Mixed Fractions

Introduction

• In mathematics, fractions are used to determine part of a whole number.

• The word ‘fraction’ is derived from the Latin word ‘fraction’ which means ‘to break’. In Ancient Rome, fractions were used only to describe the part to the whole.

• But in India fractions were represented by one number above another but without a line.

• Then Arabs were added a line between two numbers. This line is used to separate the numerator & denominator.

• Fractions can be represented in the form of decimals & percentages like 0.012 & 12 % represent the fraction $\mathrm{\frac{12}{100}}$. So let's study fractions & their types along with examples.

Fraction

• A fraction is a number representing part of a whole. The whole may be a single object or a group of objects.

• Fractions are represented in the form of $\mathrm{\frac{a}{b}}$ . The number which is above the line is known as a numerator. Whereas, the number which is below the line is called the denominator.

• / is used to separate numerator & denominator. Sometimes / a symbol is used to separate the numerator & denominator.

• The denominator represents the whole section or part that has been divided into parts & numerator represents how many sections or parts have been selected for representing fractions.

• Let's understand the concept of fractions by example. If we divided a cake into four equal slices. Then each slice will be represented in the form of a fraction is $\mathrm{\frac{1}{4}}$ . From this we can say that we are referring to one part out of 4 parts.

Types of fraction

According to numerator & denominator criteria types of fractions are given as follows.

Proper fractions

If the numerator of a fraction is less than its denominator then this type of fraction is known as proper fractions.

For example, $\mathrm{\frac{2}{5},\frac{3}{8},\frac{8}{15}}$

Improper fractions

If the numerator of a fraction is more or equal to its denominator, then this type of fraction is known as improper function.

For example, $\mathrm{\frac{4}{3},\frac{8}{5},\frac{7}{2}}$

Mixed fraction

Mixed fraction is the combination of integer part & fraction part.

For example,$\mathrm{2 \frac{1}{3} , 5 \frac{1}{3}}$

Equivalent fraction

Equivalent fractions are those fractions whose numerator & denominator are different, but they shows the same value.

For example, $\mathrm{\frac{1}{2} \& \frac{3}{6}}$ have different variables but they represent the same value that is $\mathrm{\frac{1}{2}}$.

Equivalent fractions can be calculated by multiplying & numerator & denominators of the given fraction by the same number.

For example, Equivalent factors of $\mathrm{\frac{1}{3}}$ are

$$\mathrm{\frac{1\times 2}{3\times 2}=\frac{2}{6} , \frac{1\times 3}{3\times 3}=\frac{3}{9}, \frac{1\times 4}{3\times 4}=\frac{4}{12}}$$

Unit fraction

Fraction having numerator 1 are known as unit fraction.

For example, $\mathrm{\frac{1}{4},\frac{1}{7},\frac{1}{5}}$

Like & unlike fractions

• Fractions having the same denominators are known as like fractions. For example, $\mathrm{\frac{3}{15},\frac{7}{15},\frac{13}{15}}$

• Fractions having different denominators are known as, unlike fractions. For example, $\mathrm{\frac{7}{12},\frac{5}{13},\frac{8}{15}}$

Mixed Fractions

Mixed fraction is a combination of the integer part & fraction part.

For example, $\mathrm{8\frac{1}{2}}$ here 8 is an integer & $\mathrm{\frac{1}{2}}$ is a fraction part.

Also $\mathrm{4 \frac{1}{3} ,5\frac{1}{4} ,7\frac{1}{2} ,9\frac{1}{5}}$ are some examples of mixed fractions.

This fraction can be converted into improper fractions.

Conversion of Mixed Fractions

Mixed fractions can be converted into improper fractions by using the following method −

Step 1 − Multiply the denominator of the mixed fraction along with the whole number part.

Step 2 − Add numerator to the product obtained from step 1

Step 3 − Write improper fraction obtain from step 2 in improper fraction form.

Mathematically above steps can be represented as,

$$\mathrm{\frac{(Whole \times Denominator)+Numerator}{Denominator}}$$

For example, Express $\mathrm{7\frac{1}{9}}$ as improper fraction.

