 Try this beautiful problem on balls based on Number theory, useful for ISI B.Stat Entrance.

## Problem on Balls | ISI B.Stat Entrance | Problem 128

A bag contains colored balls of which at least 90% are red. Balls are drawn from the bag one by one and their color noted. It is found that 49 of the first 50 balls drawn are red. Thereafter 7 out of every 8 balls are red. The number of balls in the bag can not be

• (a) $170$
• (b) $210$
• (c) $250$
• (d) $194$

### Key Concepts

Number theory

Percentage

Inequility

But try the problem first…

Answer: (b) $210$

Source

TOMATO, Problem 128

Challenges and Thrills in Pre College Mathematics

## Try with Hints

First hint

Let the number of balls in the bag is n.Let, m number of times 8 balls are drawn. Therefore, $n = 50 + 8m$. Red balls =$49 + 7m$

Can you now finish the problem ……….

Final Step

Percentage of red balls = $\frac{(49 + 7m)}{(50 + 8m)}≥ \frac{90}{100}$

$\Rightarrow \frac{(49 + 7m)}{(50 + 8m)}≥0.9$

$\Rightarrow 49 + 7m ≥ 45 + 7.2m$

$\Rightarrow 0.2m ≤ 4$

$\Rightarrow m ≤ 20$

$\Rightarrow n ≤ 50 + 8 \times 20 = 210$

Therefore we can say that at most $210$ balls can be drawn. So there must be at most $210$ balls in the bag, and so there cannot be $250$ balls in the bag (because $250>210$).

Therefore option (C) is the correct answer