Fractions

In the fraction $\frac{3}{5}$, 3 being the numerator is less than the denominator 5.

Further proper fractions are$\frac{1}{5},\frac{4}{7},\frac{6}{11}$ etc.

In the fraction $\frac{7}{6}$, 7 being the numerator is greater than the denominator 6.

Further improper fractions are $\frac{8}{3},\frac{11}{2},\frac{5}{3}$ etc.

The mixed fraction $1\frac{2}{3}$ consists of

1 as the whole number,

2 as the numerator of the proper fraction in the mixed fraction, and

3 as the denominator of the proper fraction in the mixed fraction.

Other examples include, $2\frac{1}{5},4\frac{2}{7},5\frac{3}{8}$ etc.

Convert $\frac{7}{3}$ to a mixed fraction.

Solution

$\frac{7}{3}=7÷3$

The quotient from the division of 7 by 3 is 2, and the remainder is 1.

Thus, 2 is your whole number that stays on the left-hand side and 1 is the remainder that becomes your new numerator and 3 is your denominator that remains unchanged.

Thus,

$\frac{7}{3}=2\frac{1}{3}$

Convert $2\frac{1}{3}$ to an improper fraction.

Solution

First, we find the product between the denominator which is 3, and the whole number which is 2, thus,

$3\times 2=6$.

Next, we find the sum between the numerator which is 1 and the product found in the first step which is 6, thus

$1+6=7$

Now your sum is 7, the new numerator is thus 7, and the denominator being 3 remains the same. Therefore,

$2\frac{1}{3}=\frac{7}{3}$.

Having 3 parts of cake from the total of 6 parts is the same as having half the cake.

The fractions $\frac{3}{5}$ and $\frac{6}{10}$ are equivalent but differ by a multiplying factor, 2.

In fact,

$\frac{3}{5}=\frac{3\times 2}{5\times 2}=\frac{6}{10}$

Similarly, we have

$\frac{6}{10}=\frac{6÷2}{10÷2}=\frac{3}{5}$

Therefore, the best way of knowing if two fractions are equivalent, but differ by a multiplying factor, is by simplifying these fractions to the simplest indivisible forms.

$2\frac{1}{5}$ is the same as $\frac{11}{5}$, the only difference is that $2\frac{1}{5}$ is a mixed fraction while $\frac{11}{5}$ is an improper fraction.

Calculate the following

$\frac{4}{5}+\frac{1}{3}$

Solution

Step 1.

We see that there is no whole number.

Step 2.

The LCD between 5 and 3 is 15, thus 15 is the new general denominator.

Step 3.

We divide the new denominator by the former denominator and multiply the result by the respective numerator. Therefore we have

$\frac{4}{5}+\frac{1}{3}=\frac{15\left(\frac{4}{5}\right)+15\left(\frac{1}{3}\right)}{15}=\frac{{}^3\cancel{15}\left(\frac{4}{\cancel{5}}\right)+\cancel{{}^{5}15}\left(\frac{1}{\cancel{3}}\right)}{15}$

Now, we compute our results to get,

$\frac{3\left(4\right)+5\left(1\right)}{15}=\frac{12+5}{15}=\frac{17}{15}=1\frac{2}{15}$

Calculate

$4\frac{1}{2}-1\frac{3}{4}$

Solution

Step 1.

We solve the whole numbers first,

$4-1=3$

We keep our result on the left-hand side.

Step 2.

The LCD between 2 and 4 is 4. Thus 4 is the new general denominator.

Step 3.

We divide the new denominator by the former denominator and multiply our result by the respective numerator. Therefore we have,

$4\frac{1}{2}-1\frac{3}{4}=3\frac{4\times \left(\frac{1}{2}\right)-4\times \left(\frac{3}{4}\right)}{4}=3\frac{{}^2\cancel{4}\times \left(\frac{1}{\cancel{2}}\right)-\cancel{4}\times \left(\frac{3}{\cancel{4}}\right)}{4}$

Now, we compute the result to get,

$3\frac{2\left(1\right)-1\left(3\right)}{4}=3\frac{2-3}{4}$

Note that 2 is less than 3, so take 1 from the whole number 3, so that your whole number is now 2. The 1 you have taken when put into the expression becomes equal to your denominator value which is 4. Add it to what is already there. Therefore, we have

$3\frac{2-3}{4}=2\frac{\mathbf{4}\mathbf{+}2-3}{4}=2\frac{6-3}{4}=2\frac{3}{4}$

Calculate

$\frac{2}{3}\times \frac{5}{7}$

Solution

Multiply both numerators and denominators together to get,

Calculate

$\frac{3}{4}\times \frac{8}{9}$

Solution

This can be solved using two methods.

