A body of 7300 troops is formed of 4 battalions so that $\frac{1}{2}$ of the first, $\frac{2}{3}$ of the second, $\frac{3}{4}$ of the third and $\frac{4}{5}$ of the fourth are all composed of the same number of men. How many men are there in the second battalion?

### Concept & Formula When fractional parts of multiple independent variables are equivalent, we equate them to a constant scaling metric factor $k$ to express each variable relative to a single shared parameter. If $\frac{1}{2}A = \frac{2}{3}B = \frac{3}{4}C = \frac{4}{5}D = k$, then: $$ A = 2k, \quad B = \frac{3}{2}k, \quad C = \frac{4}{3}k, \quad D = \frac{5}{4}k $$ ### Step-by-Step Solution * **Given:** * Total troops = $7300$ * Let the sizes of the four battalions be $B_1, B_2, B_3, B_4$. * $\frac{1}{2}B_1 = \frac{2}{3}B_2 = \frac{3}{4}B_3 = \frac{4}{5}B_4 = k$ * **Calculation:** * Express each battalion in terms of $k$: $$ B_1 = 2k $$ $$ B_2 = \frac{3}{2}k $$ $$ B_3 = \frac{4}{3}k $$ $$ B_4 = \frac{5}{4}k $$ * Sum the battalions up to equate to the total population: $$ 2k + \frac{3}{2}k + \frac{4}{3}k + \frac{5}{4}k = 7300 $$ * Find the common denominator for the fractions (LCM of 2, 3, 4 is 12): $$ \left( \frac{24 + 18 + 16 + 15}{12} \right) k = 7300 $$ $$ \frac{73}{12}k = 7300 \implies k = 1200 $$ * Compute the size of the second battalion ($B_2$): $$ B_2 = \frac{3}{2} \times 1200 = 1800 $$ ### Exam Strategy & Shortcut Write down the ratios of the battalions directly by reciprocating the coefficients: $$ B_1 : B_2 : B_3 : B_4 = 2 : \frac{3}{2} : \frac{4}{3} : \frac{5}{4} $$ Multiply through by the LCM (12) to clear out fractional remainders: $$ B_1 : B_2 : B_3 : B_4 = 24 : 18 : 16 : 15 $$ Sum of ratio parts = $24 + 18 + 16 + 15 = 73$. Since total troops = 7300, 1 unit value = $\frac{7300}{73} = 100$. Second battalion size = $18 \times 100 = 1800$. ### Common Pitfall A standard error is failing to invert the fractional coefficients properly when shifting expressions into proportional value components, leading to an incorrect aggregate sum equation. ### Final Answer **Therefore, the correct answer is 1800.**