# Original Number

#### harpazo

##### Pure Mathematics
One of the digits of a two-digit number is twice the other digit. If you interchange the digits of this two-digit number and add the resulting number to the original number, you get 99. What is the original number?

What is the set up?

#### MarkFL

##### La Villa Strangiato
Staff member
Moderator
Math Helper
Let one digit be A and the other would then be 2A. Right away we know $$1\le A\le4$$. Anyway, using the information in the problem, we can write:

$$\displaystyle (10A+2A)+(20A+A)=99$$

Combine like terms, and solve for $$A$$. Because the problem is not specific about which digit is which in the original problem, you have two options for the original number.

harpazo

#### harpazo

##### Pure Mathematics
Let one digit be A and the other would then be 2A. Right away we know $$1\le A\le4$$. Anyway, using the information in the problem, we can write:

$$\displaystyle (10A+2A)+(20A+A)=99$$

Combine like terms, and solve for $$A$$. Because the problem is not specific about which digit is which in the original problem, you have two options for the original number.
Where did 10A and 20A come from?

#### MarkFL

##### La Villa Strangiato
Staff member
Moderator
Math Helper
Where did 10A and 20A come from?
A two digit number $$AB$$ in base ten notation has a value of $$10A+B$$.

harpazo

#### harpazo

##### Pure Mathematics
A two digit number $$AB$$ in base ten notation has a value of $$10A+B$$.
I did not know this, honestly.

#### MarkFL

##### La Villa Strangiato
Staff member
Moderator
Math Helper
I did not know this, honestly.
Sure you do...you know that the number $$54$$ has a value of $$5\cdot10+4$$, right?

harpazo

#### harpazo

##### Pure Mathematics
Sure you do...you know that the number $$54$$ has a value of $$5\cdot10+4$$, right?
Yes but???

#### MarkFL

##### La Villa Strangiato
Staff member
Moderator
Math Helper
Yes but???
But what? If a number has the two digits from left to right as x and y, then the value of the number (assuming base 10) is 10x + y. If y is twice x, then the number's value is 10x + 2x. Switching the digits gives the number a value of 20x + x.

Does that make sense?

harpazo

#### harpazo

##### Pure Mathematics
But what? If a number has the two digits from left to right as x and y, then the value of the number (assuming base 10) is 10x + y. If y is twice x, then the number's value is 10x + 2x. Switching the digits gives the number a value of 20x + x.

Does that make sense?
You said:

"If y is twice x, then the number's value is 10x + 2x. Switching the digits gives the number a value of 20x + x."

How does switching the digits yield
20x + x?

Staff member