$\mathrm{ 7\frac{1}{9}=\frac{(7×9)+1}{9}=\frac{64}{9}}$

Conversion of improper fractions to mixed fractions

In improper fraction numerator is greater than or equal to the numerator, therefore we found difficulties while solving improper fraction. These fractions can be easily simplified by converting them into an improper fraction.

How to convert improper fraction to mixed fraction

Step 1 − Divide numerator with denominator.

Step 2 − Calculate the remainder.

Step 3 − Write the numbers in the following way

$$\mathrm{ Quotient\times \frac{Remainder}{Divisor}}$$

For example, express $\mathrm{\frac{11}{3}}$ as mixed fraction

$$\mathrm{\:\:\:\:\:\:\:3\\\:3)\overline{11}\:\\\:\:\:\:\underline{-9}\\\\\:\:\:\:\:\:2}$$

It can be written as 3 whole $\mathrm{\frac{2}{3}}$ more i,e $\mathrm{3\frac{2}{3}}$.

Algebra of mixed fraction

We can perform basic arithmetic operation like addition, subtraction, multiplication & division on mixed fraction

Addition of mixed fractions

Addition of a mixed fraction can be done by using following steps −

Step 1 − Express the given mixed fraction into an improper fraction.

Step 2 − Look over denominators. Check whether they are the same or not.

Step 3 − If yes, then add the numerators of fractions & then write down the result.

Step 4 − If denominators are not the same then by taking LCM make them the same.

Step 5 − Now add the numerators & obtained results.

For example, Add $\mathrm{2\frac{1}{5}\: \&\: 3\frac{2}{5}}$

Solution: For addition, we have to convert mixed fraction to improper fraction

$$\mathrm{\frac{(2×5)+1}{3}=\frac{11}{3} \: \& \: \frac{(3×5)+2}{3}=\frac{17}{3}}$$

By adding above fraction we get,

$$\mathrm{\frac{17}{3}+\frac{11}{3}=\frac{28}{3}}$$

Subtraction of mixed fractions

Steps to subtract one mixed fraction from another are same as above. Instead of adding we have to perform subtraction.

e.g. Subtract $\mathrm{2\frac{1}{3}}$ from $\mathrm{3\frac{2}{3}}$.

Solution: First, we have to convert a mixed fraction into an improper fraction.

i,e $\mathrm{\frac{(2×3)+1}{3}=\frac{7}{3} \: \& \: \frac{(3×3)+2}{3}=\frac{11}{3}}$

On subtracting we get,

$$\mathrm{\frac{11}{3} - \frac{7}{3}=\frac{4}{3}}$$

Converting $\mathrm{\frac{4}{3}}$ into a mixed fraction we get $\mathrm{1\frac{1}{3}}$.

Multiplication of mixed fraction

Multiplication of two mixed fractions can be done by using following steps:

Step 1 − Express the given fraction into improper function.

Step 2 − Multiply numerator with numerator & denominators & write down the result.

Step 3 − This result can be simplified to its lower form as an improper or converted into a mixed portion.

For example, Multiply $\mathrm{2\frac{2}{5}\: \& \: 3\frac{1}{5}}$.

Solution: First we have to convert a mixed fraction into an improper fraction.

i,e $\mathrm{\frac{(2×5)+2}{5}=\frac{12}{5}\: \& \: \frac{(3×5)+1}{5}=\frac{16}{5}}$

Multiplying $\mathrm{\frac{12}{5}\: \& \: \frac{16}{5}}$

$$\mathrm{\frac{12}{5}\times \frac{16}{5}=\frac{192}{25}}$$

Converting $\mathrm{\frac{192}{25}}$ into a mixed fraction we get $\mathrm{7\frac{17}{25}}$.

Division of mixed fraction

Division of two mixed fractions can be done by using following steps:

Step 1 − Convert the given mixed fractions into improper fractions.

Step 2 − Multiply the first fraction with inverse multiplicative of the second fraction

Step 3 − Obtained result can be simplified to its lowest form to mixed fraction.

