Mental math

Use friendly numbers: replace each number with a nearby round number, do the easy arithmetic, then fix the difference. For multiplication, expand $(aimesb)$ by splitting one factor into a round number plus or minus a small amount, such as $47imes99=47(100-1)$. For squaring a two-digit number ending in 5, write the first digit as $n$ and compute $n(n+1)$ followed by 25, so $85^2$ begins with $8imes9=72$ and ends in 25. To add $1$ through $n$, pair the first and last terms: each pair sums to $n+1$, and there are $n/2$ pairs. For adding a number close to a round value, add the round value then subtract the shortfall, as in $998+467=1000+467-2$.