In each of these sums the two numbers under the square-root signs add up to 27. Which sum is the LARGEST?

Square each sum: $(\surd a+\surd b{)}^2=a+b+2$√$(ab)=27+2$√$(ab)$. With $a+b$ fixed, the biggest sum is the one with the biggest product $ab$, and a product grows as the two numbers move closer together: $10\times 17=170,11\times 16=176,12\times 15=180$ and $13\times 14=182$. So √13 + √14 wins. The decimals confirm how tight the race is: 7.285, 7.317, 7.337 and 7.347. Picking √11 + √16 because it contains the tidy whole number 4 is the trap — a neat-looking term says nothing about the total.