Appearance
1.3 — Fractions, Decimals, Ratios and Percentages
Three people share two pizzas. How much does each get? The integers have no answer. Every real division that does not come out exactly — sharing, measuring, mixing, pricing — falls into this gap, and it is a much bigger gap than the one negatives filled. Negatives were needed for bookkeeping. Fractions are needed the moment you cut anything.
1. What a fraction is
A fraction \dfrac{a}{b} is the answer to the question "what number, multiplied by b, gives a?" Written the other way round, \dfrac{a}{b} is defined by:
\frac{a}{b} \times b = a
The top number a is the numerator (from Latin numerare, to count — it counts how many pieces) and the bottom number b is the denominator (from nominare, to name — it names what kind of piece). So \frac{3}{4} is three pieces of the kind called "quarters". The naming is not decoration; it tells you the two numbers are doing genuinely different jobs, which is why you cannot add fractions by adding tops and bottoms.
Numbers of this form, where a and b are integers and b \neq 0, are the rational numbers, written \mathbb{Q}. The word is from ratio, not from rational thought, though the second meaning is a happy accident.
And the hole from Chapter 1.2 is now closed: the rationals are closed under division, except by zero, which we showed has no meaning at all.
2. Why the same number has infinitely many names
\frac{1}{2}, \frac{2}{4}, \frac{50}{100} and \frac{17}{34} are all the same number. Cut a cake in two and take one piece, or cut it in four and take two — you are holding the same amount of cake.
The rule behind this:
\frac{a}{b} = \frac{a \times k}{b \times k} \quad \text{for any } k \neq 0
Multiplying top and bottom by the same thing changes nothing, because you are multiplying by \frac{k}{k}, which is 1. Nothing survives contact with more clarity than that: cancelling is multiplying by one.
A fraction is in lowest terms when the top and bottom share no common factor. \frac{18}{24} is not; both divide by 6, giving \frac{3}{4}. Chapter 1.5 gives the systematic way to find that 6 — it is the greatest common divisor, and Euclid found the algorithm for it 2,300 years ago.
This "many names for one number" business is not a quirk; it is the reason fraction arithmetic looks harder than it is. Most of the work in adding fractions is renaming them so they can be compared.
3. The four operations on fractions, with the reasons
Multiplication is the easy one:
\frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd}
Why: \frac{1}{2} \times \frac{1}{3} means half of a third. Take a cake, cut it in three, take one piece, cut that piece in half. You now hold one piece of a cake that has effectively been cut into six. So \frac{1}{6}. The denominators multiply because you cut twice; the numerators multiply because you take that many of them each time. Remember from Chapter 1.1 that multiplication is scaling, not repeated addition — "half a time" makes sense as a scaling and makes no sense as a repetition.
Division is multiplication by the flipped fraction:
\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}
Why? Because \frac{d}{c} is the reciprocal of \frac{c}{d} — the number that multiplies it to give 1:
\frac{c}{d} \times \frac{d}{c} = \frac{cd}{dc} = 1
Just as subtracting is adding the opposite (Chapter 1.2), dividing is multiplying by the reciprocal. Both operations are the same reduction: the inverse operation is the original operation applied to the inverse element.
There is also a plain-English check. "How many halves are there in 3?" is 3 \div \frac{1}{2}, and the answer is obviously 6 — and 3 \times \frac{2}{1} = 6 agrees. Dividing by a number smaller than one makes things bigger, which is only surprising until you say it in words.
Addition needs the denominators to match first:
\frac{a}{b} + \frac{c}{d} = \frac{ad}{bd} + \frac{cb}{db} = \frac{ad + cb}{bd}
Why: you cannot add three quarters and two thirds directly, for exactly the reason you cannot add three metres and two seconds. The denominator is the unit, and units must agree before amounts can be combined. Renaming both to twelfths — multiplying each by a disguised 1 — makes them the same kind of thing:
\frac{3}{4} + \frac{2}{3} = \frac{9}{12} + \frac{8}{12} = \frac{17}{12}
Nine twelfths plus eight twelfths is seventeen twelfths, in the same way nine apples plus eight apples is seventeen apples. Once the units match, the counting is ordinary.
Subtraction is identical, with a minus in place of the plus.
This unit-matching idea is worth carrying forward. In Volume III when you add two alternating voltages, or in Chapter 9.2 when you add two waves, the same principle governs: things can only be combined when they have been expressed in the same basis.
