Fractions, Decimals and Percentages

In this article, we look at fractions , decimals and percentages.

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Fractions
Equivalent fractions
Proper/Improper fractions
Arithmetic of Fractions

Surds

In this article, we look at surds, simplification of surds, arithmetic of surds, and many more. Reducing Surds to basic form Perfect Squares simplifying Surds RECOMMENDED ARTICLES IntroductionA...

A fraction is a mathematical expression that represents the division of one quantity by another quantity. A fraction is part of a whole or ratio between two quantities.

It consists of a numerator (the top number) and a denominator (the bottom number), separated by a horizontal line. For example, in the fraction $\frac{3}{4}$ , 3 is the numerator, and 4 is the denominator.

A valuable way to think about fraction is in terms of a unit. Assume we have a pizza. We take the whole pizza as a unit. We divide the pizza into equal pieces.

One pizza is divided into 2 equal havles: $\frac{1}{2}+\frac{1}{2}=1$

One pizza is divided into 3 equal parts: $\frac{1}{3}+\frac{1}{3}+\frac{1}{3}=1$ . This implies that $\frac{1}{3}+\frac{1}{3}=\frac{2}{3}$

One pizza is divided into 4 equal parts: $\frac{1}{4}+\frac{1}{4}+\frac{1}{4}+\frac{1}{4}=1$ . This implies that $\frac{1}{4}+\frac{1}{4}+\frac{1}{4}=\frac{3}{4}$

Equivalent Fractions

Equivalent fractions are fractions with different numerators and denominators but have same value. For example, $\frac{1}{2}, \frac{2}{4}, \frac{3}{6}, \frac{4}{8}, \frac{5}{10}$ are equivalent fractions.

Fractions with common factors in the numerator and denominator can be simplified further by cancelling out the common factors. For example,

$\frac{2}{4}=\frac{ \cancel{2} \times 1}{ \cancel{2} \times 2}=\frac{1}{2}$

$\frac{12}{20}=\frac{3 \times \cancel{4}}{5 \times \cancel{4}}=\frac{3}{5}$

If there’s no common factor between the numerator and denominator, the fraction is said to be in its $\textbf{lowest term}$ . Such fractions are called irreducible fractions. Examples include $\frac{3}{5}, \frac{13}{4}, \frac{7}{9},$ .

Proper Fractions

A fraction which has the numerator less than the denominator is called a proper fraction. Examples include $\frac{1}{2}, \frac{3}{5}, \frac{7}{9}$

Improper Fractions

A fraction which has the numerator equal or greater than the denominator is called an improper fraction. Examples include $\frac{5}{2}, \frac{9}{5}, \frac{4}{4}, \frac{24}{9},$ .

Negative Fractions

Fractions can be negative. For example, we can have $\frac{-2}{5}$ . Note that $\frac{-2}{5}=\frac{2}{-5}$ .

Arithmetic Of Fractions

Addition and Subtraction Of Fractions

If we want to add or subtract two fractions, we first replace each fraction with an equivalent one such that the two fractions have common denominator. The common denominator is called the Least Common Multiple (L.C.M) of the denominators.

Example 1: Add $\frac{1}{3}$ and $\frac{1}{5}$

Solution

$\frac{1}{3}$ and $\frac{1}{5}$ have different denominators so we find equivalent fractions for each such that they have the same denominator.

$\frac{1}{3}=\frac{1 \times 5}{3 \times 5}=\frac{5}{15}$

$\frac{1}{5}=\frac{1 \times 3}{5 \times 3}=\frac{3}{15}$

$\frac{1}{3}+\frac{1}{5}=\frac{5}{15}+\frac{3}{15}=\frac{8}{15}$

Example 2: Evaluate $\frac{1}{3} + \frac{1}{2}$

$\frac{1}{3}$ and $\frac{1}{2}$ have different denominators so we find equivalent fractions for each such that they have the same denominator.

