Answer:
39
Step-by-step explanation:
The question is this.
There is a two-digit number that you need to find.
If you add the digits of the number, you get 12.
For example, using the number 45 as an example, when you add the digits, you get 4 + 5 = 9, which is not 12. When you add the digits of the number 48, you do get 12. Now, interchange the digits. 48 becomes 84. The original number is 48. The new number is 84. Now find the difference between the two numbers. 84 - 48 = 36. The difference is 36. The difference is not 54, so 48 is not the number we are looking for. The number we are looking for must have a difference of 54 when you subtract the new number from the original number.
Method 1.
<em>Since you are given choices, you can start with the choices, go through each one at a time, and see which one works in the problem.</em>
A) 39
Interchange the digits to get 93.
93 - 39 = 54
The answer is 39, since when you interchange the digits you get 93, and 93 is 54 more than 39.
Answer: 39
Method 2.
<em>If you are not given choices, you can just guess at numbers until one works.</em>
You know the digits must add to 12. There are only a few choices of 2-digit numbers whose digits add to 12: 39, 48, 57, 66, 75, 84.
The new number must be 54 greater than the original number. Of those 2-digit numbers, you can eliminate 66, 75, and 84 since by interchanging the digits, you don't get larger numbers.
Now you start with 39.
Interchange the digits: 93
93 - 39 = 54
Answer: 39
Method 3.
<em>The method that will give you an answer without guessing is by writing two equations in two unknowns. Then you solve the system of equations and find the answer.</em>
You are dealing with a 2-digit number, so you have a units digit and a tens digit.
Let the variable u represent the units digit.
Let the variable t represent the tens digit.
The number has the form: tu
The sum of the digits is 12, so we have our first equation:
t + u = 12 <em>Equation 1</em>
The tens digit represents the number of tens in the number, so the value of the tens digit is 10t. The units digit represents the digits 0 through 9, so its value is simply t.
The original number is 10t + u
When you interchange the digits, the t becomes the units digit, and the u becomes the tens digit.
The interchanged number is
10u + t
The interchanged number is 54 more than the original number. That gives us our second equation.
10u + t = 10t + u + 54
which simplifies to
9u - 9t = 54
Divide both sides by 9 to get
u - t = 6 <em>Equation 2</em>
Our system of equations consists of Equations 1 and 2:
u + t = 12
u - t = 6
We can use the addition method. Add the equations.
2u = 18
u = 9
The units digit is 9.
u + t = 12
9 + t = 12
t = 3
The tens digit is 3.
The original number is 39.
Answer: 39