4. Decimals are fractions with the denominators hidden
A decimal is a fraction whose denominator is a power of ten, written using place value from Chapter 1.1 extended to the right of a point.
3.407 = 3 + \frac{4}{10} + \frac{0}{100} + \frac{7}{1000}
Exactly the same rule as before — each step to the right divides by ten instead of multiplying. The decimal point is not a separator between two numbers; it is a marker for where the units column is.
Why the world switched to decimals. Fractions are exact but awkward to compare: is \frac{7}{9} bigger than \frac{11}{14}? You have to do work. Decimals are comparable at a glance, digit by digit, left to right, which is why money, measurement and every computer use them. The price is that some fractions cannot be written exactly.
Terminating and repeating
Divide 1 by 8: you get 0.125 and it stops. Divide 1 by 3: you get 0.3333\ldots and it never stops. Divide 1 by 7: 0.142857142857\ldots, repeating in a block of six.
There is a clean rule for which happens. A fraction in lowest terms terminates exactly when its denominator's only prime factors are 2 and 5 — the primes that divide ten. Chapter 1.5 defines prime factors properly; for now, 8 = 2^3 terminates, 20 = 2^2 \times 5 terminates, 3 does not, 6 = 2 \times 3 does not because of the 3.
The reason is that a terminating decimal is a fraction over a power of ten, and 10^n = 2^n \times 5^n contains no other primes to cancel against.
A repeating decimal is exactly a rational number. Both directions are true, and the second is a nice piece of algebra. Take x = 0.3333\ldots Multiply by ten:
10x = 3.3333\ldots
Subtract the first from the second. The infinite tail is identical in both, so it cancels completely:
10x - x = 3.3333\ldots - 0.3333\ldots = 3
9x = 3 \quad\Rightarrow\quad x = \frac{3}{9} = \frac{1}{3}
Try the same on x = 0.\overline{142857}, where the bar marks the repeating block. Multiply by 10^6 because the block is six long:
1000000x - x = 142857 \quad\Rightarrow\quad 999999x = 142857 \quad\Rightarrow\quad x = \frac{142857}{999999} = \frac{1}{7}
The one that breaks people: 0.\overline{9} = 1
Run the same procedure on x = 0.9999\ldots: 10x = 9.999\ldots, so 10x - x = 9, so 9x = 9, so x = 1. They are not "nearly equal" or "equal in the limit" — they are two numerals for one number, exactly as \frac{1}{2} and \frac{2}{4} are.
The intuition that resists this is the feeling that 0.999\ldots is "creeping up on" 1. But the numeral does not creep; it is a completed thing, and the number it names is 1. Another way to see it: if they were different numbers, some number would lie between them. Name it. You cannot.
The consequence for computers, which is genuinely important. Computers store decimals in base two, where the terminating rule becomes "only powers of 2 terminate". The fraction \frac{1}{10} has a 5 in the denominator, so in base two it repeats forever, and the machine has to cut it off. That is the whole reason 0.1 + 0.2 does not give exactly 0.3 in almost every programming language — a fact Volume I, 1.4 covers in full, and which Chapter 10.1 of this volume revisits from the numerical-analysis side.
5. Ratios: the same idea wearing different clothes
A ratio compares two quantities without committing to their sizes. "Mix cement and sand in the ratio 1:3" means for every one unit of cement, three of sand — one bucket and three buckets, or one tonne and three tonnes.
The link to fractions: a ratio a : b says the first quantity is \frac{a}{b} times the second, and that the first is \frac{a}{a+b} of the total. In 1:3, the cement is one third of the sand and one quarter of the mixture. Confusing those two is the classic ratio mistake, and saying which total you mean out loud fixes it every time.
A proportion is the statement that two ratios are equal:
\frac{a}{b} = \frac{c}{d}
and it is the workhorse of practical arithmetic. Cross-multiplying — multiplying both sides by bd — turns it into ad = bc, which is easier to solve.
Worked example. A recipe for four people needs 300\ \text{g} of rice. For seven people:
\frac{300}{4} = \frac{x}{7} \quad\Rightarrow\quad 4x = 2100 \quad\Rightarrow\quad x = 525\ \text{g}
Worked example, the scaling kind. A map has a scale of 1 : 50000. Two towns are 7\ \text{cm} apart on the map. Real distance is 7 \times 50000 = 350{,}000\ \text{cm} = 3.5\ \text{km}.
