First a couple of things to keep in mind about percentages. A
percentage is a pure number, a fraction expressed as "parts per
hundred" (per cent). 7.75% is the same as 0.0775, for example.
Also, when we speak of one thing being a certain percentage of some
value, the word "of" means times. So $5 is 50% of $10 means simply:
$5 = 0.50 * $10
1. If we know that a number X is 7.75% of a larger number Y, that means:
X = 0.0775 * Y
To find what Y is when X is known, we should divide X by 0.0775:
Y = X / 0.0775
For example, if the smaller number is X = 15.50, and you divide it by
0.0775, you find the larger number is Y = 15.50/0.775 = 200.
We often refer to the amount that the percentage is taken "of" as the
"base", which is Y in this case.
2. Suppose you paid Y for an item, and you know what "mark-up"
percentage you like to have. Since the mark-up is an additional
amount, added to the amount to paid to get the selling price, we
would write the selling price S as:
S = Y + (p*Y) = (1 + p) * Y
where p is the mark-up percentage.
For example, if you want the mark-up to be 10% = 0.10, you would
multiply the amount you paid Y by 1 + 0.10:
S = 1.10 * Y
to get the selling price S.
3. When you know what something is selling for, S, and also what was
the amount paid for it , Y, then the same formula can be "solved" to
find the mark-up:
S = (1 + p) * Y
1 + p = S / Y
p = (S / Y) - 1 = (S - Y)/Y
That is, the mark-up is the difference S - Y between selling price and
amount paid, divided by the base (amount paid).
For example, if a lamp is sold for $124 that you paid only $80 for, we
would say the mark-up was:
p = (124 - 80)/80 = 44/80 = 0.55 = 55%
Note that if the selling price S were less than the amount you paid Y,
then the mark-up would be a negative number! In this case we usually
turn the difference around and refer to the percentage as a mark-down,
but either way the amount paid Y is the "base" of the percentage.
Please let me know if further clarification would be helpful.