Propagating Errors: Sums & Differences

Take two thermometer readings in a calorimetry experiment: the water starts at $35.0 \pm 0.5\ ^\circ\mathrm{C}$ and ends at $38.0 \pm 0.5\ ^\circ\mathrm{C}$ . Each reading is respectable — half a degree of doubt on thirty-odd degrees is barely 1.4%. Now subtract them to get the temperature rise, the number your whole heat calculation depends on: $3.0\ ^\circ\mathrm{C}$ . What is its uncertainty? Not $\pm 0.5$ . Not zero because "the errors cancel". It is $\pm 1.0\ ^\circ\mathrm{C}$ — a full 33% of the answer. Two measurements you trusted to one part in seventy have manufactured a result you can barely trust to one part in three. That silent amplification is this module's entire subject. When measured quantities are added or subtracted, their absolute uncertainties always add — in the worst case errors gang up, never cancel, and the worst case is what you must quote. JEE-Main and NEET setters build whole questions out of the one counterintuitive fact that $\Delta(a-b) = \Delta a + \Delta b$ , and CIE A-Level practical papers award or withhold marks on exactly this bookkeeping. You already speak the $\Delta x$ language from the errors module. Here you learn the first — and most misused — rule of combining errors, see why subtraction of nearly equal numbers is the great destroyer of precision, and drill the exam patterns it generates.

Key takeaways

When measurements combine by addition or subtraction, their absolute uncertainties add: $\Delta(a+b) = \Delta(a-b) = \Delta a + \Delta b$ . The sign of the operation never touches the sign of the errors.
The rule is worst-case (maximum error) reasoning: both inputs can sit at the unlucky ends of their ranges simultaneously, and you are not entitled to assume they don't.
For many terms, $Z = a + b - c + \dots$ has $\Delta Z = \Delta a + \Delta b + \Delta c + \dots$ — every term contributes, whatever its sign.
Subtracting nearly equal quantities is catastrophic: the absolute uncertainty stays the same while the value shrinks, so the relative error explodes by a factor of roughly $(a+b)/(a-b)$ .
Quote the result like a professional: uncertainty to 1 significant figure, value cut at the same decimal place, units on both.
Key idea: uncertainty measures ignorance, and ignorance accumulates — absolute errors add under both + and −, so a difference is always the most fragile number in your experiment.

The core insight

For sums and differences the absolute uncertainties of the inputs add — $\Delta(a \pm b) = \Delta a + \Delta b$ — because in the worst case both measurements err in the directions that push the result the same way; subtraction shrinks the value but never the uncertainty, which is why small differences of large numbers carry huge relative errors.

Where this lives today

The most expensive subtractions in science are running right now. The Gaia spacecraft measures stellar parallax — the same difference-of-angles Bessel fought for — at the ten-microarcsecond level, and its catalogue is really two columns: the value and the propagated uncertainty, star by star. Cosmology's "Hubble tension" is at bottom an argument about whether two error budgets (one anchored in parallax differences) genuinely fail to overlap. And LIGO detects gravitational waves by comparing two 4-km arm lengths to within $10^{-18}\ \mathrm{m}$ — a difference measurement so extreme that the whole instrument exists to cancel every noise source common to both arms before it reaches the subtraction. The same logic runs your daily technology: GPS positions come from differences of signal arrival times, so atomic-clock uncertainties propagate through those differences into the metres of your blue dot — while in every school lab, the humble $\theta_{\text{final}} - \theta_{\text{initial}}$ quietly demonstrates the whole subject.

The physical picture

Stop thinking of a measurement as a number. Writing $a \pm \Delta a$ claims the true value lives somewhere in the strip from $a - \Delta a$ to $a + \Delta a$ , best guess at the centre. That picture — measurements as intervals, not points — makes the propagation rule almost obvious. Combine two territories. If $A$ can roam its strip and $B$ can roam its strip independently, then the sum $A + B$ can reach its highest value when both roam high, and its lowest when both roam low. The strip for the sum is therefore wider than either input strip — its half-width is exactly $\Delta a + \Delta b$ . Now flip the sign: for $A - B$ the extremes happen when $A$ roams high while $B$ roams low, and vice versa. Different corners of the worst case, same total width: $\Delta a + \Delta b$ again. The operation on the central values changed; the accumulation of doubt did not.

