Ashok has a bag containing 40 cards, numbered with the integers from 1 to 40. No two cards are numbered with the same integer. Likewise, his sister Shilpa has another bag containing only five cards that are numbered with the integers from 1 to 5, with no integer repeating. Their mother, Latha, randomly draws one card each from Ashok’s and Shilpa’s bags and notes down their respective numbers. If Latha divides the number obtained from Ashok’s bag by the number obtained from Shilpa’s, what is the probability that the remainder will not be greater than 2?
- 0.8
- 0.91
- 0.73
- 0.94
- 0.87
The Correct Option is
Solution and Explanation
Ashok has cards numbered from 1 to 40. Shilpa has cards numbered from 1 to 5. When a card is selected from each, the division can yield a remainder between 0 and one less than the divisor. Since we are interested in cases where the remainder is not greater than 2, we need to check the remainders 0, 1, and 2.
Let's analyze each number from Shilpa's bag (considered as the divisor) and determine for how many numbers out of Ashok's bag the remainder when divided by this number is 0, 1, or 2.
- For number 1, all numbers 1 to 40 will have remainder 0, which is ≤ 2.
- For number 2, numbers leaving remainders 0 or 1 are favorable: numbers include all even numbers (20 numbers) and all odd numbers (20 numbers).
- For number 3, numbers leaving remainder 0, 1, or 2 are favorable. This results in 13 numbers for each remainder, totaling 39 numbers.
- For number 4, numbers leaving remainder 0, 1, or 2 are favorable. This gives 10 + 10 + 10 = 30 numbers.
- For number 5, numbers leaving remainder 0, 1, or 2 are favorable. This results in 8 + 8 + 8 = 24 numbers.
| Divisor | Favorable Outcomes |
|---|---|
| 1 | 40 |
| 2 | 40 |
| 3 | 39 |
| 4 | 30 |
| 5 | 24 |
Sum the favorable outcomes for each divisor: 40 + 40 + 39 + 30 + 24 = 173.
Total outcomes = number of cards in Ashok’s bag × number of cards in Shilpa’s bag = 40 × 5 = 200.
The probability is the ratio of favorable outcomes to total outcomes.
P(favorable) = 173/200 = 0.865.
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Top XAT Questions
- Ashok, a maths teacher in the school, asked his students to guess his date of birth by giving the following hints.
Hint 1: He was born in 1995
Hint 2: The sum of the products of his birth date multiplied by 24 and his birthday month multiplied by 60 is 1284.
Find the sum of Ashok’s birth date, birth month, and birth year. - Solve for the number of integral values of \(x: \frac{(x^2-15x+47)(x-13)}{(x-8)}<0\).
- In an examination, Akhil scored 11.11% less than the pass marks and Akshay scored 12.5% more than the pass marks. Find the percentage of pass marks if the sum of the scores of Akhil and Akshay is 25% more than the total marks.
- The ratio of paint and oil in Tank A is 3: 4, whereas in Tank B, the respective ratio is 4: 5 . Then, 96 liters of a mixture consisting of paint and oil in the ratio of 5: 1 is added to Tank A, such that the respective ratio in Tank A becomes exactly the reverse of that in Tank B. Find the amount (in liters) of the mixture from Tank B that is poured into Tank A such that the amount of paint and water in Tank A become equal.
- The points (–4, 3) and (–1, –4) are two end points of a diagonal of a parallelogram. If the other diagonal has the equation 7y + 3x + c = 0, then find the value of c.