Explanation:
ratios are special fractions. they are actually the result of the relation of both individual fractions (fraction divided by fraction).
that way, the numerator and the denominator are providing information about the balance between both types of items in the whole bag, and their sum give us a relative number of the total items.
this sum (like in our case 99 + 55 = 154) tells us that the whole content of the bag is split into 154 equal units of coins (with the same number of coins per unit).
we don't know the absolute number of coins here, but we don't need it. we only need the simplified fraction of each type of coins.
so, we know from these 154 units (of x coins each) 99 units are filled with quarters, and 55 units are filled with dimes.
so, the fraction of the total number being dimes is
55/154.
and the fraction of the total number being quarters is
99/154
again, we don't know x, and therefore, we don't know how many coins there are for real.
but for the relative information of fractions and ratios it is sufficient to see the relative size of each part.
imagine this like a pizza : we don't care how large the pizza is (small, medium or large), we only care how many slices have what toppings. that was the only question we were asked.
and now you see, how the ratio was originally created :
99/154 / 55/154 = 154×99 / (154×55) = 99/55
it could have come with e.g. x = 10 giving us
990/1540 / 550/1540
simplifying brings us always back to
99/154 / 55/154 = 154×99 / (154×55) = 99/55
so, again,
1.
the fraction of dimes is 55/154
2.
since we are still talking about relative sizes, we can again relate the number of coin units with each other.
because every coin unit has the same number of coins, the factor between the units is the same as between the absolute number of coins.
55 × f = 99
f = 99/55 = 1.8
so there are 1.8 times more quarters than dimes in the bag.