Figure

Measurements are intervals — and the worst case sets the width of the result

One more thing the picture shows: the width of the result's strip is fixed by the inputs, but how much it hurts depends on the central value. A strip of $\pm 1.0$ centred on $73.0$ (the sum of our readings) is precise; the same width centred on $3.0$ (their difference) is fog. That asymmetry makes differences dangerous and sums forgiving — and it is where this module's hardest exam questions come from.

The rule, stated properly

Everything rests on one convention, inherited from the errors module: at HS and JEE level, uncertainties combine by the maximum-error rule. You assume the least favourable alignment of errors that the stated ranges allow, so the quoted $\Delta Z$ is a guarantee, not a hope. For a sum this reads:

Z = A + B \implies Z = (a + b) \pm (\Delta a + \Delta b) \qquad (1)

where: $a, b$ — the central (best) values of the two measurements; $\Delta a, \Delta b$ — their absolute uncertainties, in the same units as the values themselves. Read it physically: doubt is a length, and laying two doubtful lengths end to end lays their doubts end to end too. If one input is exact ( $\Delta b = 0$ ), the sum simply inherits $\Delta a$ — an exact constant never adds fog. For a difference the central values subtract but the uncertainties still add:

Z = A - B \implies Z = (a - b) \pm (\Delta a + \Delta b) \qquad (2)

where: the symbols are as before — and the plus sign between $\Delta a$ and $\Delta b$ is not a typo. The worst case for $A - B$ is $A$ high and $B$ low (or the mirror image), and each of those misfortunes stretches the result in the same direction. Subtraction is a promise about central values only; it offers no discount on doubt. The extension to any chain of additions and subtractions follows the same worst-case logic term by term:

Z = a + b - c + d - \dots \implies \Delta Z = \Delta a + \Delta b + \Delta c + \Delta d + \dots \qquad (3)

where: every $\Delta$ enters with a positive sign regardless of the sign of its term in $Z$ . A perimeter measured as four separate sides carries four uncertainties; a five-term energy balance carries five. Uncertainty grows linearly with the number of measured pieces — one reason good experimenters measure a whole directly rather than as a sum of parts when they can. Sums and differences work in absolute uncertainties; products and quotients (next module) work in relative ones. Exams test whether you reach for the right currency — anchor the split now:

operation	what adds	currency
$Z = a + b$ or $a - b$	$\Delta Z = \Delta a + \Delta b$	absolute uncertainties
$Z = ab$ or $a/b$	$\dfrac{\Delta Z}{Z} = \dfrac{\Delta a}{a} + \dfrac{\Delta b}{b}$	relative (fractional) errors
$Z = a^n$	$\dfrac{\Delta Z}{Z} = n\,\dfrac{\Delta a}{a}$	relative, scaled by the power

The second and third rows are the sequel module's business; they sit here so the contrast is burned in before you ever need it.

Derivation AWhy absolute uncertainties add for both sums and differences

We want to prove that if $A = a \pm \Delta a$ and $B = b \pm \Delta b$ , then both $A + B$ and $A - B$ have absolute uncertainty exactly $\Delta a + \Delta b$ under the maximum-error convention. The strategy is pure interval arithmetic: write down the full range each quantity can occupy, push the ranges through the operation by choosing the extreme ends, and read off the half-width of the resulting interval. The thing to watch is which ends of the input ranges produce the extremes — they differ between the sum and the difference, but the resulting width does not.

Step 1. State the territories. $A$ lies somewhere in $[\,a - \Delta a,\; a + \Delta a\,]$ and $B$ lies in $[\,b - \Delta b,\; b + \Delta b\,]$ , with the two roaming independently — nothing ties $A$ 's error to $B$ 's.

