Adding Fractions, and the Mistake

Set two fractions. The tool adds them over a common denominator, and separately adds the numerators and the denominators — the classic mistake. Watch where each answer falls on the line.

Worked example — Adding Fractions, and the Mistake

Common denominator	20
Sum	23/20
Fractions inserted	0

These are the values the calculator shows when the page opens. Change any input above to recompute them.

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The classic mistake is a real operation, and it never misses 🖖

Add 3/4 and 2/5 the wrong way — tops together, bottoms together — and you get 5/9. That is not the sum, which is 23/20. But look at where 5/9 lands: at 0.5556, between 0.4 and 0.75. It is called the mediant, and it lands between the two fractions every single time. The proof is one subtraction: the mediant minus a/b works out to (bc − ad) over b(b+d), and c/d minus the mediant is the same (bc − ad) over d(b+d). Same sign, both times. So the mediant is above one fraction exactly when it is below the other, and it has nowhere to go but the gap. Drag the numbers anywhere you like; it stays in the shaded band.

A common denominator is not a ritual, it is a unit 🖖

You cannot add three sheep to four hours. Halves and thirds are units in the same way, which is why 1/2 + 1/3 is not 2/5 — the pieces are different sizes and the count means nothing until they are not. Sixths are the first size both a half and a third can be cut into exactly, and that is all the common denominator is: 3/6 + 2/6 = 5/6. This also says why the mistake feels so natural. Adding the tops and the bottoms is counting, honestly done — five pieces out of nine — but it counts pieces of two different sizes as if they were the same, so what comes out is a proportion of the whole pile rather than a total.

Keep making the mistake and you generate every fraction there is 🖖

Press Filling the whole gap. Start with 0 and 1, put the mediant between them — that is 1/2 — then do it again between each new neighbouring pair, and again. Four rounds gives 15 fractions. Not one of them can be cancelled down, and not one appears twice. Keep going and every fraction between 0 and 1 turns up exactly once, each in lowest terms, each arriving in its own place: this is the Stern–Brocot tree, published in 1858 by a German number theorist and in 1861 by a French clockmaker who was looking for gear ratios. Achille Brocot needed a fraction close to a ratio he could not build, with teeth he could actually cut, and the tree hands it to you.

Problem solved in full

Is the mediant the simplest fraction between the two? It always lands in the gap. Whether anything with a smaller denominator gets there first is a different question, and the answer is not always.

Open this problem in the tool
1. Start with the sum, done properly. Twenty is the first denominator both quarters and fifths fit into, so the two fractions become fifteen twentieths and eight twentieths.
  
  $\frac{3}{4} + \frac{2}{5} = \frac{15}{20} + \frac{8}{20} = \frac{23}{20}$
2. Now the mistake. Five over nine, and the panel agrees: it sits inside the gap, as it must.
  
  $\frac{3 + 2}{4 + 5} = \frac{5}{9} = 0.5556, \frac{2}{5} = 0.4 < \frac{5}{9} < 0.75 = \frac{3}{4}$
3. Why it must. Subtract the mediant from each end and the same quantity, bc - ad, appears on top both times. One denominator is b(b+d) and the other d(b+d), both positive, so the two differences carry the same sign and the mediant is squeezed between the two fractions.
  
  $\frac{a + c}{b + d} - \frac{a}{b} = \frac{b c - a d}{b ( b + d )}, \frac{c}{d} - \frac{a + c}{b + d} = \frac{b c - a d}{d ( b + d )}$
4. But look for something simpler. A half is in the gap too, and its denominator is 2 against the mediant's 9. So for this pair the mediant is not the simplest fraction between them, and it is not close.
  
  $\frac{1}{2} = 0.5 \in (\frac{2}{5}, \frac{3}{4}), q = 2 < 9$
5. The quantity that decides it is the one already on the table. For three quarters and two fifths it is seven, not one.
  
  $b c - a d = 4 \times 2 - 3 \times 5 = - 7, ∣ b c - a d ∣ = 7$
6. Try a pair where it is one. A third and a half: the mediant is two fifths, and nothing with a denominator below five lies between them. That is the whole condition.
  
  $\frac{1}{3}, \frac{1}{2} : b c - a d = 3 \times 1 - 1 \times 2 = 1, \frac{1 + 1}{3 + 2} = \frac{2}{5}$
Answer

The mediant is the simplest fraction between the two exactly when |bc - ad| = 1 — when the pair are neighbours in a Farey sequence. Swept over every pair with denominators up to 25: it holds for all 798 neighbouring pairs, and fails for 111,992 of the 119,636 pairs that are not. The tool never prints bc - ad, and that single number separates the case where the schoolroom mistake is the best possible answer from the case where it is merely a correct one.

$∣ b c - a d ∣ = 1 ⟺ \frac{a + c}{b + d} = q arg min {\frac{p}{q} \in (\frac{a}{b}, \frac{c}{d})}$

Learning path

Writing a number down

By the 42Math team · Reviewed against published sources. Corrections welcome.