18/46+78/456

To add the fractions \( \frac{18}{46} + \frac{78}{456} \), we need to find a common denominator.

First, let's simplify both fractions:

1. Simplify \( \frac{18}{46} \):

   • The GCD of 18 and 46 is 2.
   • So, \( \frac{18 \div 2}{46 \div 2} = \frac{9}{23} \).

2. Simplify \( \frac{78}{456} \):

   • The GCD of 78 and 456 is 78.
   • So, \( \frac{78 \div 78}{456 \div 78} = \frac{1}{6} \).

Now, we have \( \frac{9}{23} + \frac{1}{6} \).

Next, we need to find a common denominator for the fractions \( \frac{9}{23} \) and \( \frac{1}{6} \). The least common multiple of 23 and 6 is \( 138 \).

Now, convert the fractions:

• For \( \frac{9}{23} \):

  • \( \frac{9 \times 6}{23 \times 6} = \frac{54}{138} \).

• For \( \frac{1}{6} \):

  • \( \frac{1 \times 23}{6 \times 23} = \frac{23}{138} \).

Now we add the fractions:

\[ \frac{54}{138} + \frac{23}{138} = \frac{54 + 23}{138} = \frac{77}{138}. \]

Finally, we can check if \( \frac{77}{138} \) can be simplified:

• The GCD of 77 and 138 is 1, so the fraction is already in its simplest form.

Thus, the final result is:

\[ \frac{77}{138}. \]