The difference between the digits of a two digit number is 5. Also the original number is 18 more than two times the number obtained by reversing its digits. Find the original number.a)94b)61c)72d)49e)27Correct answer is option 'C'. Can you explain this answer?

Question

Answer

Solution:

Let the tens digit of the number be x and the units digit be y. So the two digit number is 10x+y.

Given, x-y=5 ...(1) [The difference between the digits of a two digit number is 5]

The number obtained by reversing the digits of the two digit number is 10y+x.

Given, 10x+y=2(10y+x)+18
=> 10x+y=20y+2x+18
=> 8x=19y+18
=> y = (8x-18)/19

As y is a digit, 8x-18 should be a multiple of 19. The values of x that satisfy this condition are 2, 4, 7 and 9. We can try these values to get the answer.

When x=2, y = (8x-18)/19 = (8x-19+1)/19 = 1, which gives the number 21. But this does not satisfy the given condition that the difference between the digits is 5.

When x=4, y = (8x-18)/19 = (8x-19+1)/19 = 2, which gives the number 42. But this also does not satisfy the given condition.

When x=7, y = (8x-18)/19 = (8x-19+1)/19 = 5, which gives the number 75. This satisfies the given condition.

When x=9, y = (8x-18)/19 = (8x-19+1)/19 = 8, which gives the number 98. But this does not satisfy the given condition that the difference between the digits is 5.

Therefore, the original number is 75, which is option (c).

Answer

Let number is 10x+y
Then x-y = 5 or y-x = 5
Now given that, 10x+y = 2(10y+x) + 18 Solve, 8x – 19y = 18
Now solve: 8x – 19y = 18 and x-y = 5. In this y = 2, x = 7
And also solve; 8x – 19y = 18 and y-x = 5. In this y come to be negative which is not possible so discard this
So number is 10*7 + 2

Answer

Solution:

Let's assume that the original two-digit number is in the form of AB, where A and B are digits.
According to the problem, we know that A - B = 5 or B - A = 5. Therefore, A can be B + 5 or A can be B - 5.

Now let's consider the second part of the problem, which states that the original number is 18 more than two times the number obtained by reversing its digits.

The number obtained by reversing AB is BA, which is equal to 10B + A.

So, according to the problem, we can write the equation as:

AB = 2(10B + A) + 18

Simplifying this equation, we get:

AB = 20B + 2A + 18

Substituting A = B + 5 or A = B - 5, we get:

B(B + 5) = 20B + 2(B + 5) + 18 or B(B - 5) = 20B + 2(B - 5) + 18

Simplifying these equations, we get:

B^2 - 13B - 16 = 0 or B^2 - 15B + 22 = 0

Solving these quadratic equations, we get B = 4 or B = 7.

If B = 4, then A = 9. Therefore, the original number is 94.

If B = 7, then A = 2. Therefore, the original number is 72.

Out of these two possible answers, only 72 satisfies the condition that the difference between the digits is 5 and the original number is 18 more than two times the number obtained by reversing its digits.

Therefore, the correct answer is option C, which is 72.

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