Math and Logic Puzzles

Relative speed = 72 - 54 = 18 kmph = 5 m/s. Total distance to cover = 100 + 200 = 300m. Time = distance/speed = 300/5 = 60 seconds = 1 minute. The faster train must cover the sum of both train lengths to completely overtake.

Let Cost Price (CP) be 100. A 20% gain means Selling Price (SP) is 120. If sold for thrice the price, the new SP is 360. The profit is 360 - 100 = 260. The gain percent is (260/100) * 100 = 260%. The stored answer '260' represents this percentage.

To find the percentage, divide 15 by 26 and multiply by 100. (15 / 26) * 100 equals approximately 57.69, which rounds to 57.7. The other options are mathematically incorrect calculations of the ratio between these two specific numbers.

Starting with Rs 8 and receiving Rs 2, the final amount is Rs 10. The increase is Rs 2, which is 2/8 = 0.25 or 25% of the original amount. Percentage increase is calculated as (increase/original amount) × 100.

Let b = number of boys, g = number of girls. Using weighted averages: 60b + 80g = 71(b + g). Solving: 60b + 80g = 71b + 71g, so 9g = 11b. Since 9/11 < 1, we have g > b. Therefore, the number of girls (Col B) is greater. Option C is incorrect because the numbers differ. Option D is incorrect because we can determine the relationship.

Total sum of 15 numbers = 15 * 8 = 120. Sum of 5 numbers = 5 * 12 = 60. The sum of the remaining 10 numbers is 120 - 60 = 60.

If 1500 is the price after a 20% discount, then 1500 represents 80% of the original price. Original price = 1500 / 0.8 = 1875. This is a standard percentage calculation.

If 20% of x = 25, then x = 125. The individual digits are 1, 2, and 5. The squares are 1, 4, and 25. The sum of squares is 1+4+25 = 30. 50% of 30 is 15.0.

To solve this question, the user needs to know that an odd number can only be obtained by adding an odd number of odd numbers. Therefore, the sum of two integers whose product is odd must be an even number.

Now, let's go through each option and explain why it is right or wrong:

A. Odd number: This option is incorrect because the sum of two integers whose product is odd cannot be odd. Adding two odd numbers together results in an even number.

B. Even number: This option is correct. The sum of two integers whose product is odd must be an even number. This is because the product of two odd numbers is odd, and odd + odd = even.

C. Prime number: This option is incorrect because the sum of two integers whose product is odd has nothing to do with prime numbers. Prime numbers are numbers that are only divisible by 1 and themselves.

D. Divisible by difference of two numbers: This option is incorrect because there is no relation between the sum of two integers whose product is odd and the difference of the two numbers.

E. Perfect square: This option is incorrect because there is no relation between the sum of two integers whose product is odd and perfect squares.

Therefore, the answer is:

The Answer is: B. Even number

When an odd number is added to itself an odd number of times, we're essentially calculating odd × odd. The product of two odd numbers is always odd. For example, if the odd number is 3 and we add it 3 times: 3 + 3 + 3 = 9, which is odd. This pattern holds for all odd numbers.

Let the integers be n, n+1, n+2. Their sum is 3n+3. For 3(n+1) to be odd, n+1 must be odd, meaning n must be even. If n is even, then n+2 is also even. Thus, the first and last integers must both be even.

If the sum of two primes x and y is odd, one must be 2 (the only even prime) and the other must be odd. So we have 2 + odd = odd. The product is 2 × odd, which is always divisible by 2. For example: if x=2 and y=5, sum=7 (odd), product=10, which is divisible by 2.

The smallest number divisible by all integers from 1 to 10 is the Least Common Multiple (LCM) of those numbers. LCM(1, 2, 3, 4, 5, 6, 7, 8, 9, 10) = 2^3 * 3^2 * 5 * 7 = 8 * 9 * 5 * 7 = 2520.

The series is of squares: 1, 4, 9... n^2. The sum of the first n squares is [n(n+1)(2n+1)]/6. For n=25: (25 * 26 * 51) / 6 = 25 * 13 * 17 = 5525. Since 17 is a prime factor in the formula (2n+1 = 51, which is 3*17), 17 is a factor.

If a person was born in the year 2000, their age in the year 2009 is calculated by subtracting the birth year from the current year: 2009 - 2000 = 9 years old.

In the standard English alphabet, the letter 'a' is the first letter and comes before 'b', which is the second letter. This is the basic alphabetical order used in English and other Latin-script languages, taught at the earliest levels of literacy.

Following the sequence A-B-C, 'b' precedes 'c'. Therefore, the statement that 'b' comes before 'c' is true.

In the alphabet, the order is B then C. Therefore, 'c' comes after 'b'. This is a true statement regarding standard sequential ordering.