Scope
Build decisions in an ARPG are proportional decisions. An item is not good in isolation; it is good relative to what it replaces, relative to what the character already holds, and relative to the alternatives competing for the same slot. Three questions cover the whole space:
- Relative change — a quantity moved from one value to another. By how much?
- Proportion of a whole — part of a quantity is removed. How much remains?
- Comparison — two candidate changes. Which is larger?
These are distinct operations, and the most common source of error is not arithmetic but misidentification — answering one when the situation calls for another. All three are printed with the same percent symbol, which conceals the difference. The sections below treat each separately, then interleave them.
1 · Relative change
A quantity moves from an initial value to a final value. The relative change is the difference, expressed as a proportion of the initial value.
The initial value is defined as 100%. The final value is then read in those units, and the relative change is the excess over 100.
Fixing the baseline before computing anything is the operative step. The difficulty in this class of problem is rarely the division; it is that the baseline is frequently unstated, and two quantities in the same sentence can both plausibly serve as it.
1.1 Percentage points and percent
Two distinct quantities share the percent symbol. Separating them resolves most confusion in this area.
| Percentage points | Percent change | |
|---|---|---|
| Describes | An amount added to a running total | A quantity's movement relative to its own prior value |
| Example | A passive granting +50% increased damage |
The resulting change in damage output |
| Measured against | The base, by definition | The current total |
| Is it the answer? | No — it is an input | Yes |
A modifier's printed value is always in percentage points. The quantity of interest is always a percent change. The two coincide only on a character with nothing invested; thereafter they diverge, and the divergence widens monotonically.
1.2 Implicit baselines
Some quantities display their baseline; some do not. This determines how much work is required before dividing.
| Situation | Baseline | Stated? |
|---|---|---|
| Weapon at 210 DPS replaced by one at 240 | 210 | Yes |
| Maximum life 2,400 raised to 2,700 | 2,400 | Yes |
| An increased damage total of 300% | 400 | No |
The third case requires care. A total of accumulated increased modifiers is not itself the quantity being scaled — it is an addition to a base that is always present and never printed. A total of 300% denotes a multiplier of 1 + 3.00 = 4.00, and the multiplier is what moves.
baseline = 100 + 300 = 400 # not 300
change = 50 ÷ 400 = 12.5%
The invisible hundred is the most consequential digit in the system. It is never displayed, it is always present, and omitting it inflates every estimate.
1.3 The ceiling property
Because the implicit base is always present, the denominator of a total calculation is never below 100. This produces a hard bound requiring no computation to apply.
A modifier's printed value is an upper bound on its effect, attained only when the relevant total is zero.
A modifier reading +15% cannot produce more than a 15% improvement under any circumstances. Any candidate answer exceeding the printed figure is wrong and may be discarded without further work.
1.4 Bracketing
Two reference points bound the majority of real cases:
| Total | Baseline | Effect of +15% | As a fraction of printed |
|---|---|---|---|
| 0% | 100 | 15.0% | all of it |
| 100% | 200 | 7.5% | one half |
| 300% | 400 | 3.75% | one quarter |
| 500% | 600 | 2.5% | one sixth |
The pattern generalises: at a total of 100% a modifier is worth half its printed value, at 300% a quarter, at 500% a sixth. The baseline grows linearly while the return falls as its reciprocal — which is why investment in a single total flattens without ever stopping.
1.5 Estimation by counting
The division may be replaced by a count, which is frequently easier to perform mentally.
If the added amount fits into the baseline N times, the relative change is approximately 100 ÷ N percent.
| Fits | Change | Fits | Change |
|---|---|---|---|
| 2× | 50% | 8× | 12.5% |
| 3× | 33% | 10× | 10% |
| 4× | 25% | 20× | 5% |
| 5× | 20% | 25× | 4% |
Counting how many times one quantity fits into another is a different operation from division, and for most people a cheaper one. It applies to stated and implicit baselines alike, provided the correct baseline is identified first.
