Three persons are walking from a place $A$ to another place $B$. Their speeds are in the ratio of 4 : 3 : 5. The time ratio to reach $B$ by these persons will be

### Concept & Inverse Proportionality of Ratios Because the distance is identical for all three persons, the time taken is inversely proportional to their respective speeds. If speeds are in ratio $x : y : z$, times are in ratio $\frac{1}{x} : \frac{1}{y} : \frac{1}{z}$. ### Step-by-Step Solution 1. The given ratio of speeds for the three persons is $4 : 3 : 5$. 2. The distance is constant (from place $A$ to place $B$). Therefore, the ratio of the time taken will be the reciprocal of the ratio of their speeds. $$Time \ Ratio = \frac{1}{4} : \frac{1}{3} : \frac{1}{5}$$ 3. To convert these fractions into a simplified integer ratio, find the Least Common Multiple (LCM) of the denominators $4$, $3$, and $5$. $$LCM(4, 3, 5) = 60$$ 4. Multiply each term of the fractional ratio by the LCM: $$Time \ Ratio = \left( \frac{1}{4} \times 60 \right) : \left( \frac{1}{3} \times 60 \right) : \left( \frac{1}{5} \times 60 \right)$$ 5. Simplify the terms: $$Time \ Ratio = 15 : 20 : 12$$ ### Exam Strategy & Shortcut For an inverse ratio of three numbers $a:b:c$, a quick shortcut to find the new ratio without writing fractions is to multiply the "other two" numbers for each spot: First term: $b \times c = 3 \times 5 = 15$ Second term: $a \times c = 4 \times 5 = 20$ Third term: $a \times b = 4 \times 3 = 12$ Resulting ratio: $15 : 20 : 12$. This avoids LCM calculation entirely! ### Common Pitfall A frequent error is simply reversing the order of the numbers from $4 : 3 : 5$ to $5 : 3 : 4$, incorrectly assuming a direct reverse works the same way for three items as it does for two. ### Final Answer Therefore, the correct answer is **15 : 20 : 12**.