A jar contains black and white marbles. If there are ten marbles in the jar, then which of the following could not be the ratio of black to white marbles?

### Concept & Logic For objects that cannot be divided into fractions (like marbles), the total number of items must be a multiple of the sum of the ratio parts. $$ \text{Total Items} = k(a + b) $$ where $a$ and $b$ are the parts of the ratio and $k$ is a positive integer. ### Step-by-Step Solution * **Given:** Total number of marbles = 10. * **Option (a):** Sum of ratio parts = $1 + 4 = 5$. Since 10 is perfectly divisible by 5 (10 / 5 = 2), this could be the ratio (2 black, 8 white). * **Option (b):** Sum of ratio parts = $1 + 10 = 11$. Since 10 is not divisible by 11, it would result in fractional marbles, which is impossible. * **Option (c):** Sum of ratio parts = $7 + 3 = 10$. Since 10 is perfectly divisible by 10, this could be the ratio. * **Option (d):** Sum of ratio parts = $9 + 1 = 10$. Since 10 is perfectly divisible by 10, this could be the ratio. ### Exam Strategy & Shortcut Quickly add the parts of each ratio in the options. Find the sum that does not evenly divide into the total number given (10). $1+10=11$, which doesn't divide 10. ### Common Pitfall A common mistake is failing to realize that items like marbles, people, or animals must always result in integer quantities when distributed according to a ratio. ### Final Answer Therefore, the correct answer is **$1 : 10$**.