Which of the following statements are true?

(i) 51 : 68 = 85 : 102

(ii) 1.5 : 2.5 = 3.6 : 6

(iii) 30 bags : 18 bags = ₹ 450 : ₹ 270

(iv) 32 kg : ₹ 36 = 8 kg : ₹ 9

(i) False
Reason —

First Ratio (51 : 68):

Let us find H.C.F. of 51 and 68:

$\begin{array}{r} 1 \ 51 \overline{) 68} \ \underline{-51} \ 17 \overline{) 51} ( 3 \ \underline{-51} \ 0 \end{array}$

H.C.F. is 17.

Simplest form: $\dfrac{51 \div 17}{68 \div 17} = {\dfrac{3}{4}}$

Second Ratio (85 : 102):

Let us find H.C.F. of 85 and 102:

$\begin{array}{r} 1 \ 85 \overline{) 102} \ \underline{-85} \ 17 \overline{) 85} ( 5 \ \underline{-85} \ 0 \end{array}$

H.C.F. is 17.

Simplest form: $\dfrac{85 \div 17}{102 \div 17} = {\dfrac{5}{6}}$

Since 3:4 ≠ 5:6, the statement is False.

(ii) True
Reason —

First Ratio = 1.5 : 2.5

Multiply by 10 to remove decimals = (1.5 x 10 : 2.5 x 10) = 15 : 25

Let us find H.C.F. of 15 and 25:

$\begin{array}{r} 1 \ 15 \overline{) 25} \ \underline{-15} \ 10 \overline{) 15} ( 1 \ \underline{-10} \ 5 \overline{) 10} ( 2 \ \underline{-10} \ 0 \end{array}$

H.C.F = 5

Simplest form: $\dfrac{15 \div 5}{25 \div 5} = {\dfrac{3}{5}}$

Second Ratio = 3.6 : 6

Multiply by 10 to remove decimals = (3.6 x 10 : 6 x 10) = 36 : 60

Let us find H.C.F. of 36 and 60:

$\begin{array}{r} 1 \ 36 \overline{) 60} \ \underline{-36} \ 24 \overline{) 36} ( 1 \ \underline{-24} \ 12 \overline{) 24} ( 2 \ \underline{-24} \ 0 \end{array}$

H.C.F. = 12.

Simplest form: $\dfrac{36 \div 12}{60 \div 12} = {\dfrac{3}{5}}$

Since both simplify to 3:5, the statement is True.

(iii) True
Reason —

First Ratio = 30 : 18

Let us find H.C.F. of 30 and 18:

$\begin{array}{r} 1 \ 18 \overline{) 30} \ \underline{-18} \ 12 \overline{) 18} ( 1 \ \underline{-12} \ 6 \overline{) 12} ( 2 \ \underline{-12} \ 0 \end{array}$

H.C.F. = 6.

Simplest form: $\dfrac{30 \div 6}{18 \div 6} = {\dfrac{5}{3}}$

Second Ratio = 450 : 270

First, cancel the zeros: 45 : 27.

Let us find H.C.F. of 45 and 27:

$\begin{array}{r} 1 \ 27 \overline{) 45} \ \underline{-27} \ 18 \overline{) 27} ( 1 \ \underline{-18} \ 9 \overline{) 18} ( 2 \ \underline{-18} \ 0 \end{array}$

H.C.F. = 9.

Simplest form: $\dfrac{45 \div 9}{27 \div 9} = {\dfrac{5}{3}}$

Since both simplify to 5:3, the statement is True.

(iv) False
Reason —

First Ratio = 32 : 36

Let us find H.C.F. of 32 and 36:

$\begin{array}{r} 1 \ 32 \overline{) 36} \ \underline{-32} \ 4 \overline{) 32} ( 8 \ \underline{-32} \ 0 \end{array}$

H.C.F. = 4.

Simplest form: $\dfrac{32 \div 4}{36 \div 4} = {\dfrac{8}{9}}$

Second Ratio = 8 : 9

This is already in its simplest form: ${\dfrac{8}{9}}$

So numerically both ratios are equal.

But a ratio should exist only between quantities of the same kind. Here the quantities are kg and ₹, which are different kinds. Therefore the statement is false.