Question

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?

• A. 0.8
• B. 0.91
• C. 0.73
• D. 0.94
• E. 0.87

Answer 1

Answer: (E)

To solve the problem of finding the probability that the remainder of dividing the number from Ashok's bag by the number from Shilpa's bag will not be greater than 2, follow these steps:

Step 1: Understanding the possible outcomes.

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.

Step 2: Calculating favorable outcomes.

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

Step 3: Total favorable outcomes.

Sum the favorable outcomes for each divisor: 40 + 40 + 39 + 30 + 24 = 173.

Step 4: Calculating total possible outcomes.

Total outcomes = number of cards in Ashok’s bag × number of cards in Shilpa’s bag = 40 × 5 = 200.

Step 5: Calculating probability.

The probability is the ratio of favorable outcomes to total outcomes.
P(favorable) = 173/200 = 0.865.

Rounding off to the nearest available option, the final probability, considering slight adjustments, corresponds to 0.87.