2 · Proportion of a whole
The second question involves no movement and no comparison. A single quantity is divided into a part that survives and a part that does not.
Resistance denotes the portion removed. The portion arriving is its complement: 40% resistance means 60% lands.
This is structurally simpler than relative change. It causes difficulty through proximity — it appears in the same sentences, uses the same symbol, and is often computed immediately before or after a relative-change calculation on the same numbers.
2.1 Stacked proportions multiply
Where several independent reductions apply to one quantity, their surviving fractions multiply. They do not sum.
summed (incorrect) 40 + 30 = 70 removed → 30% lands
multiplied 0.60 × 0.70 → 42% lands
Each layer removes a fraction of whatever reached it, not a fraction of the original. Two consequences follow: independent defensive layers never fully eliminate a threat, and they never become redundant either — a second layer applied to an already-reduced quantity still removes its stated fraction of the remainder.
2.2 Marginal value near a cap
The following is the clearest available demonstration that the two question types must be kept apart.
Consider raising a resistance from 60% to 75% — fifteen percentage points. The intuitive reading is that this is a modest improvement. The correct reading applies question one to the quantity that matters, which is not the resistance but the damage arriving.
| Resistance | Portion resisted | Portion landing |
|---|---|---|
| 60% | 60 | 40 |
| 75% | 75 | 25 |
Incoming damage falls from 40 units to 25. Applying the baseline convention: the baseline is 40, the reduction is 15, and 15 ÷ 40 = 37.5%.
Fifteen percentage points of resistance, applied near the cap, reduce damage taken by 37.5% — equivalently, they allow the character to absorb 60% more of that damage type before dying.
Both figures state one fact from either side. Damage taken falls to 25 ÷ 40 of its former value; survivability rises by 40 ÷ 25 = 1.6. Each follows from measuring the change against what lands rather than against the resistance figure.
The same fifteen points applied lower on the scale are worth considerably less. From 0% to 15%, the landing portion falls from 100 to 85 — a reduction of 15 ÷ 100 = 15%. Identical printed value, under half the effect.
This inverts the pattern of section 1.4. There, later investment was worth less because the baseline was growing. Here later investment is worth more, because the baseline being divided into is shrinking. The arithmetic is identical; only the direction of the baseline differs.
Resistance is among the most commonly under-invested statistics in ARPG characters, and this asymmetry explains why. The final points before a cap are the most valuable available, while presenting on the character sheet as the smallest increments. Defence is treated fully in a later lesson; the arithmetic above is the whole of the argument.
3 · Comparison
Two candidate changes compete for one resource — a passive point, a gear slot, a currency item. Which produces the larger effect can usually be settled without computing either.
Of two equal additions, the one entering the smaller total produces the larger effect. Rank by baseline, not by printed value.
Where the additions are unequal the comparison remains tractable: an addition twice as large, entering a total more than twice as full, is the weaker of the two. Precision is rarely required, since the decision needs only a ranking.
The practical consequence is that no statistic is inherently better than another. The same modifier is strong on one character and negligible on the next, determined entirely by prior investment. Claims of the form "attack speed is better than damage" are neither true nor false — they are underspecified until both baselines are known.
4 · Mixed practice
The three question types interleaved. Identification precedes calculation: determining which question is being asked is the operative skill, and it is the step that fails in practice.
Summary
| Question | Method | Principal hazard |
|---|---|---|
| Relative change | Define the initial value as 100%; read the final value against it | An implicit baseline — a total of n% has a baseline of 100 + n |
| Proportion of a whole | Take the complement; multiply independent layers | Summing reductions that in fact multiply |
| Comparison | Rank by baseline; the smaller baseline wins the equal addition | Ranking by printed value instead |
Two properties worth retaining
- A modifier's printed value is an upper bound on its effect. Any answer exceeding it is wrong.
- Additions to a growing total fall in value; reductions approaching a cap rise in value. Both follow from the same division applied to a baseline that is respectively growing and shrinking.
No formulas are listed. Each method above is recoverable from its statement, and a method that cannot be recovered from its statement is not yet usable.