What is 0.03703703703 as a fraction

The Correct Answer and Explanation is :

The correct answer is: (\frac{1}{27})

To convert the repeating decimal (0.03703703703) into a fraction, we can follow these steps:

Step 1: Define the decimal

Let (x = 0.03703703703).

Step 2: Identify the repeating part

The repeating part of the decimal is (0.037), which has three digits. We can express (x) in terms of this repeating segment:

[
x = 0.03703703703\ldots
]

Step 3: Multiply to eliminate the decimal

To eliminate the repeating decimal, we multiply (x) by (1000) (since there are three digits in the repeating segment):

[
1000x = 37.03703703703\ldots
]

Step 4: Set up the equation

Now, we have two equations:

(x = 0.03703703703\ldots) (1)
(1000x = 37.03703703703\ldots) (2)

Step 5: Subtract the first equation from the second

Subtract equation (1) from equation (2):

[
1000x – x = 37.03703703703\ldots – 0.03703703703\ldots
]

This simplifies to:

[
999x = 37
]

Step 6: Solve for (x)

Now, we can solve for (x):

[
x = \frac{37}{999}
]

Step 7: Simplify the fraction

Next, we check if the fraction (\frac{37}{999}) can be simplified. The number 37 is a prime number, and 999 can be factored as (3^3 \times 37). Thus, we can simplify:

[
\frac{37}{999} = \frac{1}{27}
]

Final Result

Therefore, the repeating decimal (0.03703703703) can be expressed as the fraction (\frac{1}{27}).

Explanation

Converting decimals to fractions involves understanding repeating patterns. By setting up an equation and eliminating the decimal through multiplication, we isolate the repeating portion and find a relationship that allows us to express it as a fraction. Simplifying the fraction may involve identifying common factors, as seen with the prime number 37 in this case. This process demonstrates how rational numbers can be represented in different forms, providing clarity on the relationship between decimals and fractions.