$\frac{1}{3}=\frac{1 \times 2}{3 \times 2}=\frac{2}{6}$

$\frac{1}{2}=\frac{1 \times 3}{2 \times 3}=\frac{3}{6}$

$\frac{1}{3}+\frac{1}{2}=\frac{2}{6}+\frac{3}{6}=\frac{5}{6}$

The general rule of addition and subtraction are:

$\frac{a}{b}+\frac{c}{d}=\frac{ad + bc}{bd}$

$\frac{a}{b}-\frac{c}{d}=\frac{ad - bc}{bd}$

Multiplication

Fraction times whole number : If you multiply a fraction by a whole number, the whole number multiplies the numerator of the fraction. Example,

$2 \times \frac{4}{9}$ is $\frac{2}{1} \times \frac{4}{9}=\frac{8}{9}$

In the above, if we use the pizza scenario, $\frac{4}{9}$ is a pizza divided into 9 equal parts and 4 taken out of the 9. Twice the selection (4) gives 8 out of the 9.

Given $n$ as a whole number, the general rule of multiplying a whole number by a fraction is

$n \times \frac{a}{b} = \frac{na}{b}$

Fraction times fraction: If you multiply two fractions, you multiply the numerators separately, then the denominators separately. You then simplify the resultant fraction to its lowest term if there is a common factor in the numerator and denominator.

Example,

$\frac{3}{2} \times \frac{5}{4}=\frac{15}{8}$

Given any two fractions $\frac{a}{b}$ and $\frac{c}{d}$

$\frac{a}{b} \times \frac{c}{d}=\frac{ac}{bd}$

Division

Fraction divided by whole number : If we divide a fraction by a whole number, we divide the numerator by the whole number and leave the denominator.

Example 1 : $\frac{14}{9} \div 2 =\frac{7}{9}$

In the above, we divide the 14 by 2 to get 7 and leave the denominator, 9, as it is.

Example 2 : $\frac{15}{7} \div 3=\frac{5}{7}$

In the above, we divide the 15 by 3 to get 5 and leave the denominator, 7, as it is.

Example 3 : $\frac{8}{3} \div 8=\frac{1}{3}$

In the above, we divide 8 by 8 to get 1 and leave the denominator, 3, as it is.

When the numerator is NOT divisible by the whole number

If the numerator of the fraction is not divisible by the whole number, an equivalent fraction of the main fraction is used.

Example 1: Simplify $\frac{5}{3} \div 2$

5 is not divisible by 2 so we use an equivalent fraction of $\frac{5}{3}$ such that the numerator can be divided by 2. If we multiply both the numerator, 5, and denominator, 3, by 2, we get $\frac{10}{6}$ which is an equivalent fraction of $\frac{5}{3}$ . The 10 is divisible by 2. That is,

$\frac{10}{6} \div 2 = \frac{5}{6}$

NOTE: If you study the result $\frac{5}{6}$ carefully, you realize that after dividing the numerator of the equivalent fraction, $\frac{10}{6}$ , by the whole number, what you get is the original fraction with its denominator multiplied by the whole number. That is, $\frac{5}{3} \div 2= \frac{5}{3 \times 2} = \frac{5}{6}$

Example 2: Simplify $\frac{7}{9} \div 5$

7 is not divisible by 5 so we use an equivalent fraction of $\frac{7}{9}$ such that the numerator can be divided by 5. If we multiply both the numerator, 7, and denominator, 9, by 5, we get $\frac{35}{45}$ which is an equivalent fraction of $\frac{7}{9}$ . The 35 is divisible by 5. That is,

$\frac{35}{45} \div 5 = \frac{7}{45}$

NOTE: If you study the result $\frac{7}{45}$ carefully, you realize that after dividing the numerator of the equivalent fraction, $\frac{35}{45}$ , by the whole number, 5, what you get is the original fraction, $\frac{7}{9}$ , with its denominator multiplied by the whole number. That is, $\frac{7}{9} \div 5= \frac{7}{9 \times 5} = \frac{7}{45}$

