A tin of oil was $\frac{5}{8}$ full. When 10 bottles of oil was taken out and 8 bottles of oil was poured into it, it was $\frac{3}{5}$ full. How many bottles of oil can the tin contain?

### Concept & Logic This problem tests the ability to calculate net change and map it to fractional differences in capacity. Instead of working with the extracted and added volumes separately, find the **net change** in the number of bottles. Equate this net change to the difference between the initial and final fractions of the tin. Let the total capacity in bottles be $x$. ### Step-by-Step Solution * **Given:** Initial state = $\frac{5}{8}$ full. Final state = $\frac{3}{5}$ full. Action: Take out 10 bottles, add 8 bottles. * **Calculation:** Determine the net change in oil volume: Net oil removed = $10 \text{ (out)} - 8 \text{ (in)} = 2$ bottles. * Set up the equation reflecting this decrease: $$ \frac{5}{8}x - 2 = \frac{3}{5}x $$ $$ \frac{5}{8}x - \frac{3}{5}x = 2 $$ * Find the LCM for 8 and 5, which is 40: $$ \frac{25x}{40} - \frac{24x}{40} = 2 $$ $$ \frac{1x}{40} = 2 $$ * Solve for $x$: $$ x = 2 \times 40 = 80 $$ ### Exam Strategy & Shortcut Use the **Denominator Divisibility Check**. The total capacity must be a common multiple of the given denominators, 8 and 5. Their LCM is 40. Looking at the options (20, 30, 40, 80), both 40 and 80 are multiples of 40. You can quickly test 40: Initial = 25, Final = 24. Difference is 1 bottle (incorrect, we need a net difference of 2). Therefore, test 80: Initial = 50, Final = 48. Difference is 2 bottles. 80 is the correct answer. ### Common Pitfall The most frequent mistake is setting up the equation as $\frac{5}{8}x - 10 + 8x$, mistakenly treating the 8 bottles added as a fraction of $x$ rather than absolute units, or failing to realize that taking 10 and adding 8 simply means a net loss of 2. ### Final Answer Therefore, the correct answer is **80**.