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 ```1. I know that a number is 7.75% of a larger number. What is a simple formula to find the larger number? 2. I know what I paid for an item. I know what I want my mark-up % to be. What is the simple formula to find the selling price? 3. I know what something is selling for and I know what I paid for it. What is the simple formula to find what the mark-up % is?```
 ```Hi, tobes-ga: 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. regards, mathtalk-ga```