Question

Let S be the sample space of all five digit numbers. It p is the probability that a randomly selected number from S, is multiple of 7 but not divisible by 5, then 9p is equal to

• A. 1.0146
• B. 1.2085
• C. 1.0285
• D. 1.1521

Answer 1

The Correct Option is C

To solve this problem, we need to calculate the probability \( p \) that a randomly selected five-digit number is a multiple of 7 but not divisible by 5. We then multiply this probability by 9 to find \( 9p \). 

1. Determine the range of five-digit numbers. The smallest five-digit number is 10000 and the largest is 99999.
2. Calculate the total number of five-digit numbers, which is: \(99999 - 10000 + 1 = 90000\)
3. Find the first and last five-digit numbers that are multiples of 7:
   • The smallest five-digit number divisible by 7 is calculated as follows: \(10000 \div 7 \approx 1428.57\). Rounding up gives us 1429, so \(1429 \times 7 = 10003\).
   • The largest five-digit number divisible by 7 is calculated as follows: \(99999 \div 7 \approx 14285.57\). Rounding down gives us 14285, so \(14285 \times 7 = 99995\).
4. Calculate the number of multiples of 7: \(14285 - 1429 + 1 = 12857\)
5. Next, find the multiples of 35 (since they are both multiples of 7 and 5):
   • The smallest five-digit number divisible by 35 is: \(10000 \div 35 \approx 285.71\). Rounding up gives 286, \(286 \times 35 = 10010\).
   • The largest five-digit number divisible by 35 is: \(99999 \div 35 \approx 2857.11\). Rounding down gives 2857, \(2857 \times 35 = 99995\).
6. Calculate the number of multiples of 35: \(2857 - 286 + 1 = 2572\)
7. Calculate the number of five-digit numbers that are multiples of 7 but not multiples of 5: \(12857 - 2572 = 10285\)
8. Determine the probability \( p \): \(p = \frac{10285}{90000}\)
9. Compute \( 9p \): \(9p = 9 \times \frac{10285}{90000} = \frac{92565}{90000} \approx 1.0285\)

The value of \( 9p \) is approximately 1.0285, which corresponds to the correct answer.

Answer 2

The correct answer is (C):
Among the 5 digit numbers,
First number divisible by 7 is 10003 and last is 99995.
⇒ Number of numbers divisible by 7.
= \(\frac{99995-10003}{7} + 1\)
= 12875
First number divisible by 35 is 10010 and last is 99995.
⇒ Number of numbers divisible by 35 =\(\frac{ 99995-10010}{35}+1 = 2572\)
Hence number of number divisible by 7 but not by 5
= 12857 – 2572
= 10285
9P =\( \frac{10285}{90000} × 9\)
= 1.0285