Step 2. Take the sum. The largest possible value of $A + B$ needs both inputs at their tops: $(a + \Delta a) + (b + \Delta b) = (a + b) + (\Delta a + \Delta b)$ . The smallest needs both at their bottoms: $(a + b) - (\Delta a + \Delta b)$ . Why both at once: independence means no law of nature forbids the thermometer and the ruler from both erring high on the same afternoon. Maximum-error accounting must allow it.

Step 3. Read off the half-width. The sum's interval is centred on $a + b$ and extends $\Delta a + \Delta b$ each way, so $\Delta(A+B) = \Delta a + \Delta b$ .

Step 4. Take the difference. The largest value of $A - B$ occurs with $A$ high and $B$ low: $(a + \Delta a) - (b - \Delta b) = (a - b) + (\Delta a + \Delta b)$ . The smallest occurs with $A$ low and $B$ high: $(a - b) - (\Delta a + \Delta b)$ . Why the ends swap: subtracting a small $B$ helps $Z$ just as much as having a large $A$ — the minus sign recruits $B$ 's downward error into an upward push on the result.

Step 5. The difference's interval is centred on $a - b$ with the same half-width $\Delta a + \Delta b$ . Both rules are one rule: under $\pm$ , doubts add. The multi-term form of equation (3) follows by applying this pairwise, one term at a time. Verified: with $a = 38.0, \Delta a = 0.5, b = 35.0, \Delta b = 0.5$ — sum extremes $38.5 + 35.5 = 74.0$ and $37.5 + 34.5 = 72.0$ , i.e. $73.0 \pm 1.0$ ; difference extremes $38.5 - 34.5 = 4.0$ and $37.5 - 35.5 = 2.0$ , i.e. $3.0 \pm 1.0$ . Both half-widths equal $0.5 + 0.5 = 1.0$ exactly.

The subtraction catastrophe

Equations (1) and (2) look symmetric, but their consequences are wildly asymmetric, because value and uncertainty move in opposite directions under subtraction. Add our two thermometer readings and the relative error is $1.0/73.0 \approx 1.4\%$ — as good as the inputs. Subtract them and it is:

\frac{\Delta Z}{Z} = \frac{\Delta a + \Delta b}{a - b} = \frac{0.5 + 0.5}{38.0 - 35.0} = \frac{1.0}{3.0} \approx 33\% \qquad (4)

where: the numerator is the (unchanged) combined absolute uncertainty and the denominator is the (shrunken) central value. The fraction is a ratio of a fixed fog to a collapsing signal: as $a - b \to 0$ the relative error grows without bound, even though nothing about your instruments got worse. Experimentalists call this catastrophic cancellation — the digits that agreed between $a$ and $b$ carried all the confidence, and subtraction threw exactly those digits away.

Go deeper · Derivation B · 1 minHow big the blow-up is — the amplification factor

Derivation BHow big the blow-up is — the amplification factor

We want a formula for how much subtraction amplifies relative error, so you can smell a catastrophe before computing it. The route: assume both inputs carry roughly the same relative error $r$ , express the difference's relative error in terms of $r$ and the values, and extract the multiplying factor. Watch for the appearance of the ratio $(a+b)/(a-b)$ — the "how close is the cancellation" dial.

Step 1. Let both measurements have relative error $r$ , so $\Delta a = r\,a$ and $\Delta b = r\,b$ .

Step 2. Apply the rule for a difference: $\Delta Z = \Delta a + \Delta b = r\,(a + b)$ .

Step 3. Divide by the central value $Z = a - b$ :

\frac{\Delta Z}{Z} = r \cdot \frac{a + b}{a - b}

The relative error of the difference is the input relative error times an amplification factor $(a+b)/(a-b)$ . Why this factor: $a + b$ measures how large the numbers are, $a - b$ how little survives the subtraction. Nearly equal inputs make the factor huge; well-separated inputs make it harmless (for $b \ll a$ it tends to 1 and the difference is as trustworthy as $a$ itself).

