  = 4×5⁄7×5 = 20⁄35

  ∴3⁄5 – 4⁄7 = 21⁄35 – 20⁄35 = (21-20)/35 = 1⁄35

Addition and Subtraction of Mixed fractions

→ Write them as a whole part plus proper fraction or entirely as improper fraction.

→ Add the whole parts and fractional parts separately and then combine them to get final result.

→ Add 4\frac{3}{5} & 3\frac{4}{7}

4\frac{3}{5} = 4+ 3⁄5, 3\frac{4}{7} = 3+ 4⁄7

Adding the whole parts = 3+4= 7

Adding the fractional parts  3⁄5 & 4⁄7

    LCM of 5 & 7 = 35

    3⁄5  = 3×7⁄5×7 = 21⁄35

    4⁄7  = 4×5⁄7×5 = 20⁄35

⇒ 3⁄5 + 4⁄7 = 21⁄35 + 20⁄35 = (21+20)⁄35  = 41⁄35 = 1\frac{6}{35}

∴ 4\frac{3}{5} +  3\frac{4}{7} = 7 + 1 + \frac {6}{35} = 8\frac{6}{35}

Fractions – Problems with Solutions

⏪ Multiplication & Division of IntegersMultiplication & Division of Fractions ⏩

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