For example, Divide $\mathrm{1\frac{1}{5}\: by\: 3\frac{4}{5}}$.

Solution: First, we have to convert the mixed fraction into an improper fraction.

$$\mathrm{\frac{(1×5)+1}{5}=\frac{6}{5} \: \& \: \frac{(3×5)+4}{5}=\frac{19}{5}}$$

$$\mathrm{\frac{6}{5}\times \frac{5}{19}=\frac{6}{19}}$$

Solved examples

1) Add $\mathrm{2\frac{4}{5}\: \& \: 3\frac{5}{6}}$.

Solution :$\mathrm{2\frac{4}{5}+3\frac{5}{6}=2+\frac{4}{5}+3+\frac{5}{6}}$

Now, $\mathrm{\frac{4}{5}+\frac{5}{6}=\frac{4\times 6}{5\times 6}+\frac{5\times 5}{6\times 5}}$(Since LCM of 5 & 6 is 30)

$$\mathrm{=\frac{24}{30}+\frac{25}{30}=\frac{49}{30}=\frac{30+19}{30}=1+\frac{19}{30}}$$

Therefore, $\mathrm{5+\frac{4}{5}+\frac{5}{6}=5+1+\frac{19}{30}=6\frac{19}{30}}$

Hence, $\mathrm{2\frac{4}{5}+3\frac{5}{6}=6\frac{19}{30}}$

2) Subtract $\mathrm{4\frac{2}{5}\: from\: 2\frac{1}{5}}$.

Solution : First we have to convert mixed fraction into improper fraction.

$$\mathrm{i,e\:\: \frac{(4×5)+2}{5}=\frac{22}{5}\: \& \: \frac{(2×5)+1}{5}=\frac{11}{5}}$$

Subtracting $\mathrm{\frac{11}{5}}$ from $\mathrm{\frac{22}{5}}$ we get,

i,e $\mathrm{\frac{22}{5}-\frac{11}{5}=\frac{11}{5}}$

Converting $\mathrm{\frac{11}{5}}$ into mixed fraction i.e $\mathrm{2\frac{1}{5}}$ .

Conclusion

This tutorial covers the topic fraction , types of fraction & mixed fraction along with solved examples. a fraction is a number representing part of a whole. The whole may be a single object or a group of objects. Different types of fractions are proper fraction, improper fraction, mixed fraction, unit fraction, like & unlike fraction & equivalent fraction. Multiple fractions are the combination of the whole number part & fraction part. Mixed fractions can be converted into an improper fraction. In daily life concept of the fraction is used to determine body mass index (BMI), dividing pizza into equal pieces. Also, fractions are used to determine the ingredients for recipes. Also, it is used in photography & medicine. This article will surely help you to understand the topic fraction & mixed fraction.

FAQs

1.Where are fractions used in mathematics?

Fractions are used in −

• Determine the part of a whole number or object.

• Calculate decimal & percentage.

• Ratio & proportion

• Probability

• Algebraic equations

2.What are the parts of fractions?

Fraction has two parts, numerator & denominator. The numerator indicates the number placed above the fractional bar & the denominator is the number placed below the fractional bar. For example,$\mathrm{\frac{6}{7}}$ here numerator value is 6 & denominator is 7.

3.Can we represent 0.75 as a fraction?

Yes. we can represent 0.75 as a fraction $\mathrm{\frac{3}{4}}$. Conversion of decimal to a fraction is given as follows:$\mathrm{0.75=\frac{75}{100}=\frac{3}{4}}$.

4.Explain comparing of fractions.

Comparing fractions is finding the larger & smaller fraction between two or more fractions.

For example, Consider two fractions $\mathrm{\frac{3}{16}\: \& \: \frac{7}{19}}$. By observing these fractions we can say that $\mathrm{\frac{3}{16}}$ is greater than $\mathrm{\frac{7}{19}}$.

5.Can zero be a denominator to any fraction?

No. A fraction having zero at the denominator place is invalid or undefined.