write 0.3737373737 (repeating decimal) as a ratio of two relatively prime integers
= 37/99
To write the repeating decimal 0.3737373737 as a ratio of two relatively prime integers, we can use a simple algebraic approach.
Let's assign a variable to the repeating decimal, such as x. Hence, we have:
x = 0.373737...
Multiplying both sides of this equation by 100 (to shift the decimal point two places to the right), we get:
100x = 37.373737...
Now, let's subtract the original equation from the second equation:
100x - x = 37.373737... - 0.373737...
This simplifies to:
99x = 37
Dividing both sides of the equation by 99, we have:
x = 37 / 99
Therefore, the repeating decimal 0.373737... can be written as the ratio 37/99. Since 37 and 99 do not share any common factors other than 1, they are relatively prime.