Cross-Multiplication Grid

Put the four numbers in a square, draw the cross, circle the one you want. What is left tells you what to multiply and what to divide by.

Worked example — Cross-Multiplication Grid

Read off the grid	250 × 20 ÷ 100
Answer	50

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

Loading interactive simulation...

You do not remember which two to multiply — you look 🖖

That is the whole argument for this method, and it is not a mathematical one. There are four questions here — find the part, the whole, the percent, the new value — and taught as formulas they are four things to keep straight under pressure. Taught as this square they are one: circle the number you want, and the two it is not joined to by the cross multiply, while the one it is joined to divides. That never changes, whichever corner you circle. A picture you can rebuild on the back of a receipt beats a formula you have to recall correctly, and that is why people who learned it this way still use it decades later.

The square is two similar triangles 🖖

A stick 2 m tall throws a shadow 3 m long; the tree beside it throws 18 m. Same sun, same angle, so the two triangles have the same shape — and equiangular triangles have proportional sides, which is Euclid, Elements VI.4. Put the four numbers in the square and the tree is 2 × 18 ÷ 3 = 12 m, measured without leaving the ground. This is also the method’s honest limit. It works exactly when one quantity is a fixed multiple of another, because that is when the two triangles exist. Nothing similar stands behind Kepler’s T² ∝ a³, so no arrangement of a 2×2 grid will give you an orbital period.

For a thousand years this was where arithmetic stopped 🖖

Brahmagupta set the rule down in 628 CE, and not as an equation — there is no unknown in it. He named three quantities, pramāṇa, phala and icchā, with the observation that two of them are the same kind of thing and the third is not. That is the constraint the square enforces by keeping like under like. The same formulation is still recognisable centuries later in Arabic sources and then in the Italian abbacus books — the merchants’ arithmetic of Fibonacci’s Liber abaci (1202) and the reckoning schools that followed it. And it stayed the end of the syllabus rather than the beginning: asked in December 1859 to describe his own schooling, Abraham Lincoln wrote that he could “read, write, and cipher to the Rule of Three; but that was all.”

The rise that gets you back is not the fall you took 🖖

A 20% fall is not undone by a 20% rise, and the difference is not a rounding error — it is a curve that turns upward and becomes vertical. At a 50% fall you need a 100% gain to recover; at an 80% fall you need 400%; at a 90% fall you need 900%. The loss scales linearly, but the effort to recover from it runs away to infinity.

Problems solved in full

What 7 cost when 3 apples cost 2.4 5 steps
3 apples cost 2.4. Work out what 7 cost — and then work out what you had to assume to do it.
1. Both rows of the square describe the same rate, and that is the entire claim being made. The top row is the pair you were given, 3 beside 2.4; the bottom row is the pair you want, 7 beside the unknown. So the fraction 3/2.4 and the fraction 7/D are the same number.
  
  $\frac{3}{2.4} = \frac{7}{D}$
2. Fractions are awkward to solve with, so clear them. Multiply both sides by 2.4 and by D at once and each denominator cancels on its own side. What survives is the cross: the two diagonals of the square multiply to the same thing.
  
  $\frac{3}{2.4} = \frac{7}{D} \times 2.4 D ⟹ 3 \times D = 2.4 \times 7$
3. Now the unknown is out of its fraction. It sits multiplied by the number diagonally opposite it, and the two numbers it needs are the ones on the diagonal that does not touch it.
  
  $D = \frac{2.4 \times 7}{3}$
4. 2.4 × 7 = 16.8, and 16.8 ÷ 3 = 5.6.
  
  $D = \frac{16.8}{3} = 5.6$
5. Do it once the long way as well, because it shows what the cross rule really is. 2.4 ÷ 3 = 0.8 is the price of one apple, and 7 × 0.8 = 5.6. The cross is the unit rate with the division postponed to the end, which is why it also works when the unit rate is an ugly number.
  
  $\frac{2.4}{3} = 0.8 ⟹ 7 \times 0.8 = 5.6$
Answer

The grid prints 5.6. The arithmetic is the easy half; what it rests on is not. Cross-multiplying assumes the cost is a fixed multiple of the count — that the rate is the same at 3 as at 7, and that 0 apples cost 0. Price the same apples at 0.6 for the bag plus 0.6 each and 3 still cost exactly 2.4, so the question looks untouched, but 7 now cost 4.8 and the grid is 0.8 too high. One line catches it: double an amount and check whether the other side doubles. Fixed charges are everywhere — a delivery fee, a standing charge, a minimum fare — and every one of them breaks the square while leaving the wording of the question completely unchanged.

