Lesson Word problems on reversing digits of numbers

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Word problems on reversing digits of numbers

Problem 1

Sum of the digits of a two digit number is 10. When we interchange the digits, it is found that the resulting number is greater than the original number by 36.
Find the number.

Solution

Let a and b are the digits of your number n, so that n = 10a + b.

The number after interchanging digits is 10b + a.

Then you have the system of two equations for the unknowns a and b:

system%28a%2Bb+=+10%2C%0D%0A%0D%0A%2810b+%2Ba%29+-+%2810a+%2B+b%29+=+36%29

system%28a%2Bb+=+10%2C%0D%0A%0D%0A%2810b+%2Ba%29+-+%2810a+%2B+b%29+=+36%29

.

Now, simplify it:

system+%28a%2Bb=+10%2C%0D%0A%0D%0A9b+-+9a+=+36%29

.

Simplify it one more time:

system+%28a%2Bb=+10%2C%0D%0A%0D%0Ab+-+a+=+4%29

.

Solve it.

The solution is b=7, a=3.

Hence, the number is 10a + b = 10*3+7 = 37.

Answer. 37.

Problem 2

When a two-digit number is divided by the product of the two digits, the quotient is 2.
If 27 is added to the number, the original number turns into a new number with the digits being swapped around. Find the number.

Solution

Let a and b be the first (the left) and the second (the right) digit of the number respectively, so that the number is 10a + b.

The first condition says that %2810a+%2B+b%29%2F%28ab%29

,   or

10a + b = 2ab.         (1)

The number with swapped digits is 10b + a, so the second condition says that

10a + b + 27 = 10b + a,   or   9a - 9b = -27,   or   a-b = -3.

So, we have a system of two equations

system%2810a+%2B+b+=+2ab%2C%0D%0Aa-b+=+-3%29

system%2810a+%2B+b+=+2ab%2C%0D%0Aa-b+=+-3%29

.

Express b from the last equation, b = a + 3, and substitute it into the previous equation. You will get

10a + a + 3 = 2a*(a+3), or

2a%5E2

.

Solve this quadratic equation using the quadratic formula. You will get the roots a%5B1%5D

and

.
The negative root doesn't suit, because we are looking for the digit, which should be non-negative integer less than 10. The root a%5B1%5D

= 3 is good.

So, the number is 36.

Please check yourself that this number satisfies all conditions of the problem.

Answer. 36.

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