Percentage increase: which number do I divide by?

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1. Great question! The rule of thumb is: for percentage change, always compare to where you started. That means divide by the old value. The formula to memorize is: percentage change = (new − old) ÷ old × 100%. Reason: “percent” answers “how many out of each 100 of the original?” If a price goes from 240 to 300, the change is 60, and 60 ÷ 240 = 0.25, so it’s a 25% increase. Dividing by 300 instead would be asking a different question-“what percent is the change of the new amount?”-which isn’t how we measure increases. A small analogy: imagine your backpack went from 240 g to 300 g; the extra 60 g is felt relative to the weight you were already carrying, not the heavier weight after you’ve added the stuff. One more neat check: if you go up 25% from 240 you get 300, but going down 25% from 300 lands at 225-so the base matters, and for changes we base it on the starting point.

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2. Great question. I used to second‑guess this too, especially when I was practicing for a shop-inventory project and kept getting two different “percents” depending on which number I divided by. It finally clicked when I separated two ideas: percent change versus percent of the final total.

   Here’s the rule of thumb I rely on:
   – For percent increase (or decrease), divide by the original (starting) value.
   – In symbols: percent change = (new − old) ÷ old × 100%.

   Why the original? Because “increase by r%” literally means “add r percent of the original amount.” If New = Old + r% of Old, then New = Old × (1 + r). Solving for r gives r = (New − Old) ÷ Old.

   Apply it to your example:
   – Old = 240, New = 300
   – Change = 60
   – Percent increase = 60 ÷ 240 = 0.25 = 25%

   So the correct percent increase is 25%, not 20%.

   What does 60 ÷ 300 = 20% represent, then?
   – That’s the fraction of the final amount that the increase accounts for. It answers a different question: “What percent of the final price is the added amount?”
   – This comes up in finance language as margin vs. markup:
   – Markup compares the increase to the original cost: 60 ÷ 240 = 25% markup.
   – Margin compares the increase to the final price: 60 ÷ 300 = 20% margin.
   Both are valid ratios, but only the first is the standard “percent increase.”

   Sanity checks I use to keep myself from mixing them up:
   – Doubling test: If a number doubles, that’s a 100% increase. Old = 50, New = 100, change = 50. Dividing by Old gives 50 ÷ 50 = 100%. Dividing by New gives 50 ÷ 100 = 50%, which contradicts the idea of “doubling,” so that reminds me the base must be the old value for percent change.
   – Reverse test: If you go from 300 down to 240, the change is still 60, but now the starting value is 300. Percent decrease = 60 ÷ 300 = 20%. Notice how the percent depends on which direction you’re moving and therefore which starting point you use.

   A small flow I keep in my notes:
   – Percent increase: (New − Old) ÷ Old × 100%
   – Percent decrease: (Old − New) ÷ Old × 100%
   – If you know a percent increase r and want the new value: New = Old × (1 + r)
   – If you know New and r and want Old: Old = New ÷ (1 + r)

   Personal anecdote:
   When I first learned this, I ran a little “lemonade stand” spreadsheet to practice. I set cost at 240 and tried a selling price of 300. I proudly wrote “20% markup” because 60 ÷ 300 felt so tidy. My friend asked, “Would you really say you marked up your cost by 20%? Or that you’re making a 20% margin on the final price?” That phrasing snapped it into place:
   – Markup on cost (percent increase from cost to price): divide by cost → 60 ÷ 240 = 25%.
   – Profit margin (share of the final price that is profit): divide by final price → 60 ÷ 300 = 20%.
   Same numbers, different bases, different meanings.

   A quick mental cue you might like:
   – Percent change compares to where you started.
   – Percent-of-final (like margin) compares to where you ended.

   If you stick to “compare to where you started” for increases/decreases, you won’t go wrong.

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3. I totally get the second-guessing-percentages can feel slippery! The sturdy rule I lean on is: compare the change to where you started. In other words, for percent increase or decrease, divide by the old value (the starting value). The formula is percent change = (new − old) ÷ old × 100%. It’s like asking, “How big is the extra (or the drop) compared to my original pile?” If you started with 240 and gained 60, you want to know what fraction of 240 that 60 is.

   So for your example, from 240 to 300: the change is 60, and you divide by the starting value 240. That’s 60 ÷ 240 = 0.25 = 25% increase. Your 60 ÷ 300 = 0.2 is telling a different story: it’s the change as a fraction of the final amount, which isn’t the standard “percent increase.” A neat double-check is to rebuild the new value: a 25% increase on 240 means 240 × 1.25 = 300, which matches. Also, note the symmetry: if you go backward from 300 down to 240, that’s a 60 drop out of the old value 300, so it’s a 20% decrease-this is where dividing by 300 does make sense.

   One more quick example: suppose a score goes from 50 to 65. Change is 15. Divide by the starting 50: 15 ÷ 50 = 0.30, so that’s a 30% increase. If later it falls from 65 back to 50, the change is 15 again, but now you divide by the old value 65: 15 ÷ 65 ≈ 0.2308, so about a 23.08% decrease. Same 15, different bases-always the starting value.

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