Simpler explanation :
Statement 1 -
Break it as ->
The amount of water _ increasing
The amount of salt _ increasing
in both statements above , we have subject ( the water / the salt ) are singular subjects.
Also as per quantity rule , " The amount (quantity) always takes a singular verb.
so " is " correct
statement 2 :
The amount of Both water and salt continue to increase.
we can use quantity rule here but but see usage of " both " along with "and" .Its combining water and salt and gives a compound plural subject, so use a plural verb " continue " here.
Note 1 :"continues" is a singular verb
Note 2 :Correlative conjunctions always join grammatically equal elements
Note 3 : Here our correlative conjunction "both" along with "and" joining equal elements i.e a noun ( water ) with another noun (sea )
Note 4 :
" Both ... and " indicates a parallel structure with correlative conjunctions.
Correlative conjunctions work together in pairs, and there are four of these conjunctive duos: either—or, neither—nor, both—and, not only—but also.
Important subject verb agreement for correlative conjunctions :Subject-verb agreement with correlative conjunctions is entirely different. We have to be wide awake.
Since correlative conjunctions join grammatically equal elements of a sentence, the situation often arises that we have two subjects: one singular, one plural. Now what? What form does the verb take?
It’s all about proximity. The subject closest to the verb determines the verb form. If the subject closest to the verb is singular, the verb takes the singular form. If the subject closest to the verb is plural, the verb takes the plural form. This holds true for all correlative conjunctions except both…and. (Don’t you just love how there always has to be an exception?)
Not only our dog but also our cats need to be inoculated.
Not only our cats but also our dog needs to be inoculated.
Neither my colleagues nor my boss is coming to my birthday party.
Neither my boss nor my colleagues are coming to my birthday party.
The correlative conjunction both…and doesn’t require much thought as far as subject-verb agreement goes. The verb is always plural.
Both Peter and Paul are on the bus.
Both the boys and the girls are on the bus.
Hope this should clear the answer for second statement now