To find the solution, we can translate the given statement into a mathematical equation and then solve it. Firstly, let's denote the unknown number as 'x'.
The phrase 'One third of a number' can be represented as 'x/3'.
The phrase 'increased by 7' can be represented as '+ 7'.
The phrase 'is 13' signifies an equation, which can be represented as '= 13'.
So, the entire sentence 'One third of a number increased by 7 is 13' translates into the equation 'x/3 + 7 = 13'.
Next, we need to solve this equation for 'x'.
Step 1: To isolate x, let's first get rid of the '7' on the left side by subtracting 7 from both sides. This gives us 'x/3 = 13 - 7'.
Step 2: To further isolate x, let's get rid of the division by 3 on the left side, by multiplying both sides by 3. The equation then becomes 'x = 3 * (13 - 7)'.
Replacing 13 - 7 with 6 and then multiplying 3 by 6 gives us 'x = 18'.
So, the initial value of the number 'x' is 18.