Poisson Distribution - How does it come to whatever it is right now? The Question

Two cafes are side by side and are open 7 days a week.
(a) The first cafe sells cappuccinos at 5 dollars each and has 30 customers a
day. Half the customers buy a cappuccino, the rest don't. What is the
distribution of dollar sales made by the first cafe in a week?
(b) The second cafe sells cafe lattes at 6 dollars each. Customers arrive at
a rate of 60 a day and all the customers buy a cafe latte. What is the
distribution of dollar sales made by the second cafe in a week?
(c) Now the Poisson distribution is approximately Gaussian when the
rate is larger, and this applies moderately well when the rate is 15
or more. Give Gaussian approximations to the distributions of weekly
dollar sales for each of the two cafes.

My Understanding
I have the answers for this but I just don't know how it has been calculated. As in the whole process of coming to the conclusion of those answers and would appreciate if someone helps me understand or gives me step by step solution in order to understand.
The solutions given:
(a) The number of sales per week is Poisson with rate 105. The dollar sales is 5 times this, i.e. distribution is Poisson with rate parameter of 525 dollars per week.
(b) The number of sales per week is Poisson with rate 420. The dollar sales is 6 times this, i.e. distribution is Poisson with rate parameter of 2520 dollars per week.
(c) The question isn't very clear about when to apply the Gaussian approximation. One possible answer is to apply the normal approximation to the Poisson distribution of sales per week. In this case the sales rates respectively are, and. Now we scale to get dollars. The first is, and the second is
.Note since we already applied the Gaussian approximation to the sales rates, we have to square the dollars as the units for the variance are sales, so to get variance of total dollars we compute (sales times dollars per sale)=(sales)(dollars per sale)
$N(105, 105) N(420, 420) N(525, 105.5.5) = N(525, 2625) N(2520, 420.6.6) = N(2520, 15120)
σ2 2 2 2$
 A: Though the question is badly worded, assume that the number of coffee bought at Cafe A per week is poisson distributed with $\lambda=105$ and sells for 5 dollars and the number of coffees bought per week at Cafe B is poisson distributed with $\lambda=420$ and sells for $6 each. (This isn't what the problem actually says.)
The solution for a and b isn't necessarily correct per se because a poisson multiplied by a constant doesn't follow a poisson distribution. (Though it follows poisson probabilities) Therefore the answer for a is $P(d)=f(\frac d 5;\lambda=105)$ where d=0,5,10,...,525 is dollars sold and f is the pmf of a poisson distribution.
How they got c is that a poisson distribution approximately follows a gaussian distribution with mean lambda and and variance lambda. So for cafe A, the distribution of purchases per week is about $N(105,105).$ Multiply this by the dollar amount to get $5*N(105,105)=N(105*5,105*5^2)=N(525,2625)$. (The mean is multiplied by 5, but the variance is multiplied by 5 squared.) The distribution of purchases per week for cafe B can be calculated similarly, and is $N(105,105). 5*N(420,420)=N(420*6,420*6^2)=N(2520,15120)$.
