Express each of the following ratios in simplest form :

(i) 35 paise : ₹ 1

(ii) 1 kg : 375 g

(iii) 60 cm : 1 m

(iv) 400 m : 1 km

(v) 8 months : $1\dfrac{2}{3}$ years

(vi) $6\dfrac{2}{3}$ hours : 1 day

(i) 35 paise : ₹ 1

Convert rupee into paise: ₹ 1 = 100 paise.

The ratio is 35 : 100.

Let us find H.C.F. of 35 and 100:

$\begin{array}{r} 2 \ 35 \overline{) 100} \ \underline{-70} \ 30 \overline{) 35} ( 1 \ \underline{-30} \ 5 \overline{) 30} ( 6 \ \underline{-30} \ 0 \end{array}$

H.C.F. = 5

$\dfrac{35 \div 5}{100 \div 5} = \dfrac{7}{20} = 7 : 20$

Hence, the answer is 7 : 20

(ii) 1 kg : 375 g

Convert kg into g: 1 kg = 1000 g.

The ratio is 1000 : 375.

Let us find H.C.F. of 1000 and 375:

$\begin{array}{r} 2 \ 375 \overline{) 1000} \ \underline{-750} \ 250 \overline{) 375} ( 1 \ \underline{-250} \ 125 \overline{) 250} ( 2 \ \underline{-250} \ 0 \end{array}$

H.C.F. = 125

$\dfrac{1000 \div 125}{375 \div 125} = \dfrac{8}{3} = 8 : 3$

Hence, the answer is 8 : 3

(iii) 60 cm : 1 m

Convert m into cm: 1 m = 100 cm.

The ratio is 60 : 100.

Let us find H.C.F. of 60 and 100:

$\begin{array}{r} 1 \ 60 \overline{) 100} \ \underline{-60} \ 40 \overline{) 60} ( 1 \ \underline{-40} \ 20 \overline{) 40} ( 2 \ \underline{-40} \ 0 \end{array}$

H.C.F. = 20

$\dfrac{60 \div 20}{100 \div 20} = \dfrac{3}{5} = 3 : 5$

Hence, the answer is 3 : 5

(iv) 400 m : 1 km

Convert km into m: 1 km = 1000 m.

The ratio is 400 : 1000.

Let us find H.C.F. of 400 and 1000:

$\begin{array}{r} 2 \ 400 \overline{) 1000} \ \underline{-800} \ 200 \overline{) 400} ( 2 \ \underline{-400} \ 0 \end{array}$

H.C.F. = 200

$\dfrac{400 \div 200}{1000 \div 200} = \dfrac{2}{5} = 2 : 5$

Hence, the answer is 2 : 5

(v) 8 months : $1\dfrac{2}{3}$ years

Convert $1\dfrac{2}{3}$ years to improper fraction: $\dfrac{5}{3}$ years.

Convert years into months: $\dfrac{5}{3} \times 12 = 20$ months.

The ratio is 8 : 20.

Let us find H.C.F. of 8 and 20:

$\begin{array}{r} 2 \ 8 \overline{) 20} \ \underline{-16} \ 4 \overline{) 8} ( 2 \ \underline{-8} \ 0 \end{array}$

H.C.F. = 4

$\dfrac{8 \div 4}{20 \div 4} = \dfrac{2}{5} = 2 : 5$

Hence, the answer is 2 : 5

(vi) $6\dfrac{2}{3}$ hours : 1 day

Convert $6\dfrac{2}{3}$ hours to improper fraction: $\dfrac{20}{3}$ hours.

Convert 1 day into hours: 24 hours.

The ratio is $\dfrac{20}{3}$ : 24.

Multiply both sides by 3 to remove the denominator:

$\Big(\dfrac{20}{3} \times 3\Big) : 24 \times 3 = 20 : 72$

Let us find H.C.F. of 20 and 72:

$\begin{array}{r} 3 \ 20 \overline{) 72} \ \underline{-60} \ 12 \overline{) 20} ( 1 \ \underline{-12} \ 8 \overline{) 12} ( 1 \ \underline{-8} \ 4 \overline{) 8} ( 2 \ \underline{-8} \ 0 \end{array}$

H.C.F. = 4

$\dfrac{20 \div 4}{72 \div 4} = \dfrac{5}{18} = 5 : 18$

Hence, the answer is 5 : 18