The difference between a two-digit number and the number

[nkmungila1]

The difference between a two-digit number and the number obtained by interchanging the digits is 36. What is the difference between the sum and the difference of the digits of the number if the ratio between the digits of the number is 1 : 2 ?

A. 4
B. 8
C. 16
D. 20
E. 24

[Brent@GMATPrepNow]

The difference between a two-digit number and the number obtained by interchanging the digits is 36. What is the difference between the sum and the difference of the digits of the number if the ratio between the digits of the number is 1 : 2 ?

A. 4
B. 8
C. 16
D. 20
E. 24

The ratio between the digits of the number is 1 : 2
This information significantly LIMITS the possible values of the 2-digits. In fact, the number must be one of the following: 12, 24, 36 or 48
So, rather that get all algebraic, the fastest approach is probably just testing the 4 possibilities to see which one satisfies the other information.

We'll start with 12
Reverse the digits to get 21
21 - 12 = 9. No good. The difference is supposed to be 36

How about 24?
Reverse the digits to get 42
42 - 24 = 18. No good. The difference is supposed to be 36

How about 36?
Reverse the digits to get 63
63 - 36= 27. No good. The difference is supposed to be 36

How about 48?
Reverse the digits to get 84
84 - 48= 36. BINGO!!

What is the difference between the sum and the difference of the digits of the number?
SUM of digits = 4 + 8 = 12
DIFFERENCE of digits = 8 - 4 = 4
So, the difference between the sum and the difference of the digits = 12 - 4 = 8

Answer: B

Cheers,
Brent

[EconomistGMATTutor]

The difference between a two-digit number and the number obtained by interchanging the digits is 36. What is the difference between the sum and the difference of the digits of the number if the ratio between the digits of the number is 1 : 2 ?

A. 4
B. 8
C. 16
D. 20
E. 24

Hi nkmungila1,
Let's take a look at your question.

Let the 'x' be at ones place and 'y' be at tens place in the two digit number, then the number will be:
x + 10y
By interchanging the digits 'x' will be at tens place and 'y' will be at ones place, then the number will be:
10x + y

The question states that "The difference between a two-digit number and the number obtained by interchanging the digits is 36."
(x+10y) - (10x + y) = 36
x + 10y - 10x - y = 36
x - 10x + 10y - y = 36
-9x + 9y = 36
9(-x + y) = 36
- x + y = 4 ----- (i)

The ratio between the digits of the number is 1 : 2
We can write it as
y = 2x ------- (ii)

Plugin y = 2x in (i)
- x + y = 4
- x + 2x = 4
x = 4

To find y, y = 2x
y = 2(4) = 8

The number is 48.

Sum of the digits = 4+8 = 12
Difference of digits = 8 - 4 = 4

Difference of the sum and difference of digits = 12 - 4 = 8

Therefore, Option B is correct.

I am available if you'd like any followup.

[Scott@TargetTestPrep]

The difference between a two-digit number and the number obtained by interchanging the digits is 36. What is the difference between the sum and the difference of the digits of the number if the ratio between the digits of the number is 1 : 2 ?

A. 4
B. 8
C. 16
D. 20
E. 24

Solution:

Let’s say we randomly pick a two-digit number (with tens digit > units digit), e.g., 51. Reversing the tens and units digits, we have 15 and 51 - 15 = 36. So we have a difference of 36, however, the ratio between the digits is not 1 : 2.

Let’s pick another two-digit number, say 72. Reversing the tens and units digits, we have 27 and 72 - 27 = 45. This time, we have neither a difference of 36 nor a ratio of 1 : 2. However, not all is lost.

Notice that 51 - 15 = 36 = 4 x 9 and 72 - 27 = 45 = 5 x 9. We see that in the first example, the difference between the units and tens digits is 4, the difference between the two numbers is 4 x 9, and in the second example, the difference between the units and tens digits is 5, the difference between the two numbers is 5 x 9. Since we want the difference of the two numbers to be 36, which is 4 x 9, we want the difference between the units and tens digits to be 4. Since we also want the ratio between the digits of the number to 1 : 2, we can see that the units digit must be 4 and the tens digit must be 8. In other words, the number and its digit-reversing counterpart are 84 and 48 (notice that 84 - 48 = 36). So we have the sum of the digits = 8 + 4 = 12 and the difference of the digits = 8 - 4 = 4 and the difference between the sum of the digits and the difference of the digits is 12 - 4 = 8.

Alternate Solution:

Since the ratio between the two digits is 1:2, the number can be 12, 24, 36 or 48. Reversing the digits and subtracting the original number from it, we get: 21 - 12 = 9; 42 - 24 = 18; 63 - 36 = 27 and 84 - 48 = 36. Thus, the number we are looking for is 48; the sum of the digits is 4 + 8 = 12 and the difference of the digits is 8 - 4 = 4. Thus, the difference between the two is 12 - 4 = 8.

Answer: B