Method 1.

Just multiply directly the numerators and denominators.

$\frac{3}{4}\times \frac{8}{9}=\frac{3\times 8}{4\times 9}=\frac{24}{36}$

The next step to take is to divide both the numerator and denominator by their greatest common divisor (GCD). The GCD between 24 and 36 is 12. Hence,

$\frac{3}{4}\times \frac{8}{9}=\frac{24}{36}=\frac{24÷12}{36÷12}=\frac{2}{3}$

Method 2.

You can directly simplify the fractions while multiplying them as seen below.

$\frac{{}^1\cancel{3}}{{}^1\cancel{4}}\times \frac{{}^2\cancel{8}}{{}^3\cancel{9}}=\frac{1\times 2}{1\times 3}=\frac{2}{3}$

Calculate

$\frac{2}{5}÷\frac{3}{7}$

Solution

Step 1.

Inverse the fraction at the right of the division sign and change the division sign to multiplication sign to get,

$\frac{2}{5}÷\frac{3}{7}=\frac{2}{5}\times \frac{7}{3}$

Step 2.

Multiply the numerators and denominators.

$\frac{2}{5}\times \frac{7}{3}=\frac{14}{15}$

Calculate,

$\frac{5}{14}÷\frac{100}{28}$

Solution

Step 1.

Inverse the fraction at the right of the division sign and change the division sign to multiplication sign, to get,

$\frac{5}{14}÷\frac{100}{28}=\frac{5}{14}\times \frac{28}{100}$

Step 2.

We now simplify to get,

$\frac{5}{14}\times \frac{28}{100}=\frac{{}^1\cancel{5}}{{}^1\cancel{14}}\times \frac{{}^2\cancel{28}}{{}^{20}\cancel{100}}=\frac{1\times 2}{1\times 20}=\frac{1\times \cancel{2}}{1\times \cancel{{}^{10}20}}=\frac{1}{10}$

What is the sum of one-third of a pie and half of a pie?

SolutionOne-third of a pie is $\frac{1}{3}$ and half of a pie is $\frac{1}{2}$.

The sum means the addition of both fractions. Thus their sum is given by

$\frac{1}{3}+\frac{1}{2}$

Their LCD is 6, thus, we have

$\frac{1}{3}+\frac{1}{2}=\frac{6\left(\frac{1}{3}\right)+6\left(\frac{1}{2}\right)}{6}=\frac{{}^2\cancel{6}\left(\frac{1}{\cancel{3}}\right)+\cancel{{}^{3}6}\left(\frac{1}{\cancel{2}}\right)}{6}=\frac{2\left(1\right)+3\left(1\right)}{6}=\frac{2+3}{6}=\frac{5}{6}$

If one-fifth of a man's income is used on accommodation and half of the rest is used on groceries, what fraction of his income is used on groceries?

Solution

Step 1.

First, the fraction of the rest of the man's income after deducing that of the accommodation is

$1-\frac{1}{5}=\frac{1}{1}-\frac{1}{5}=\frac{5\left(\frac{1}{1}\right)-5\left(\frac{1}{5}\right)}{5}=\frac{5\left(\frac{1}{1}\right)-\cancel{5}\left(\frac{1}{\cancel{5}}\right)}{5}=\frac{5-1}{5}=\frac{4}{5}$

Step 2.

The rest of the man's income is four-fifth. We are told that he used half of the rest for groceries. Therefore, the fraction of the groceries is

$\frac{4}{5}\times \frac{1}{2}=\frac{{}^2\cancel{4}}{5}\times \frac{1}{\cancel{2}}=\frac{2}{5}$.