Percentages and fractions: recognize, do not compute
On a floor, percentages are done in your head and fast, and the trick is never to calculate what you can recognize. An eighth, ten plus five plus two and a half, a rise that a smaller fall undoes: hold a few shapes and most questions are recall.
- Take a percentage of a number by decomposing it into pieces you know
- Compute a percentage change from the starting value, and undo one from the ending value
- Chain a rise and a fall by multiplying factors, not adding percentages
- Convert fractions and basis points instantly
The intuition
17.5% of 4,800: ten percent is 480, five percent is 240, two and a half percent is 120, so 840. Nobody multiplied 4,800 by 0.175. 12.5% is an eighth, 37.5% is three eighths, 4.5% is half of 9%. A price from $80 to $100 is up 25%, but from $100 back to $80 is down 20%, because the fall is measured from the bigger number. And after rising 25% to $500, the original was $400, not $375: you divide by 1.25, you do not take 25% off.
Successive changes multiply. Up 20% then down 20% is 1.2 × 0.8 = 0.96, down 4%, not flat; the shortcut is a − b − ab ÷ 100. Basis points are hundredths of a percent, so one basis point on a million dollars is $100, and 25 bp on $40 million is $100,000. Know sevenths, ninths, elevenths, twelfths and sixteenths cold and most floor arithmetic becomes recognition, leaving your attention for the actual question.
x% of N = N × x ÷ 100, built from 10%, 5%, 1% and the fractions. % change = (new ÷ old − 1) × 100, always from the start. Original before a rise of a% = new ÷ (1 + a ÷ 100). Rise a% then fall b%: (1 + a)(1 − b) − 1, or a − b − ab ÷ 100. Fall that undoes a rise of a% = a ÷ (100 + a). 1 bp on $1M = $100.
Why it works
- The conventions here: a speed drill. Percentage changes are relative to the starting value. Basis points are hundredths of a percent. Exact answers, with the 1% grading tolerance absorbing sensible mental rounding.
- Decompose. Every percentage is a sum of 10%s, 5%s, 1%s and halves of those, or a fraction you know. 17.5% = 10 + 5 + 2.5; 45% = 50 − 5; 62.5% = five eighths.
- Why the undo is smaller than the rise. A rise of a% takes 100 to 100 + a. The fall back is a out of 100 + a, so a ÷ (100 + a): 25% up needs 20% down; 50% up needs 33% down. Losses hurt more than equal gains help.
- Why reverse-percent divides. After a rise the new value is old × (1 + a). Taking a% off the new value subtracts a% of a bigger number and lands too low. Divide by the factor instead.
- Why successive changes multiply. Each change is a factor on whatever came before. Order does not matter for the endpoint, only for the path.
- Fractions to know cold. Sevenths (14.29%, the digits 142857 repeating), ninths (11.11%), elevenths (9.09%, and k/11 is 9.09 × k), twelfths (8.33%), sixteenths (6.25%). Compare two fractions by cross-multiplying, no decimals needed.
| 17.5% of 4,800: 480 + 240 + 120 | 840 |
| $80 to $100, and back | +25%, then −20% |
| Worth $500 after a 25% rise: 500 ÷ 1.25 | $400 (not $375) |
| 3/8 as a percentage | 37.5% |
| Up 20% then down 20%: 1.2 × 0.8 − 1 | −4% |
| 25 basis points on $40M: 40 × 25 × $100 | $100,000 |
| Fall that undoes a 25% rise: 25 ÷ 125; rise that undoes a 25% fall: 25 ÷ 75 | 20%; 33.3% |
The formulas
Built from tens, fives, ones and fractions.
From the starting value, always.
Divide by the factor; do not take the percentage off.
Factors multiply; the cross term is the correction.
The fall is measured from the bigger number; basis points are hundredths of a percent.
Worked example
Multiply the two factors, subtract one, and check against the shortcut. The follow-up asks whether the order of the two years matters.
See it move
Same drill. Change the rise and the fall, the fraction, and the basis-point charge.
- Raise the rise. The net change rises, and so does the fall that would undo it, but by less than the rise did.
- Raise the fall. The net change falls, one point of the risen value per point.
- Raise the numerator, or lower the denominator. The fraction's percentage rises. The numerator is capped one below the denominator, so it never passes 100%.
- Raise the basis points or the notional. The dollar amount scales with either: 1 bp on $1M is $100, and everything else is multiplication.
Run it backwards
A position is worth a stated amount after rising a stated percentage. What was it worth before?
The new value is the old times (1 + the rise). Divide by that factor. Taking the percentage off the new value subtracts a share of a bigger number and lands too low.
The same move undoes any percentage change: divide by the factor. It is the reason a 25% rise needs only a 20% fall to reverse.
Traps
Say it in the interview
“A fund rises 20% one year and falls 20% the next. Where is it?”
Check yourself
6 fresh questions, with new numbers. Answer each one correctly to finish the chapter. Get one wrong and you will see the full working, then you can try it again with new numbers.
Answers within 1% are marked right. Type the number; $, %, x and M are fine. First tries count toward Learned: the topic is Learned once every chapter is done and 75% of first tries were right.
- Decompose percentages into tens, fives, ones and fractions you know.
- Changes are measured from the start; reverse them by dividing by the factor.
- Successive changes multiply: a − b − ab ÷ 100.
- The fall that undoes a rise is smaller: a ÷ (100 + a). 1 bp on $1M is $100.