Table of Contents
- Fraction
- Operations on Fractions
- Equivalent Fractions
- Examples on Equivalent Fractions
- Simplification of a Fraction
- Examples on Simplification of a Fraction
- Addition / Subtraction of two Fractions
- Examples on Addition / subtraction of fractions with same denominator
- Examples on Addition / subtraction of fractions with different denominator
- Multiplication of two Fractions
- Examples on multiplying two Fractions
- Division of fractions
- Examples on division of Fractions
- Frequently Asked Questions
Fraction
Fractions are parts of a whole or a collection that are equal in size. A fraction is made up of two parts.
The numerator is the number at the top of the line. It specifies how many equal parts of the entire collection or whole are taken. The denominator is the number below the line. It displays the total number of equal parts that can be divided into the whole or the total number of equal parts in a collection.
For example, One mango is divided into 5 parts that are equal in size. As a result, each part is \(\frac{1}{5}\).
Operations on Fractions
The fraction \(\frac{p}{q}\) is composed of a numerator ‘p’ and a denominator ‘q’.
Equivalent Fractions
It’s important to remember that the same fraction can be represented in a variety of ways. For example the fractions \(\frac{1}{3}\) and \(\frac{3}{9}\) are equivalent. When the numerator and denominator of a fraction are multiplied or divided by the same number, the fraction remains equivalent.
Examples on Equivalent Fractions
1) \(\frac{3}{4} = \frac{3\ \times \ 5}{4\ \times \ 5} = \frac{15}{20}\)
2) \(\frac{6}{7} = \frac{6\ \times \ 4}{7\ \times \ 4} = \frac{24}{28}\)
3) \(\frac{2}{3} = \frac{2\ \times \ 5}{3\ \times \ 5} = \frac{10}{15}\)
4) \(\frac{4}{6} = \frac{4\ \div \ 2}{6\ \div \ 2}=\frac{2}{3}\)
5) \(\frac{24}{30}=\frac{24\ \div \ 6}{30\ \div \ 6} = \frac{4}{5}\)
Simplification of a Fraction
If the numerator and denominator have no common factor, the fraction is written in its simplified form. In other words, in the simplified form of a fraction, it is impossible to find a number that is a factor of both the numerator and the denominator. If we don’t recognise the largest common factor for the numerator and denominator at the same time, a simplification can be done in several steps.
Examples on Simplification of a Fraction
1) The fraction \(\frac{150}{100}\) is not written in its simplified form.
Since there are numbers that divide both 150 and 100. The greatest common factor of 150 and 100 is 50, where
\(\frac{150}{100}=\frac{150\div 50}{100 \div 50}=\frac{3}{2}\)
Since, we divided the numerator and denominator by the same number 50, the fraction \(\frac{3}{2}\) is equivalent to \(\frac{150}{100}\). In addition, \(\frac{3}{2}\) is the simplified form of \(\frac{150}{100}\) since no other common factor exists for 3 and 2.
2) The fraction \(\frac{108}{144}\) is not written in its simplified form.
Since there are numbers that divide both 108 and 144, the greatest common factor of 108 and 144 is 36, where;
\(\frac{108}{144}=\ \frac{108\ \div\ 36}{144\ \div\ 36}\ =\ \frac{3}{4}\)
Since, we divided the numerator and denominator by the same number 36, the fraction \(\frac{3}{4} \) is equivalent to \(\frac{108}{144}\). In addition, \(\frac{3}{4} \)is the simplified form of \(\frac{108}{144}\) since no other common factor exists for 3 and 4.
Addition / Subtraction of Two Fractions
- Find the denominator of the result by finding the least common multiple of the denominators.
- Using the least common multiple, write both original fractions as equivalent fractions.
- The numerators should be added (or subtracted).
- With the denominator, write the result.
The rule for adding and subtracting fractions is \(\frac{p}{q}\pm\ \frac{r}{q}\ =\ \frac{p\ \pm\ r}{q}\)
The ‘plus or minus’ symbol \(‘\pm’\) indicates that this rule applies to both additions and subtraction. It’s worth noting that the fraction addition and subtraction rule only works if both fractions have the same denominator. In most of the cases, this will not be applicable. The fractions must be rewritten into equivalent fractions with a common multiple. Before proceeding with addition or subtraction, it may be useful to simplify fractions whenever possible. Finding a common denominator will be easier as a result of this simplification.
Examples on Addition / Subtraction of Fractions with Same Denominator
1) Find \(\frac{1}{4}\ +\frac{3}{4}\ = \) ?
