Question

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.
(a) 9:1
(b) 7:3
(c) 1:10
(d) 1:4

Answer 1

You need to check each and every option whether the option is a possible ratio or not. Interpret the ratios, for example for the ratio 9:1, the number of black balls is 9k and the number of white balls is k. Then use the condition that the total number of balls is 10 and the constraint that the number of balls must be integers and cannot be fractions to get the answer.

Complete step-by-step answer :
For solving the above question, we need to check each and every option one by one. The constraints for the ratio to be possible is that the number of balls that we get for the used ratio must be positive integers and cannot be fractions.
So, let us start with option (a) 9:1. If we interpret the ratio, we will find that the number of white balls is k and the number of black balls is 9k. Also, it is given that the total number of balls is 10.
$\begin{aligned} & \therefore 9k+k=10 \\\ & \Rightarrow k=1 \\\ \end{aligned}$
If we use this value of k, the number of black balls is 9 and white balls is 1, which are integers and possible. So, option (a) is a possible ratio.
Now let us move to the next option, i.e., option (b). For option (c) the number of balls are 7k and 3k and the sum is 10.
$\begin{aligned} & \therefore 7k+3k=10 \\\ & \Rightarrow k=1 \\\ \end{aligned}$
If we use this value of k, the number of black balls is 7 and white balls is 3, which are integers and possible. So, option (b) is a possible ratio.
Similarly, if we check for option (c), the number of balls are 10k and 1k. If we add, we get
$\begin{aligned} & \therefore 10k+k=10 \\\ & \Rightarrow k=\dfrac{10}{11} \\\ \end{aligned}$
In this case the number of balls are non-integers, i.e., $\dfrac{10}{11}\text{ and }\dfrac{1}{11}$ . so, this is not a possible ratio.
If we similarly check for option (d), we will find the number of balls to be 8 and 2, which is possible.
So, we can conclude that the answer to the above question is option (c).

Note :Remember whenever you interpret a ratio a:b, take the elements to be ak and bk where k is real, because there is a possibility that when the ratio between them is taken some factors might have been cancelled from numerator and the denominator which is compensated by multiplying k with each.