Hint: The largest possible numbers could start with 6 and 8, therefore the first digit of the answer is 1.

Answer:

Remembering that:

   E + E = E
   O + O = E
   E + O = O

To discuss individual letters it's easiest to represent the sum as:

   A B C
   D E F +
--------
 G H I J

The largest values for A and D are 6 and 8, which makes G = 1.

Since column 2 is E + O = O there can be no carry from column 1 (since E + O + 1 is always even). Therefore C and F are 3 and 5 (but we don't yet know which is which), therefore J = 8.

There can't be a carry from column 2 (as A + D is even) therefore E can't be 9 as this would force a carry.

Therefore I = 9. Hence B can't be 0. Therefore H = 0.

The last remaining odd number makes E = 7. Making B = 2.

Therefore A and D are 4 and 6 (but we don't yet know which is which).

Since the top row's digits have to add to 9 the top number must be 423.

This makes the sum 423 + 675 = 1098. QED.

Hint: The word begins with the letter A.

Answer: Art.

C...ON  = CARTON
HE...   = HEART
E...H   = EARTH
M...YR  = MARTYR
QU...ER = QUARTER
ST...   = START

Hint: HELIUM = H?? + L??

Answer:

HELIUM = HUM + LIE
LATEST = LET + SAT
ALMOND = OLD + MAN
ELDEST = SET + LED
SAFETY = FAT + YES
RADIUM = AIR + MUD
DESERT = RED + SET
WEALTH = THE + LAW
TRICKY = KIT + CRY
FUNNEL = ELF + NUN

Hint: Adrian had been teaching 6 years longer than Betty.

Answer:
Adrian had been teaching 21 years.
Betty had been teaching 15 years.
Charles had been teaching 7 years.

Adrian and Betty had been teaching for a total of 36 years.

Charles and Betty had been teaching for a total of 22 years.

Charles and Adrian had been teaching for a total of 28 years.

Let Adrian = A, Betty = B and Charles = C, then:

   A + B = 36       [1]
   C + B = 22       [2]
   C + A = 28       [3]

If we use [3] - [2] we have:

   A - B = 6        [4]

If we use [1] + [4] we have:

   2A = 42
    A = 21

By [1] B = 15. By [3] C = 7. QED.