Where this becomes a big idea. Two quantities are directly proportional when their ratio is constant: double one, double the other. Written y = kx, and its graph is a straight line through the origin. Two quantities are inversely proportional when their product is constant: double one, halve the other. Written y = \frac{k}{x}, and its graph is a hyperbola. Almost every physical law you will meet in Volume IV has one of these two shapes at its core, and recognising which one you are in tells you what happens when you change something.
6. Percentages, and the mistake everyone makes
A percentage is a fraction with the denominator fixed at 100. The word is per centum — "per hundred". 37\% means \frac{37}{100} = 0.37.
There is nothing more to the definition, and its entire value is standardisation: fixing the denominator makes any two proportions instantly comparable, which is exactly the same reason decimals beat fractions.
Finding a percentage of something is multiplication: 18\% of 250 is 0.18 \times 250 = 45.
Increasing by a percentage is multiplying by 1 plus it. A 15\% rise on 200 is 200 \times 1.15 = 230. Decreasing by 15\% is 200 \times 0.85 = 170.
Now the mistake.
Percentage changes do not add, and they do not undo each other
A stock rises 50\% then falls 50\%. Most people say you are back where you started. You are not.
100 \times 1.5 = 150, \qquad 150 \times 0.5 = 75
You have lost a quarter of your money. The reason is that the two percentages are taken of different bases — the rise is 50\% of 100, the fall is 50\% of 150. Percentages are multiplicative, and multiplication of 1.5 and 0.5 gives 0.75, not 1.
To actually undo a 50\% fall you need a 100\% rise. To undo a 20\% fall you need a 25\% rise. The recovery is always larger than the loss, which is the single most useful piece of financial arithmetic in this chapter.
The other trap is confusing a percentage with a percentage point. If an interest rate moves from 4\% to 6\%, that is a rise of two percentage points, and a rise of fifty percent. Both statements are true, they mean different things, and newspapers routinely conflate them.
7. Compound growth, in the small space it needs
If money grows by r each year — where r is written as a decimal, so 8\% is r = 0.08 — then after n years an amount P has become:
A = P (1 + r)^n
Each year multiplies by the same factor, and multiplying by the same thing repeatedly is exactly what a power means (Chapter 1.4).
₹100,000 at 8\% for 10 years:
100000 \times 1.08^{10} = 100000 \times 2.1589 = 215{,}892
That is ₹215,892.
The money more than doubles, though 8 \times 10 = 80 suggests only 80\% growth. The gap is the interest earning interest.
The rule of 72, which is worth carrying in your head forever: the number of years to double is roughly \frac{72}{\text{percentage rate}}. At 8\%, about nine years. At 6\%, twelve. At 12\%, six. Chapter 1.4 shows where the 72 comes from — it is 100 \ln 2 \approx 69.3, rounded up to 72 because 72 divides nicely by 2, 3, 4, 6, 8, 9 and 12.
The same formula with r negative describes depreciation, radioactive decay and the cooling of your tea, and Chapter 6.1 shows why all three obey it.
Every formula above, built from scratch
None of the results in this chapter are worth memorising, because each one can be rebuilt in under a minute from something simpler. What follows is that rebuilding, one result at a time, so the formula and the reason for it sit on the same page as the explanation that needed them.
Fractions
Adding fractions
\frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd}
Read it aloud. "a over b plus c over d equals a-d plus b-c, all over b-d."
Where it comes from. A fraction \frac{a}{b} means: cut a whole into b equal pieces and take a of them. You cannot add pieces of different sizes any more than you can add three apples to four oranges and call the answer seven of anything. So the first job is to make the pieces the same size.
Cut each of the b pieces into d smaller ones. Now the whole is in bd pieces, and what used to be a pieces is now ad of them:
\frac{a}{b} = \frac{a \times d}{b \times d} = \frac{ad}{bd}
Do the same to the other fraction, cutting each of its d pieces into b:
\frac{c}{d} = \frac{c \times b}{d \times b} = \frac{bc}{bd}
Both are now counts of the same size of piece — the bd-th of a whole. Counting them together gives ad + bc of those pieces, which is \frac{ad+bc}{bd}. That is the formula, and it is nothing more than "make the pieces the same size, then count".
