Let's let m_79m79 and m_81m81 represent the masses of the two isotopes.
The average atomic mass is the weighted average of the isotopic masses.
You multiply each isotopic mass by its relative abundance (percentage as a decimal fraction) in the mixture.
Thus,
"0.506 90"m_79 + "0.493 10"m_81 = "79.904 u"0.506 90m79+0.493 10m81=79.904 u
Now,
m_81/m_79 = 1.0356m81m79=1.0356
So,
m_81 = 1.0356m_79m81=1.0356m79
∴ "0.506 90"m_79 + "0.493 10" × 1.0356m_79 = "79.904 u"0.506 90m79+0.493 10×1.0356m79=79.904 u
"0.506 90"m_79 + "0.510 65"m_79 = "79.904 u"0.506 90m79+0.510 65m79=79.904 u
"1.017 55"m_79 = "79.904 u"1.017 55m79=79.904 u
m_79 = "79.904 u"/"1.017 55" = "78.525 u"m79=79.904 u1.017 55=78.525 u
Using the correct mass ratio
Your ratio of the two atomic masses is incorrect.
The correct ratio is 1.0253.
Using this number, we get
∴ "0.506 90"m_79 + "0.493 10" × 1.0253m_79 = "79.904 u"0.506 90m79+0.493 10×1.0253m79=79.904 u
"0.506 90"m_79 + "0.505 57"m_79 = "79.904 u"0.506 90m79+0.505 57m79=79.904 u
"1.012 48"m_79 = "79.904 u"1.012 48m79=79.904 u
m_79 = "79.904 u"/"1.012 48" = "78.919 u"m79=79.904 u1.012 48=78.919 u