Solution:
\(\frac{1}{4}+\frac{3}{4}\ =\frac{1\ +\ 3}{4}\) (Adding the numerators and denominator remains the same)
\(=\ \frac{4}{4}\) (Adding 1 and 3)
\(=\ 1\) (Simplified)
2) Find \(\frac{6}{4}+\frac{5}{4}\ = \) ?
Solution:
\(\frac{6}{4}+\frac{5}{4}\ =\frac{6+5}{4}\) (Adding the numerators and denominator remains the same)
\(=\ \frac{11}{4}\) (Adding 6 and 5)
3) Find \(\frac{8}{5}\ +\frac{7}{5}\ = \)?
Solution:
\(\frac{8}{5}+\frac{7}{5}\ =\frac{8+7}{5}\) (Adding the numerators and denominator remains the same)
\(=\frac{15}{5}\) (Adding 8 and 7)
\(=3\) (Simplified)
4) Find \(\frac{7}{2}\ -\frac{9}{2}= \) ?
Solution:
\(\frac{7}{2}-\frac{9}{2}\ =\frac{7-9}{2}\) (Subtracting the numerators and denominator remains the same)
\(=\frac{-2}{2}\) (Subtracting 7 and 9)
\(=-1\) (Simplified)
5) Find \(\frac{6}{4}-\frac{5}{4}= \)?
Solution:
\(\frac{6}{4}-\frac{5}{4}\ =\frac{6-5}{4}\) (Subtracting the numerators and denominator remains the same)
\(=\frac{1}{4}\) (Subtracting 6 and 5)
6) Find \(\frac{2}{5} -\frac{3}{5}= \) ?
Solution:
\(\frac{2}{5}-\frac{3}{5}\ =\frac{2-3}{5}\) (Subtracting the numerators and denominator remains the same)
\(=\frac{-1}{5}\) (Subtracting 2 and 3)
Examples on Addition / Subtraction of Fractions with Different Denominator
1) Find \(\frac{2}{5}+\frac{3}{6}= ?\)
Solution:
The common denominator of 5 and 6 is \(5 \times 6 = 30\)
30 will therefore be the common denominator.
\(\frac{2}{5}+\frac{3}{6}\ =\frac{2\times6}{5\times6}+\frac{3\times 5}{5\times 6} \) (Finding common denominator)
\(=\frac{12}{30}+\frac{15}{30}\) (Writing common denominator of 5 and 6 as \(5 \times 6 = 30\))
\(=\frac{27}{30}\) (Adding 12 and 15)
2) Find \(\frac{1}{2}+\frac{7}{3}= \)?
Solution:
The common denominator of 2 and 3 is \(2 \times 3 = 6\).
6 will therefore be the common denominator.
\(\frac{1}{2}+\frac{7}{3}=\frac{1\times 3}{2\times 3}+\frac{7\times 2}{2\times 3}\) (Finding common denominator)’
\(=\frac{3}{6}\ +\frac{14}{6}\) (Writing common denominator of 2 and 3 as \(2 \times 3 = 6\))
\(=\frac{17}{6}\) (Adding 3 and 14)
3) Find \(\frac{2}{7}-\frac{3}{8}= \)?
Solution:
The common denominator of 7 and 8 is \(7 \times 8 = 56\).
56 will therefore be the common denominator.
\(\frac{2}{7}+\frac{3}{8}=\frac{2\times 8}{7\times 8}\ -\frac{3\times 7}{7 \times 8}\) (Finding common denominator)
\(=\frac{6}{56}\ -\frac{21}{56}\) (Writing common denominator of 7 and 8 as \(7 \times 8 = 56\))
\(=\frac{-15}{56}\) (Subtracting 6 and 21)
4) Find \(\frac{1}{3}-\frac{7}{4}= \) ?
Solution:
The common denominator of 3 and 4 is \(3 \times 4 = 12.\)
30 will therefore be the common denominator.
\(\frac{1}{3}-\frac{7}{4}=\frac{1\times 4}{3\times 4}\ -\frac{7\times 3}{3\times 4}\) (Finding common denominator)
\(=\frac{4}{12}-\frac{21}{12}\) (Writing common denominator of 3 and 4 as \(3\times 4 = 12\))
\(=\frac{-17}{12}\) (Subtracting 4 and 21)
Multiplication of Two Fractions
- Multiply the numerator by the numerator and the denominator by the denominator.
