Operations with Percentages

Part 1: Percents, decimals, and fractions are the same thing

A percent, a decimal, and a fraction are just three different outfits for the same number. Learning to change between them is the key skill, and the moves are short.

Percent to decimal: move the dot two places left

To turn a percent into a decimal, divide by 100. The quick way: slide the decimal point two places to the left and drop the percent sign.

  75%   =  75.  ->  0.75
  8%    =  08.  ->  0.08
  150%  =  150. ->  1.50  =  1.5
   6.5% =  6.5  ->  0.065

If there aren't enough digits to move past, just add zeros. That's why 8% becomes 0.08, not 0.8.

Decimal to percent: move the dot two places right

Going the other way, multiply by 100, which means slide the decimal point two places to the right and add a percent sign.

  0.42  ->  42.   =  42%
  0.9   ->  90.   =  90%
  0.05  ->  05.   =  5%
  1.25  ->  125.  =  125%

Percent to fraction: put it over 100, then simplify

Since percent means "out of 100," write the number over 100 and reduce the fraction to lowest terms.

  40%  =  40/100  =  2/5    (divide top and bottom by 20)
  25%  =  25/100  =  1/4    (divide by 25)
  50%  =  50/100  =  1/2    (divide by 50)
  75%  =  75/100  =  3/4    (divide by 25)

A few of these are worth memorizing because they come up constantly: 50% = 1/2, 25% = 1/4, 10% = 1/10, and 20% = 1/5.

Part 2: Finding a percent of a number by hand

"What is 15% of 80?" The word of means multiply. You could write 0.15 × 80, but there's a friendlier mental-math route that needs no calculator: the 10% trick.

• Find 10% by moving the decimal one place left. 10% of any number is easy.
• Find 5% by taking half of the 10% you just found.
• Find 1% by moving the decimal two places left, if you need it.
• Add or stack these friendly pieces to build the percent you actually want.

Worked example #1: Find 15% of 80

15% is just 10% + 5%. So build it from those two friendly pieces.

  10% of 80  =  8        (move dot one place left)
   5% of 80  =  4        (half of 8)
  ---------------------------
  15% of 80  =  8 + 4  =  12

Worked example #2: Find 20% of 45 (a tip)

20% is simply double 10%. Find 10%, then multiply by 2.

  10% of 45  =  4.5      (move dot one place left)
  20% of 45  =  4.5 x 2  =  9

Worked example #3: Find 35% of 200

35% is 10% + 10% + 10% + 5%, or more simply 30% + 5%. Either way, start from 10%.

  10% of 200  =  20
  30% of 200  =  20 x 3  =  60
   5% of 200  =  10        (half of 20)
  ---------------------------------
  35% of 200  =  60 + 10  =  70

Handy shortcut: a percent "of" a number can be flipped. 4% of 50 is the same as 50% of 4, and 50% of 4 is just 2. When one order looks ugly, try the other.

Part 3: Simple percent increase and decrease

A price going up by a percent, or a discount taking a percent off, is just two steps you already know: find the percent of the number, then add it on (increase) or take it away (decrease).

Worked example #4: A $60 jacket is 25% off

First find 25% of 60, then subtract it from the original price. Remember 25% is one quarter.

  Discount:  25% of 60  =  60 / 4  =  15
  New price: 60 - 15     =  45

Worked example #5: A $40 fee goes up 10%

Find 10% of 40, then add it to the original.

  Increase:  10% of 40  =  4
  New fee:   40 + 4      =  44

Quick sanity check: an increase should make the number bigger and a decrease should make it smaller. If your answer went the wrong way, you added when you should have subtracted (or the reverse).

Tips that make percentages painless

• Always start from 10%. It is the easiest percent to find, and almost every other percent can be built from it by halving, doubling, or adding.
• Memorize the famous few. 50% = half, 25% = one quarter, 10% = one tenth, 20% = one fifth. These cover a huge share of real problems.
• Estimate to check. 18% of 50 should be a little under 10 (because 20% of 50 is 10). If your answer is wildly off, you slipped a decimal point.
• Watch the decimal point. Most percent mistakes are not math mistakes, they are decimal-point mistakes. Two places for percent-to-decimal, one place for 10%.

Your turn: practice problems

Grab a pen and use the 10% trick. No peeking until you've tried each one.

1. Write 30% as a decimal and as a fraction in lowest terms.
2. Find 15% of 60.
3. Find 20% of 150.
4. A $80 coat is 25% off. What is the sale price?