Key property: If $0 < k < 1$, then $k < \frac{1}{k}$
Example: $0.4 < \frac{1}{0.4}$

Important: To apply the above property, we need not actually calculate the sum of the fractions in Quantity A. Instead, we can use some number sense to approximate the value.

We know that $\frac{1}{4}+\frac{1}{4}+\frac{1}{4}+\frac{1}{4} = 1$

Since $\frac{1}{5}$, $\frac{1}{6}$ and $\frac{1}{7}$ are each LESS THAN $\frac{1}{4}$, we can be certain that $\frac{1}{4}+\frac{1}{5}+\frac{1}{6}+\frac{1}{7}$ equals some number less than 1.

So we have:
QUANTITY A: some number that's between 0 and 1
QUANTITY B: 1/(some number that's between 0 and 1)

When we apply the above property, we see that Quantity A < Quantity B

Answer: B

Could i conclude that B is bigger simply because 1 over a fraction will flip the fraction therefore making B a sum of whole numbers while A is clearly a sum that is equal to some number less than 1?

Exactly

You are correct. There is no need of any calculation here.

Actually in both quantities, the numbers are equal but in B we do have a fraction that will flip into a whole number

Suppose that A is 1/x and B is 1/1/x then A will be 1/x and B will become X

So B > A