Four children $A$, $B$, $C$ and $D$ divide a bag of sweets. $A$ takes $\frac{1}{3}$ of them, $B$ $\frac{2}{5}$th of the remainder and the rest is equally shared between $C$ and $D$. What fraction of the sweets did $C$ or $D$ get?

### Concept & Logic This problem is solved by tracking the remaining portion of a whole after sequential fractional deductions. When a whole (1) is reduced by a fraction, you multiply the remaining portion by the next fraction to find subsequent shares. Finally, dividing the ultimate remainder among a set number of people gives the individual share. ### Step-by-Step Solution * **Given:** Total fraction of sweets = 1. $A$ takes $\frac{1}{3}$ of the total. * **Calculation:** Find the remainder after $A$ takes their share: $$ \text{Remainder} = 1 - \frac{1}{3} = \frac{2}{3} $$ * Calculate $B$'s share, which is $\frac{2}{5}$ of the remainder: $$ B\text{'s share} = \frac{2}{5} \times \frac{2}{3} = \frac{4}{15} $$ * Find the new remainder after $B$ takes their share: $$ \text{New Remainder} = \frac{2}{3} - \frac{4}{15} $$ $$ \text{New Remainder} = \frac{10}{15} - \frac{4}{15} = \frac{6}{15} = \frac{2}{5} $$ * This remaining $\frac{2}{5}$ is shared equally between $C$ and $D$. Divide it by 2 to find the individual share for $C$ (or $D$): $$ \text{Share of } C = \frac{2}{5} \div 2 = \frac{2}{5} \times \frac{1}{2} = \frac{1}{5} $$ ### Exam Strategy & Shortcut Use the **LCM Assumption Method**. The denominators in the problem are 3 and 5. Their LCM is 15. Assume the total number of sweets in the bag is 15. $A$ takes $\frac{1}{3}$ of 15 = 5 sweets. Remaining sweets = 10. $B$ takes $\frac{2}{5}$ of 10 = 4 sweets. Remaining sweets = $10 - 4 = 6$ sweets. $C$ and $D$ share the 6 sweets equally, so each gets 3 sweets. The fraction $C$ got of the total is $\frac{3}{15}$, which simplifies to $\frac{1}{5}$. This avoids messy fraction arithmetic! ### Common Pitfall A very common mistake is calculating $B$'s share as $\frac{2}{5}$ of the *total* instead of $\frac{2}{5}$ of the *remainder*. Always pay close attention to whether fractions apply to the original amount or the leftover amount. ### Final Answer Therefore, the correct answer is **$\frac{1}{5}$**.