Check whether the following numbers are in proportion or not :

(i) 30, 40, 45, 60

(ii) $2, 2\dfrac{1}{2}, 3, 3\dfrac{1}{2}$

(iii) 0.8, 3, 2.4, 9

(iv) $\dfrac{1}{5},\dfrac{1}{8},\dfrac{1}{4},\dfrac{1}{10}$

(v) $\dfrac{1}{2},\dfrac{1}{5},\dfrac{1}{6},\dfrac{1}{15}$

(i) 30, 40, 45, 60

The given numbers are 30, 40, 45, 60.

We have:

$30 : 40 = \dfrac{30}{40} = \dfrac{3}{4}$

$45 : 60 = \dfrac{45}{60} = \dfrac{45 \div 15}{60 \div 15} = \dfrac{3}{4}$

∴ (30 : 40) = (45 : 60)

Hence, 30, 40, 45 and 60 are in proportion.

(ii) $2, 2\dfrac{1}{2}, 3, 3\dfrac{1}{2}$

The given numbers are $2, 2\dfrac{1}{2}, 3, 3\dfrac{1}{2}$.

Convert mixed to improper fraction:

$2\dfrac{1}{2} = \dfrac{5}{2}$

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

We have:

$2 : \dfrac{5}{2} = \dfrac{2}{5/2} = 2 \times \dfrac{2}{5} = \dfrac{4}{5}$

$3 : \dfrac{7}{2} = \dfrac{3}{7/2} = 3 \times \dfrac{2}{7} = \dfrac{6}{7}$

$\therefore (2 : 2\dfrac{1}{2}) ≠ (3 : 3\dfrac{1}{2})$.

Hence, $2, 2\dfrac{1}{2}, 3, 3\dfrac{1}{2}$ are not in proportion.

(iii) 0.8, 3, 2.4, 9

The given numbers are 0.8, 3, 2.4, 9.

Multiply by 10 to convert decimals to whole numbers:

We have:

$0.8 : 3 = \dfrac{0.8 \times 10}{3 \times 10} = \dfrac{8}{30} = \dfrac{4}{15}$

$2.4 : 9 = \dfrac{2.4 \times 10}{9 \times 10} = \dfrac{24}{90} = \dfrac{24 \div 6}{90 \div 6} = \dfrac{4}{15}$

∴ (0.8 : 3) = (2.4 : 9)

Hence, 0.8, 3, 2.4 and 9 are in proportion.

(iv) $\dfrac{1}{5},\dfrac{1}{8},\dfrac{1}{4},\dfrac{1}{10}$

The given numbers are $\dfrac{1}{5},\dfrac{1}{8},\dfrac{1}{4},\dfrac{1}{10}$.

We have:

$\dfrac{1}{5} : \dfrac{1}{8} = \dfrac{1}{5} \times \dfrac{8}{1} = \dfrac{8}{5}$

$\dfrac{1}{4} : \dfrac{1}{10} = \dfrac{1}{4} \times \dfrac{10}{1} = \dfrac{10}{4} = \dfrac{5}{2}$

$∴ \dfrac{1}{5} : \dfrac{1}{8} ≠ \dfrac{1}{4} : \dfrac{1}{10}$

Hence, $\dfrac{1}{5},\dfrac{1}{8},\dfrac{1}{4},\dfrac{1}{10}$ are not in proportion.

(v) $\dfrac{1}{2},\dfrac{1}{5},\dfrac{1}{6},\dfrac{1}{15}$

The given numbers are $\dfrac{1}{2},\dfrac{1}{5},\dfrac{1}{6},\dfrac{1}{15}$.

We have:

$\dfrac{1}{2} : \dfrac{1}{5} = \dfrac{1}{2} \times \dfrac{5}{1} = \dfrac{5}{2}$

$\dfrac{1}{6} : \dfrac{1}{15} = \dfrac{1}{6} \times \dfrac{15}{1} = \dfrac{15}{6} = \dfrac{5}{2}$

$∴ \dfrac{1}{2} : \dfrac{1}{5} = \dfrac{1}{6} : \dfrac{1}{15}$.

Hence, $\dfrac{1}{2},\dfrac{1}{5},\dfrac{1}{6},\dfrac{1}{15}$ are in proportion.