Step 4. Put in the thermometer numbers: $(38.0 + 35.0)/(38.0 - 35.0) = 73.0/3.0 \approx 24$ . An input relative error of about $1.4\%$ becomes $24 \times 1.4\% \approx 33\%$ — matching equation (4).

Step 5. Draw the experimental moral. If the factor is big, don't subtract two big measurements — measure the difference directly. A thermometer strapped to the calorimeter measuring the rise, an extensometer reading a wire's extension, a balance comparing two masses in one weighing: each replaces a catastrophic subtraction with a single small measurement whose $\Delta$ is small because the value is small. Verified: $r = 0.5/36.5 \approx 0.0137$ (using the mean input as the representative value); $0.0137 \times 73.0/3.0 = 0.0137 \times 24.33 \approx 0.333 = 33\%$ , agreeing with the direct computation $1.0/3.0$ exactly.

Figure

Catastrophic cancellation — same instruments, exploding relative error

Limiting cases

Case	Result	Meaning
$\Delta b = 0$ (exact constant)	$\Delta Z = \Delta a$	exact numbers never add fog — only measured inputs count
$b \to a$ with fixed $\Delta$ 's	$\dfrac{\Delta Z}{Z} \to \infty$	catastrophic cancellation: the value dies, the doubt survives
$b \ll a$ in $a - b$	$\dfrac{\Delta Z}{Z} \approx \dfrac{\Delta a + \Delta b}{a}$	well-separated values subtract safely — amplification factor near 1
$n$ equal terms, each $\pm\,\delta$	$\Delta Z = n\,\delta$	measuring a length as $n$ segments multiplies the uncertainty by $n$
independent random errors (statistics view)	$\sqrt{(\Delta a)^2 + (\Delta b)^2} \le \Delta a + \Delta b$	quadrature gives a smaller, statistically fair estimate; the max rule is the safe ceiling exams use

The last row previews university labs, where independent errors add in quadrature because random errors rarely conspire perfectly. The maximum rule is deliberately conservative — it never understates the doubt, which is why boards and JEE mark schemes standardise on it.

Common trap“Errors subtract when quantities subtract”

The single most common wrong answer in this topic. Students see $Z = a - b$ and write $\Delta Z = \Delta a - \Delta b$ — which is not just inaccurate but nonsense: it can produce zero or negative uncertainty, a claim of perfect knowledge manufactured from two imperfect measurements. Uncertainty is a measure of ignorance, and no arithmetic operation on your data can destroy ignorance. The subtraction in $a - b$ acts on the central values only; the error ranges of $A$ and $B$ are both still fully in play, and the worst-case alignment ( $A$ high, $B$ low) stretches the result by the sum of the half-widths. If your computed $\Delta$ ever came out smaller than the larger input $\Delta$ — let alone negative — you have used the phantom minus-sign rule. Physically sensible check: a result can never be known better than the muddier of the ingredients that made it.

Worked examples

Board level — two rods and the CIE quoting drill

A student measures two metal rods with a ruler of uncertainty $\pm 0.1\ \mathrm{cm}$ per reading: rod A is $23.5\ \mathrm{cm}$ , rod B is $18.2\ \mathrm{cm}$ . Find (a) the total length when the rods are placed end to end, (b) the difference in their lengths, and (c) the percentage uncertainty in each result. Quote all answers correctly.

Step 1. Identify the rule before touching numbers: both (a) and (b) are $\pm$ combinations, so absolute uncertainties add — equations (1) and (2). Each reading carries $\Delta = 0.1\ \mathrm{cm}$ .

Step 2. Total length: $L_+ = 23.5 + 18.2 = 41.7\ \mathrm{cm}$ , with $\Delta L = 0.1 + 0.1 = 0.2\ \mathrm{cm}$ . Quote: $\boxed{L_+ = 41.7 \pm 0.2\ \mathrm{cm}}$ — uncertainty already 1 significant figure, value ending at the same decimal place.

