Percentage calculator

The three formulas, worked through

The third one trips people up because the base is always the starting value. Going back down from 65 to 50 is (50 − 65) ÷ 65 × 100 = −23.08%, not −30%: the same 15 is a smaller share of a bigger number. A change from 0 has no percentage at all, since nothing can be divided by zero, so that line shows a dash.

The middle line answers scores and progress too: 18 right answers out of 24 is 75%, and 450 hours of a 1,800-hour target is 25% done.

Percent or percentage points?

When a rate is itself a percentage, “up 1%” becomes ambiguous. If a savings rate goes from 4% to 5%, it rose by one percentage point, but by 25 percent of its old value: (5 − 4) ÷ 4 × 100. News about interest rates, unemployment, polls and tax rates mixes the two up all the time, and the gap between them is large.

A good habit: say “points” for the distance between two percentages, and “percent” for how much bigger or smaller one is relative to the other. In Octoolo, typing 4 and 5 into the From … to … line gives the relative change, +25%. The difference in points is a plain subtraction you can do in the calculator.

A single point can cost real money. A $25,000 five-year loan at 7% instead of 6% is only one point apart, yet its total interest grows from $3,999.20 to $4,701.80, about 18% more, as the loan calculator shows when you try both rates.

Three traps with percentages

• Up and down are not symmetric. A price that rises 50% and then falls 50% ends at 75% of where it started: 100 becomes 150, then 75.
• Removing a percentage is a division. A $1,080 total that includes 8% tax was $1,000 before tax: 1,080 ÷ 1.08. Taking 8% off $1,080 gives $993.60, which is wrong.
• Percentages of percentages multiply. 20% off and then another 20% off is 36% off in total, because the second discount applies to 80% of the price: 0.8 × 0.8 = 0.64. The discount calculator page goes through stacked sales in detail.

In the calculator, the % key handles the first two for you: 100 + 50% − 50% gives 75, and 1080 ÷ 108% gives 1,000, the price before an 8% tax.

Percentages you meet every week

The three lines cover most of what people look up:

• A raise or a price rise. Pay going from $52,000 to $54,600 is +5% in the third line.
• A share of a budget. $450 of rent out of $1,800 of income is 25% in the second.
• A fee or a deposit. A 7.5% fee on $2,400 is $180 in the first.
• Anything measured twice. Weight, test scores, website visits or fuel prices: put the old and new values in the third line.

Tips have their own tool with the split built in, the tip calculator. All of these sit in one app on your PC with no ads, so a quick percentage does not come with a page full of banners.