Question

Which of the following statements are true?

(i) 16 : 24 = 20 : 30
(ii) 21 : 6 = 35 : 10
(iii) 12 : 18 = 28 : 12
(iv) 51 : 58 = 85 : 102
(v) 40 men : 200 men = Rs 5 : Rs 25
(vi) 99 kg : 45 kg = Rs 44 : Rs 20

Answer 1

(i) 16 : 24 = 20 : 30
$$\begin{gathered} \frac{16}{24}=\frac{2}{3}(\mathrm{Divi}\mathrm{ding}\mathrm{numerator}\mathrm{and}\mathrm{denominator}\mathrm{by}8) \\ \frac{20}{30}=\frac{2}{3}(\mathrm{Divi}\mathrm{ding}\mathrm{numerator}\mathrm{and}\mathrm{denominator}\mathrm{by}10) \\ \therefore \frac{16}{24}=\frac{20}{30} \\ \mathrm{Thus},\mathrm{the}\mathrm{statement}\mathrm{is}\mathrm{true}. \end{gathered}$$

(ii) 21 : 6 = 35 : 10
$$\begin{gathered} \frac{21}{6}=\frac{7}{2}(\mathrm{Divi}\mathrm{ding}\mathrm{numerator}\mathrm{and}\mathrm{denominator}\mathrm{by}3) \\ \frac{35}{10}=\frac{7}{2}(\mathrm{Divi}\mathrm{ding}\mathrm{numerator}\mathrm{and}\mathrm{denomenator}\mathrm{by}5) \\ \therefore \frac{21}{6}=\frac{35}{10} \\ \mathrm{Thus},\mathrm{the}\mathrm{statement}\mathrm{is}\mathrm{true}. \end{gathered}$$

(iii) 12 : 18 = 28 : 12

$$\begin{gathered} \frac{12}{18}=\frac{2}{3}(\mathrm{Divi}\mathrm{ding}\mathrm{numerator}\mathrm{and}\mathrm{denominator}\mathrm{by}6) \\ \frac{28}{12}=\frac{7}{3}(\mathrm{Divi}\mathrm{ding}\mathrm{numerator}\mathrm{and}\mathrm{denominator}\mathrm{by}4) \\ \therefore \frac{16}{24}\mathrm{and}\frac{20}{30}\mathrm{are}\mathrm{not}\mathrm{equal}. \\ \mathrm{Thus},\mathrm{the}\mathrm{statement}\mathrm{is}\mathrm{not}\mathrm{true}. \end{gathered}$$

(iv) 51 : 58 = 85 : 102

$$\begin{gathered} \frac{51}{58} \\ \mathrm{And}\frac{85}{102}=\frac{5}{6}(\mathrm{Dividing}\mathrm{numerator}\mathrm{and}\mathrm{denominator}\mathrm{by}17) \\ \therefore \frac{51}{58}\mathrm{is}\mathrm{not}\mathrm{equal}\mathrm{to}\frac{85}{102}. \\ \mathrm{Thus},\mathrm{the}\mathrm{statement}\mathrm{is}\mathrm{not}\mathrm{true}. \end{gathered}$$

(v) 40 men : 200 men = Rs 5 : Rs 25

$$\begin{gathered} \frac{40}{200}=\frac{1}{5}(\mathrm{Dividing}\mathrm{numerator}\mathrm{and}\mathrm{denominator}\mathrm{by}40) \\ \frac{5}{25}=\frac{1}{5}(\mathrm{Dividing}\mathrm{numerator}\mathrm{and}\mathrm{denominator}\mathrm{by}5) \\ \therefore \frac{40}{200}=\frac{5}{25} \\ \mathrm{Thus},\mathrm{the}\mathrm{statement}\mathrm{is}\mathrm{true}. \end{gathered}$$

(vi) 99 kg : 45 kg = Rs 44 : Rs 20

$$\begin{gathered} \frac{99}{45}=\frac{11}{5}(\mathrm{Dividing}\mathrm{numerator}\mathrm{and}\mathrm{denominator}\mathrm{by}9) \\ \frac{44}{20}=\frac{11}{5}(\mathrm{Dividing}\mathrm{numerator}\mathrm{and}\mathrm{denominator}\mathrm{by}4) \\ \therefore \frac{99}{45}=\frac{11}{5} \\ \mathrm{Thus},\mathrm{the}\mathrm{statement}\mathrm{is}\mathrm{true}. \end{gathered}$$