Example 3: Simplify $\frac{9}{11} \div 7$

9 is not divisible by 7 so we use an equivalent fraction of $\frac{9}{11}$ such that the numerator can be divided by 7. If we multiply both the numerator, 9, and denominator, 11, by 7, we get $\frac{63}{77}$ which is an equivalent fraction of $\frac{9}{11}$ . The 63 is divisible by 7. That is,

$\frac{63}{77} \div 7 = \frac{9}{77}$

NOTE: If you study the result $\frac{9}{77}$ carefully, you realize that after dividing the numerator of the equivalent fraction, $\frac{63}{77}$ , by the whole number, 7, what you get is the original fraction, $\frac{9}{11}$ , with its denominator multiplied by the whole number, 7. That is, $\frac{9}{11} \div 7= \frac{9}{11 \times 7} = \frac{9}{77}$

From the above examples, we can generally state that given a fraction $\frac{a}{b}$ divided by a whole number $n$ , that is, $\frac{a}{b} \times n$ , we multiply the denominator by the whole number to get $\frac{a}{bn}$ . In other words, we multiply the fraction by the reciprocal of the whole number.

$\Rightarrow \frac{a}{b} \times n=\frac{a}{bn} = \frac{a}{b} \times \frac{1}{n}$

For example, $\frac{4}{7} \div 3 = \frac{4}{7} \times \frac{1}{3}=\frac{4}{21}$

In other words, dividing a fraction $\frac{a}{b}$ by $n$ is the same as multiplying $\frac{a}{b}$ by $\frac{1}{n}$ .

Dividing fraction by fraction

If we divide one fraction by another fraction, the process is similar to dividing a fraction by a whole number. We break the process into two steps.

Example 1: Simplify $\frac{3}{5} \div \frac{4}{7}$

First, $\frac{4}{7}$ is $4 \times \frac{1}{7}$ . That is, $\frac{4}{7}$ comprises the whole number $4$ and the fraction, $\frac{1}{7}$ .

Therefore, $\frac{3}{5} \div \frac{4}{7}$ can be broken down into two as $\frac{3}{5} \div 4$ , followed by the result divided by $\frac{1}{7}$

Now, let’s look at the $\frac{3}{5} \div 4$ first.

For $\frac{3}{5} \div 4$ , 3 is not divisible by 4 so we look for an equivalent fraction by multiplying the numerator and denominator by 4. This gives $\frac{12}{20}$ .

Thus, $\frac{12}{20} \div 4= \frac{3}{20}$ .

Second, we now have $\frac{3}{20} \div \frac{1}{7}$ . However, recall that dividing by a number is same as multiplying by the reciprocal of that number. Hence, $\frac{3}{20} \div \frac{1}{7}=\frac{3}{20} \times\frac{7}{1}=\frac{21}{20}$

In summary, $\frac{3}{5} \div \frac{4}{7}=\frac{3}{5} \times\frac{7}{4}=\frac{21}{20}$

The general rule is that dividing a fraction $\frac{a}{b}$ by another fraction $\frac{c}{d}$ is same as multiplying the fraction $\frac{a}{b}$ by $\frac{d}{c}$ , the reciprocal of the dividing fraction.

$\Rightarrow \frac{a}{b} \div \frac{c}{d}=\frac{a}{b} \times \frac{d}{c}$

Comparing Fractions – arranging them in ascending and descending order of magnitude

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Fractions, Decimals and Percentages

Douglas Tawiah Dwumor

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Surds

Equivalent Fractions

Proper Fractions

Improper Fractions

Negative Fractions

Arithmetic Of Fractions

Addition and Subtraction Of Fractions

Multiplication

Division

When the numerator is NOT divisible by the whole number

Dividing fraction by fraction

Comparing Fractions – arranging them in ascending and descending order of magnitude

Join Our Telegram Group

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Surds

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