He is wrong because he should not add the denominator of the fraction too, just the numerator and the full number. Besides this, he is correct.
Therefore, to correct his calculations,
[tex]5 \frac{3}{4} + 2 \frac{1}{4}
=7 \frac{3+1}{4}
= 7 \frac{4}{4}
=8[/tex]
