Question

Express each of the following ratios in its simplest forms:
$\left(iii\right)$ $2\frac{1}{3}:3\frac{1}{2}:1\frac{1}{4}$

Answer 1

Convert the mixed fraction into an improper fraction into an improper fraction

An improper fraction is obtained by adding the whole part with a fractional Part of a mixed fraction.
Therefore,
$2\frac{1}{3}=2+\frac{1}{3}=\frac{3\times 2+1}{3}=\frac{6+1}{3}=\frac{7}{3}$
$3\frac{1}{2}=3+\frac{1}{2}=\frac{3\times 2+1}{2}=\frac{6+1}{2}=\frac{7}{2}$
$1\frac{1}{4}=1+\frac{1}{4}=\frac{4\times 1+1}{4}=\frac{4+1}{4}=\frac{5}{4}$

Find $\text{LCM}$ of the denominators $(3,2,4)$
Factors of $2=2\times 1$
Factors of $3=3\times 1$
Factors of $4=2\times 2$

Therefore, $\text{LCM}$ of $3,2,4$ is
$3\times 2\times 2=12$

Find the simplest form of given ratio
We have,
$2\frac{1}{3}:3\frac{1}{2}:1\frac{1}{4}$
$=\frac{7}{3}:\frac{7}{2}:\frac{5}{4}$
$=\frac{7}{3}\times 12:\frac{7}{2}\times 12:\frac{5}{4}\times 12$
[since $\text{LCM}$ of $(3,2,4)=12$]
$=7\times 4:7\times 6:5\times 3=28:42:15$
$\Rightarrow 2\frac{1}{3}:3\frac{1}{2}:1\frac{1}{4}=28:42:15$

Hence, the simplest form of the ratio
$2\frac{1}{3}:3\frac{1}{2}:1\frac{1}{4}$ is $28:42:15$