Sure, let's solve the problem step-by-step.
We need to express \(4 \cdot \overline{346}\) as a fraction \(\frac{p}{q}\). Here's how we can do it:
1. Understand the repeating decimal notation:
[tex]\[
4 \cdot \overline{346}
\][/tex]
This represents the number obtained by multiplying 4 by \(346.346346346\ldots\).
2. Represent the repeating decimal \(346.346346\ldots\) as a fraction:
Let \(x = 346.346346346\ldots\).
To eliminate the repeating part, we'll set up an equation by multiplying \(x\) by 1000 (because the decimal repeats every three digits):
[tex]\[
1000x = 346346.346346\ldots
\][/tex]
Now, subtract the original \(x\) from this equation:
[tex]\[
1000x - x = 346346.346346\ldots - 346.346346\ldots
\][/tex]
This simplifies to:
[tex]\[
999x = 346000
\][/tex]
Hence, we have:
[tex]\[
x = \frac{346000}{999}
\][/tex]
3. Simplify the fraction \(\frac{346000}{999}\):
We need to find the greatest common divisor (gcd) of 346000 and 999 to simplify the fraction. The gcd of 346000 and 999 is 1, so the fraction is already in its simplest form.
4. Multiply the fraction by 4:
Now, multiply the fraction by 4 to get \(4 \cdot \overline{346}\):
[tex]\[
4 \cdot \frac{346000}{999} = \frac{4 \cdot 346000}{999} = \frac{1384000}{999}
\][/tex]
Therefore, the final expression in the form \(\frac{p}{q}\) is:
[tex]\[
\frac{1384000}{999}
\][/tex]