Step 3. Difference: $L_- = 23.5 - 18.2 = 5.3\ \mathrm{cm}$ , and the uncertainty is again $0.1 + 0.1 = 0.2\ \mathrm{cm}$ , not $0.1 - 0.1 = 0$ . Quote: $\boxed{L_- = 5.3 \pm 0.2\ \mathrm{cm}}$ .

Step 4. Percentage uncertainties expose the asymmetry. Sum: $0.2/41.7 \approx 0.5\%$ . Difference: $0.2/5.3 \approx \boxed{3.8\%}$ — nearly eight times worse from identical instruments, because the central value shrank while the fog did not.

Step 5. The sentence that earns the quality mark: "The difference has a much larger percentage uncertainty because the absolute uncertainties add while the value itself is small." CIE practical mark schemes reward exactly this recognition, and dock the answer $5.3 \pm 0.0$ or $\pm 0.2$ quoted as $\pm 0.20$ (two significant figures of doubt is a claim you cannot support).

Competitive level — series resistors and the temperature-rise ambush (JEE-Main/NEET pattern)

Two resistors $R_1 = 100 \pm 3\ \Omega$ and $R_2 = 200 \pm 4\ \Omega$ are connected in series. Find the equivalent resistance with its uncertainty. Then, in the same experiment, a thermometer with $\pm 0.5\ ^\circ\mathrm{C}$ per reading records a water bath warming from $35.0\ ^\circ\mathrm{C}$ to $38.0\ ^\circ\mathrm{C}$ ; find the percentage error in the temperature rise.

Step 1. Series resistance is a pure sum: $R_s = R_1 + R_2 = 300\ \Omega$ and $\Delta R_s = \Delta R_1 + \Delta R_2 = 3 + 4 = 7\ \Omega$ . Quote $\boxed{R_s = 300 \pm 7\ \Omega}$ (relative error $7/300 \approx 2.3\%$ — comparable to the inputs, as sums always are).

Step 2. Refuse the tempting wrong turns: $\sqrt{3^2 + 4^2} = 5\ \Omega$ is the quadrature answer — correct statistics, wrong convention for this exam; and adding percentage errors ( $3\% + 2\%$ ) is the product rule applied to a sum. The setters place both among the options deliberately.

Step 3. Temperature rise: $\Delta\theta = 38.0 - 35.0 = 3.0\ ^\circ\mathrm{C}$ with uncertainty $0.5 + 0.5 = 1.0\ ^\circ\mathrm{C}$ .

Step 4. Percentage error: $\dfrac{1.0}{3.0} \times 100 \approx \boxed{33\%}$ . Each reading was good to about $1.4\%$ ; the rise — the physically meaningful number — is good to one part in three. In a specific-heat calculation this 33% would pass straight into the final answer via the product rule downstream.

Step 5. MCQ triage: options like 1.4%, 2.8%, 17%, 33% test whether you divided the combined $\pm 1.0$ by the difference (correct), by one reading (17%), or recycled the input error (1.4%). Compute $\Delta Z$ first, then choose the denominator with your eyes open. Timing note: 90 seconds for both parts. The only decisions are "operation is ± so absolute errors add" (five seconds, made once) and the denominator in the percentage. Everything else is single-digit arithmetic — this question is free marks for the prepared and a trap-field for everyone else.

Exam traps

The phantom minus sign. Writing $\Delta(a-b) = \Delta a - \Delta b$ . It feels symmetric, it is always wrong, and one option in the MCQ will be built from it. Anchor: doubts add under both signs.

Paying in the wrong currency. Adding percentage errors for a sum or difference — that is the products rule. For $\pm$ , work in absolute uncertainties; convert to percent only at the end, if asked. (The reverse crime — adding absolute errors for a product — loses the same mark in the sequel module.)

Over-quoting the uncertainty. $\pm 0.2\ \mathrm{cm}$ , never $\pm 0.20\ \mathrm{cm}$ ; uncertainty to 1 significant figure and the value's last digit at the same decimal place. This is the reporting grammar from the significant-figures module, and CIE 9702 practical papers apply it mercilessly.