A worked case. \frac{2}{3} + \frac{1}{4}. Here a=2, b=3, c=1, d=4. So ad = 2 \times 4 = 8, bc = 3 \times 1 = 3, bd = 12. The answer is \frac{8+3}{12} = \frac{11}{12}. Check it roughly: two-thirds is about 0.667, a quarter is 0.25, the total should be about 0.917, and 11/12 = 0.9167. Correct.
The formula never gives the smallest possible denominator, only a denominator that works. \frac{1}{6} + \frac{1}{6} comes out as \frac{6+6}{36} = \frac{12}{36}, which is right but ugly; it reduces to \frac{1}{3}. If you want the tidy answer directly, use the lowest common multiple of b and d instead of bd — the smallest number both denominators divide into. Section 7 shows how to find it.
Multiplying and dividing
\frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd}, \qquad \frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} = \frac{ad}{bc}
Where multiplication comes from. "Two-thirds of three-quarters" means: take three-quarters of the whole, then take two-thirds of that. Cutting into quarters and then cutting each quarter into thirds leaves you with twelve pieces, which is 3 \times 4. And you keep 2 \times 3 = 6 of them. So \frac{2}{3} \times \frac{3}{4} = \frac{6}{12} = \frac12. In general the denominators multiply because the cuts compound, and the numerators multiply because the takings compound.
Where division comes from. Division asks: how many of this fit into that? Turn it into a question about multiplication. If \frac{a}{b} \div \frac{c}{d} = x, then by the meaning of division x \times \frac{c}{d} = \frac{a}{b}. Multiply both sides by \frac{d}{c}:
x \times \frac{c}{d} \times \frac{d}{c} = \frac{a}{b} \times \frac{d}{c}
On the left, \frac{c}{d} \times \frac{d}{c} = \frac{cd}{dc} = 1, so the left side is just x. Hence x = \frac{a}{b} \times \frac{d}{c}. "Flip the second one and multiply" is not a trick anybody invented; it is forced by the fact that a fraction times its own upside-down version is 1.
Turning a repeating decimal into a fraction
0.\overline{d_1d_2\ldots d_k} = \frac{d_1d_2\ldots d_k}{\underbrace{99\ldots9}_{k \text{ nines}}}
Read it aloud. "A decimal that repeats a block of k digits forever equals that block, over k nines."
Where it comes from. Take x = 0.\overline{27} = 0.272727\ldots. The repeating block has length 2, so multiply by 10^2 = 100, which shifts the decimal point past exactly one block:
100x = 27.272727\ldots
Now subtract the original. The infinite tail is identical in both, so it cancels completely:
100x - x = 27.2727\ldots - 0.2727\ldots = 27
99x = 27 \quad \Rightarrow \quad x = \frac{27}{99} = \frac{3}{11}
The nines appear because 10^k - 1 is always a string of k nines. This is also the honest proof that 0.\overline{9} = 1: with k=1, x = \frac{9}{9} = 1. Not "nearly 1", not "1 minus something tiny". The same 1.
Percentages, interest and growth
Percentage change
\text{percentage change} = \frac{\text{new} - \text{old}}{\text{old}} \times 100
The denominator is always the starting value, and this is where most real-world errors live. A price going from £80 to £100 is a 25% rise (20/80). Going back from £100 to £80 is a 20% fall (20/100). The same £20 is a different percentage because the base changed. A 25% rise followed by a 25% fall does not return you to where you began: 100 \to 125 \to 93.75.
Compound interest
A = P\left(1 + \frac{r}{n}\right)^{nt}
Read it aloud. "The amount A equals the principal P, times one plus r over n, all raised to the power n-t" — where P is what you started with, r is the annual rate as a decimal, n is how many times a year the interest is added, and t is the number of years.
Where it comes from. One compounding period adds a fraction r/n of the current balance, so the balance is multiplied by \left(1 + \frac{r}{n}\right). Not added to — multiplied. Do that once and you have P(1 + r/n). Do it again and you have P(1+r/n)^2, because the second period's interest is charged on the first period's interest too. Over t years there are nt such periods, giving nt multiplications, which by the product law of powers is the exponent nt.
A worked case. £1,000 at 6% compounded monthly for 5 years: P = 1000, r = 0.06, n = 12, t = 5.