$c = 0.8 n ⟹ c (7) = 5.6; c = 0.6 + 0.6 n ⟹ c (3) = 2.4, c (7) = 4.8$
A 20% rise and a 20% fall applied to 200 5 steps
200 goes up by 20%, then down by 20%. The rise writes 120 against 100, so it lands on 200 × 120 ÷ 100 = 240. Work out where the fall leaves it, and why the answer is not 200.
1. The bottom row is the change written as a ratio against 100, and this is where the corner gets filled in wrongly. A 20% fall is 80 against 100, not 20, because what remains still holds 80% of what was there. Put 20 in that corner and the square returns 48, which is the size of the cut and not the new price.
  
  $100 - 20 = 80 ⟹ \frac{A}{240} = \frac{80}{100}$
2. With 80 in place the cross runs exactly as it did for the apples: the new value times 100 equals 240 times 80.
  
  $A \times 100 = 240 \times 80 = 19200$
3. Divide by the 100: 192, which is 8 below the number the whole thing started from.
  
  $A = \frac{19200}{100} = 192, 200 - 192 = 8$
4. The cause is that the two percentages are measured against different amounts. The rise added 40, which is 20% of 200. The fall removed 48, which is 20% of 240. Equal percentages, unequal money — and the bigger one was always going to be the subtraction, because it was taken from the bigger number.
  
  $240 - 200 = 40 = 0.20 \times 200, 240 - 192 = 48 = 0.20 \times 240$
5. As multipliers it is one line, and the line explains more than the grid does. 1.2 × 0.8 = 0.96, and multiplication does not care about order, so falling first and rising afterwards also lands on 192.
  
  $\frac{120}{100} \times \frac{80}{100} = 1.2 \times 0.8 = 0.96, 200 \times 0.96 = 192$
Answer

The grid prints 192. Behind it is a piece of algebra worth memorising: (1 + p)(1 − p) = 1 − p², so a rise and an equal fall always leave you short by p² of the original, whichever way round they happen. Here p = 0.2, the loss is 0.04 × 200 = 8, and it is small enough to look like rounding. At p = 0.5 the same round trip costs 25% of everything. The fall that genuinely undoes a 20% rise is not 20% at all — undoing means dividing by 1.2, and 1 ÷ 1.2 = 0.8333, a fall of 16.67%. That is the same square with the unknown moved to the before-corner: 240 × 100 ÷ 120 = 200.

$(1 + p) (1 - p) = 1 - p^{2}; \frac{1}{1.2} = 0.8333 ⟹ - 16.67%$

Learning path

One number for many: summary statistics

Leads to Weighted average the asymmetry of percentage changes: a 50% fall requires a +100% gain to recover.

References (5)

Why the square works: Euclid, Elements, Book VI, Proposition 4 — "In equiangular triangles the sides about the equal angles are proportional." David E. Joyce's edition, Clark University.
Brahmagupta and the transmission: Jens Høyrup, "Sanskrit-Prakrit Interaction in Elementary Mathematics as Reflected in Arabic and Italian Formulations of the Rule of Three — And Something More on the Rule Elsewhere", in Selected Essays on Pre- and Early Modern Mathematical Practice. Springer, 2019, pp. 131–156.
The date of the Brāhmasphuṭasiddhānta: J. J. O'Connor and E. F. Robertson, "Brahmagupta", MacTutor History of Mathematics Archive, University of St Andrews.
Merchant arithmetic in Latin Europe: Laurence Sigler (trans.), Fibonacci's Liber Abaci: Leonardo Pisano's Book of Calculation. Springer, 2002. ISBN 978-0-387-40737-1.
The Lincoln quotation: Abraham Lincoln, autobiography written for Jesse W. Fell, 20 December 1859 — "I could read, write, and cipher to the Rule of Three; but that was all."

By the 42Math team · Updated 2026-08-11

Reviewed against published sources. Corrections welcome. Editorial policy · Corrections

All Mathematics lessons →

Example problems

7 apples - 3 apples cost 2.4, so what do 7 cost? Circle D, and the grid hands you 7 × 2.4 ÷ 3 = 5.6.
20% of 250 - 20% of 250. Circle A and read it off: 250 × 20 ÷ 100 = 50. The 100 in the corner is the whole that the 20 is a share of.
A 20% rise - 200 rises by 20%. The change becomes 120 against 100, and A comes out at 200 × 120 ÷ 100 = 240.
Working backwards - A price is 240 after a 20% rise — what was it before? Circle B: 240 × 100 ÷ 120 = 200. Taking 20% off 240 gives 192, and that is the wrong answer to a different question.
Then a 20% fall - Now drop that 240 by 20%: 240 × 80 ÷ 100 = 192. A 20% rise and then a 20% fall leaves you at 192, not back at the 200 you started from.
Recovering a 50% fall - A 200 price falls by 50% to 100. Getting back to 200 takes a 100% rise (+100.0%), not 50% — double the percentage lost.