Mixed units feeding one $\Delta$ . A length in cm minus a length in mm: convert before adding uncertainties. $\Delta$ 's are lengths too — they convert with the same factor as the values.

Quadrature at the wrong moment. $\sqrt{(\Delta a)^2 + (\Delta b)^2}$ is the statistically honest combination for independent random errors and is standard in university labs — but JEE/NEET and board mark schemes use the maximum-error rule unless the paper says otherwise. Know both; deploy by context.

Not recomputing after a difference. In multi-part numericals, a difference computed early (an extension, a temperature rise, a mass loss) carries a large relative error that must be propagated forward, not quietly replaced by the raw instrument error. Setters chain questions precisely to catch this.

Origins · 1 minFriedrich Bessel and how this was discovered

Friedrich Bessel

1784–1846 · Königsberg

Bessel turned error bookkeeping into a precision weapon. A self-taught accountant's apprentice who recomputed the orbit of Halley's comet at 20, he became the first astronomer to measure the distance to a star — the 1838 parallax of 61 Cygni, a mere 0.314 arcseconds teased out as the difference between nearly equal angular positions taken months apart. The measurement lived or died on exactly this module's mathematics: the difference was tiny, the individual uncertainties were not, and only his obsessive error budgets — temperature, instrument flexure, observer bias — shrank the fog enough for the signal to stand clear. He had already shown in 1823 that each observer carries a systematic "personal equation" in timing star transits, turning a famous 1796 sacking at Greenwich into a lesson: errors have structure, and structure can be measured, budgeted and subtracted.

History

1796

— Astronomer Royal Maskelyne dismisses assistant Kinnebrook for transit timings 0.8 s from his own; the discrepancy is real and personal, not carelessness.

1823

— Bessel identifies the "personal equation": each observer carries a measurable systematic timing offset — errors themselves become objects of study.

1838

— Bessel publishes the parallax of 61 Cygni, 0.314″, the first stellar distance: a tiny difference of large angles defended by meticulous error budgets.

1997

— The Hipparcos catalogue delivers milliarcsecond parallaxes for 118,000 stars, every one an exercise in differencing with propagated uncertainties.

2022

— Gaia DR3 quotes parallaxes to tens of microarcseconds for over a billion stars; the error budget per star is the mission's core product.

Going further

Texts. J. R. Taylor, An Introduction to Error Analysis, Ch. 2–3 — the definitive treatment of both the maximum rule and quadrature, with the independence argument spelled out · CIE 9702 syllabus, Practical Assessment appendix (the quoting conventions applied here) · NTA NEET syllabus, Unit 1 "Errors in measurements" — the line this module's exam patterns serve · NCERT Physics Part 1, older editions' combination-of-errors section (the $100 \pm 3,\ 200 \pm 4$ series-resistor example is the classic).

Provenance. All rules are the standard maximum-error convention. A verifier should recompute: sum extremes $38.5+35.5 = 74.0$ , $37.5+34.5 = 72.0$ → $73.0 \pm 1.0$ (1.4%); difference extremes $38.5-34.5 = 4.0$ , $37.5-35.5 = 2.0$ → $3.0 \pm 1.0$ (33.3%); amplification $73.0/3.0 = 24.3$ and $0.0137 \times 24.33 \approx 0.333$ ; rods $23.5+18.2 = 41.7 \pm 0.2$ (0.48%) and $23.5-18.2 = 5.3 \pm 0.2$ (3.77%); resistors $300 \pm 7\ \Omega$ (2.33%) with quadrature alternative $\sqrt{9+16} = 5\ \Omega$ ; quiz items $20.0 \pm 0.5$ , $\pm 0.5$ , $300 \pm 7$ , $33\%$ , and $50.0-48.0 = 2.0 \pm 1.0 = 50\%$ .