A = 1000\left(1 + \frac{0.06}{12}\right)^{60} = 1000(1.005)^{60} = 1000 \times 1.3489 = £1348.85
Simple interest would have given 1000 + 1000(0.06)(5) = £1300. The £48.85 difference is interest earning interest.
Where e comes out of a bank account
Push the compounding faster and faster — daily, hourly, every second — and the balance does not run away to infinity. It closes in on a specific number. Set P=1, r=1, t=1 and watch:
| Compounded | n | (1 + 1/n)^n |
|---|---|---|
| yearly | 1 | 2 |
| quarterly | 4 | 2.4414 |
| monthly | 12 | 2.6130 |
| daily | 365 | 2.7146 |
| every second | 31,536,000 | 2.7182818 |
The limit is the number e, about 2.718281828:
e = \lim_{n\to\infty}\left(1 + \frac{1}{n}\right)^n
and the general continuous-growth formula that follows from it is
A = Pe^{rt}
This is the same e that turns up in 5.3 as the only base whose exponential is its own derivative, and the two facts are the same fact seen from two sides: continuous growth means the rate of increase is proportional to the current size.
Averages, and the inequality between them
Three different averages of two positive numbers a and b:
\text{arithmetic } A = \frac{a+b}{2}, \qquad \text{geometric } G = \sqrt{ab}, \qquad \text{harmonic } H = \frac{2ab}{a+b}
The arithmetic mean is what everyone calls "the average". The geometric mean is the right average for things that multiply, like growth rates. The harmonic mean is the right average for rates measured per unit of something fixed, like speed over a fixed distance.
The inequality.
H \le G \le A, \qquad \text{with equality only when } a = b
Proof of G \le A, from a square that cannot have negative area. Whatever a and b are, \left(\sqrt{a} - \sqrt{b}\right)^2 \ge 0, because squaring any real number gives something that is not negative. Expand it:
a - 2\sqrt{ab} + b \ge 0
Move the middle term across:
a + b \ge 2\sqrt{ab}
Divide by 2:
\frac{a+b}{2} \ge \sqrt{ab}
That is A \ge G. And the only way \left(\sqrt a - \sqrt b\right)^2 can equal zero is \sqrt a = \sqrt b, meaning a = b. So the two averages agree exactly when the numbers agree, and the further apart the numbers are, the bigger the gap.
Proof of H \le G. Apply the result you have just proved to \frac1a and \frac1b:
\frac{\frac1a + \frac1b}{2} \ge \sqrt{\frac{1}{ab}}
Take the reciprocal of both sides. Taking reciprocals of positive numbers reverses the direction of an inequality — if 3 \ge 2 then \frac13 \le \frac12 — so:
\frac{2}{\frac1a + \frac1b} \le \sqrt{ab}
The left-hand side simplifies: \frac1a + \frac1b = \frac{a+b}{ab}, so the whole thing is \frac{2ab}{a+b} = H. Hence H \le G.
Why the harmonic mean is the honest one for speed. Drive 60 km at 30 km/h and 60 km back at 60 km/h. The naive average is 45 km/h. The truth: the outward trip takes 2 hours, the return takes 1 hour, so you covered 120 km in 3 hours, which is 40 km/h. And \frac{2 \times 30 \times 60}{30+60} = \frac{3600}{90} = 40. The harmonic mean got it right because time, not distance, is what the two legs share unevenly.
8. Where this shows up in your life
Every loan and every salary negotiation. An EMI, a credit card's monthly rate, an inflation figure, a hike expressed as a percentage — all of this chapter.
Every recipe scaled up, every paint mixed, every dose of medicine by body weight. Proportion, directly.
Every screen you look at. A 16:9 aspect ratio is a ratio in the literal sense; the pixel dimensions can be anything as long as the ratio holds.
Every discount sign that says "50% off, extra 20% off". That is not 70\% off. It is 0.5 \times 0.8 = 0.4, so you pay 40\% and the discount is 60\%. The shop is relying on you adding percentages that multiply.
Every floating-point bug you will ever debug. 0.1 + 0.2 !== 0.3 is the terminating-decimal rule of Section 4, in base two.
Powers appeared twice in this chapter without being defined — in 10^3 for place value and in (1+r)^n for compound growth. The next chapter defines them properly, and then inverts them, which produces the logarithm.