Hasan writes down a two-digit number. He then writes the same two-digit number next to his original number to form a four-digit number. What is the ratio of his four-digit number to his two-digit number? 
Overseas Competition->Knowledge Point->Number Theory Modules->Place Value and Number Bases->Applying the Principle of Place Value
Let Hasan\textquotesingle s two-digit number be \textquotesingle$$ab$$\textquotesingle, which is equal to $$10 a + b$$. The four-digit number he forms is therefore \textquotesingle$$abab$$\textquotesingle, which is equal to $$1000 a + 100 b + 10 a + b$$ and hence to $$100 (10 a + b)+ 10 a + b = 101 \times (10 a + b)$$. Therefore the ratio of his four-digit number to his two-digit number is $$101:1$$. 