Prerequisites. Absolute, Relative & Percentage Errors (the $\Delta x$ language and mean absolute error) · Accuracy, Precision & Uncertainty. Leads to. Propagating Errors: Products, Quotients & Powers (the relative-error rule and the max-percent-error staple) · Error-Question Patterns (the technique drill that turns both rules into 30-second MCQ answers) · Significant Figures, Rounding & Reporting (the quoting grammar used throughout).

Why do the errors add even when I subtract the quantities? It feels like they should cancel.

Cancellation would require the two errors to be correlated — the second measurement erring in the same direction as the first by the same amount. Independent measurements owe each other nothing: on any given day $A$ can read high while $B$ reads low, and $A - B$ then misses by the sum of both errors. The maximum-error convention quotes the widest miss the stated ranges permit, so the $\Delta$ 's add. Only when errors genuinely are shared — the same miscalibrated scale used for both readings, with the offset identical — does a subtraction cancel anything, and that is a systematic-error argument (from the accuracy-and-precision module), not a licence to subtract random uncertainties.

My lab manual says to combine errors as the square root of the sum of squares. Which rule is right?

Both, in their own jurisdictions. Quadrature — $\sqrt{(\Delta a)^2 + (\Delta b)^2}$ — is the statistically fair combination for independent random errors: perfect conspiracies are rare, so the typical combined error is smaller than the worst case ( $5\ \Omega$ rather than $7\ \Omega$ for our series resistors). The linear rule $\Delta a + \Delta b$ is the guaranteed ceiling and is the convention in HS boards, JEE, NEET and CIE mark schemes. Use the maximum rule in exams unless the paper explicitly says otherwise; expect to graduate to quadrature in university laboratories. Note the reassuring inequality: quadrature never exceeds the linear sum, so the exam rule is always the safe claim.

How many decimal places should I keep in the uncertainty itself?

One significant figure is the standard quote (two only when the leading digit is 1, e.g. $\pm 0.14$ ), and then the central value is cut at the same decimal place: $41.7 \pm 0.2$ , never $41.68 \pm 0.2$ or $41.7 \pm 0.23$ . The logic comes from the significant-figures module: uncertainty is an estimate of doubt, and claiming three digits of doubt pretends you know your own ignorance to a precision no school experiment supports. Mismatched endings — value quoted finer than the uncertainty, or coarser — are among the most reliably penalised slips in CIE practical papers.

The percentage errors of my inputs were about 1% each. How can the answer's percentage error be 33%? Did I make a mistake?

No mistake — you subtracted nearly equal values, and percentage error is not conserved through subtraction. The absolute uncertainties behaved themselves (they added to $\pm 1.0$ ); it is the central value that collapsed from ~36 to 3, and a fixed fog over a small value is a large fraction of it. The amplification factor is $(a+b)/(a-b) \approx 24$ for readings of 38 and 35, so ~1.4% inputs honestly produce a ~33% output. If that precision is unacceptable, the fix is experimental design, not arithmetic: measure the difference directly with an instrument whose whole range is the difference.

For Z = a + b − c, do I subtract Δc since c is subtracted?

No — all three add: $\Delta Z = \Delta a + \Delta b + \Delta c$ . Run the worst case: to push $Z$ as high as possible you want $a$ and $b$ high and $c$ low, and each misfortune is allowed by the stated ranges, so each contributes its full half-width. The sign of a term decides which end of its range hurts, never whether it hurts. Sanity check for any propagation formula: every $\Delta$ appears with a plus sign, and $\Delta Z$ is at least the largest single input $\Delta$ .

Why does an exam treat one thermometer measurement of a temperature change as two readings with two uncertainties?

Because physically it is two readings: you read the scale at the start and again at the end, and each act of reading carries the instrument's least-count uncertainty independently. So a rise measured with a least count of 1 °C is uncertain by $\pm 1$ from each reading — $\pm 2$ in the worst-case convention (read the stem; some papers define it otherwise). The same doubling hides in burette volumes, rulers laid against an object's two ends, and stopwatch start/stop. Count the readings, not the instruments.