Directions: The following table gives the percentage distribution of population of five states, P, Q, R, S and T on the basis of poverty line and also on the basis of sex. Study the table and answer the questions based on it. | State | Percentage of Population below Poverty line | Proportion of Males and Females Below Poverty line M : F | Proportion of Males and Females Above Poverty Line M : F | |---|---|---|---| | P | 35 | 5 : 6 | 6 : 7 | | Q | 25 | 3 : 5 | 4 : 5 | | R | 24 | 1 : 2 | 2 : 3 | | S | 19 | 3 : 2 | 4 : 3 | | T | 15 | 5 : 3 | 3 : 2 | If the population of males below poverty line for State Q is 2.4 million and that for State T is 6 million, then the total populations of states Q and T are in the ratio :

### Concept & Algebraic Ratios To find the ratio of their total populations, calculate the absolute total population for each state separately by scaling up from the given specific demographic segments. $$ \text{Total Pop} = \frac{\text{Segment Population}}{\% \text{ in Category} \times \text{Demographic Fraction}} $$ ### Step-by-Step Solution 1. **Find Total Population of State Q (Let it be $Q_{pop}$):** - \% below poverty line = $25\%$. - Male fraction below poverty = $\frac{3}{3+5} = \frac{3}{8}$. - $Q_{pop} \times 0.25 \times \left(\frac{3}{8}\right) = 2.4$ - $Q_{pop} \times \left(\frac{1}{4}\right) \times \left(\frac{3}{8}\right) = 2.4 \Rightarrow Q_{pop} \times \left(\frac{3}{32}\right) = 2.4$ - $Q_{pop} = \frac{2.4 \times 32}{3} = 0.8 \times 32 = 25.6$ million. 2. **Find Total Population of State T (Let it be $T_{pop}$):** - \% below poverty line = $15\%$. - Male fraction below poverty = $\frac{5}{5+3} = \frac{5}{8}$. - $T_{pop} \times 0.15 \times \left(\frac{5}{8}\right) = 6$ - $T_{pop} \times \left(\frac{0.75}{8}\right) = 6 \Rightarrow T_{pop} = \frac{48}{0.75} = \frac{4800}{75} = 64$ million. 3. **Calculate Ratio $Q_{pop} : T_{pop}$:** - Ratio = $25.6 : 64$ - Multiply both sides by 10: $256 : 640$ - Divide both by $64$: $4 : 10$ - Simplify: $2 : 5$. ### Exam Strategy & Shortcut Instead of calculating the actual populations, set up the ratio fractionally to cancel common factors immediately: $$ \frac{Q_{pop}}{T_{pop}} = \frac{ \frac{2.4}{0.25 \times (3/8)} }{ \frac{6}{0.15 \times (5/8)} } = \frac{ 2.4 \times 8 \times 0.15 \times 5 }{ 6 \times 8 \times 0.25 \times 3 } $$ The $8$s cancel out: $$ \frac{2.4 \times 0.75}{6 \times 0.75} = \frac{2.4}{6} = \frac{24}{60} = \frac{2}{5} $$ ### Common Pitfall A common mistake is fully calculating large numbers and making an arithmetic error during division (like $4800/75$). Leveraging the direct ratio formula (Strategy section) inherently avoids messy division. ### Final Answer Therefore, the correct answer is **2 : 5**.