- Divide the numerator by the numerator.
The rule for multiplying two fractions is \(\frac{p}{q} \times \frac{r}{s} = \frac{p\times r}{q\times s} = \frac{p\bullet r}{q\bullet s}\)
It’s worth noting that, unlike sums, the multiplication rule has no restrictions on the denominator values. This implies that they don’t have to be common. Before multiplying, it’s a good idea to simplify fractions. In addition to simplifying each fraction separately, the denominator of one fraction may be simplified with the numerator of the other fraction if both have common factors.
Examples on Multiplying Two Fractions
1) Find \(\frac{1}{6}\times \frac{5}{7}\ = \)?
Solution: \(\frac{1}{6}\times \frac{5}{7}\ =\frac{1\times 5}{6\times 7} \) (Multiply the numerators and denominators)
\(=\frac{5}{42}\) (Simplified)
2) Find \(\frac{17}{6} \times \frac{8}{7} =\) ?
Solution:
\(\frac{17}{6} \times \frac{8}{7}=\frac{17 \times 8}{6\times 7}\) (Multiply the numerators and denominators)
\(=\frac{136}{42}\) (Simplified)
3) Find \(\frac{1}{8}\times \frac{9}{4}=\) ?
Solution: \(\frac{1}{8} \times \frac{9}{4}=\frac{1\times 9}{8\times 4}\) (Multiply the numerators and denominators)
\(=\frac{9}{32}\) (Simplified)
4) Find \(\frac{2}{3}\times \frac{5}{7} =\)?
Solution:
\(\frac{2}{3}\times \frac{5}{7}=\frac{2\times 5}{3\times 7}\) (Multiply the numerators and denominators)
\(=\frac{10}{21}\) (Simplified)
5) Find \(\frac{16}{3} \times \frac{51}{11} =\) ?
Solution:
\(\frac{16}{3} \times \frac{51}{11}=\frac{16 \times 51}{3\times 11}\) (Multiply the numerators and denominators)
\(=\frac{816}{33}\) (Multiples of 3)
\(=\frac{272}{11}\) (simplified)
Division of Fractions
The term “division” refers to the equitable distribution of a resource. When it comes to fractions, dividing them is almost the same as multiplying them. We multiply the first fraction by the reciprocal (inverse) of the second fraction to divide fractions.
Step 1: Invert the second fraction (the one you’d like to divide by) (this is now a reciprocal).
Step 2: Multiply the first fraction by the reciprocal of that fraction.
The division rule of two fractions allows us to transform a division into a multiplication is
\(\frac{p}{q}\div\frac{r}{s}=\frac{p\times s}{q\times r} =\frac{p.s}{q.r}\)
Examples on Division of Fractions
1) Divide \(\frac{1}{5} \div \frac{1}{6} = \)?
Solution:
\(\frac{1}{5} \div \frac{1}{6}\)
\(= \frac{1 \times 6}{5 \times 1}\)
\(= \frac{6}{5}\) (Multiply the first fraction by the reciprocal formed by turning the second fraction upside down.)
2) Divide \(\frac{2}{3} \div \frac{1}{6} =\) ?
Solution:
\(\frac{2}{3} \div \frac{1}{6} = \frac{2 \times 6}{3 \times 1}\)
\(= \frac{12}{3}\)
\(= 4\) (Multiply the first fraction by the reciprocal formed by turning the second fraction upside down.)
3) Divide \(\frac{21}{56} \div \frac{17}{36} =\) ?
Solution:
\(\frac{21}{56} \div \frac{17}{36} = \frac{21 \times\ 36}{56 \times 17}\)
\(= \frac{756}{952}\) (Multiply the first fraction by the reciprocal formed by turning the second fraction upside down.)
Fractions are used all over the place. Fractions are used in baking, construction, sewing, and even the stock market. Whether you realize it or not, fractions will always be a part of your life.
When adding fractions, if the denominators are not the same, you must find the least common multiple (LCM) to find the common denominator. Subtracting Fractions: If the denominators are different, you must find the least common multiple to find the common denominator.
A fraction’s numerator is the number at the top. A fraction’s denominator is the number at the bottom. To multiply fractions, start at the left and work your way to the right (numerator times numerator, denominator times denominator). Reduce your response to the simplest terms possible
Students must be able to add, subtract, multiply, and divide fractions with ease. The use of the four operations, including efficient written methods, applied to simple fractions; proper, improper, and mixed numbers, is supported by this resource list.