Sources checked 29 October 2026. The processing company and all quality figures are hypothetical. The projection is arithmetic under stated assumptions, not a forecast of improvement or investment returns.
At a hypothetical processing company, a manager proposes reducing the share of faulty units by 10% each quarter. The presentation promises a 40% reduction after four quarters. The quality analyst asks which amount each quarterly percentage applies to.
The starting batch has 1,000 faulty units among 10,000 inspected: a 10% faulty-unit rate. If every improvement removes 10% of the rate remaining from the preceding quarter, the first change takes it to 9%. The second starts from 9%, so it takes the rate to 8.1%, not 8%.
That small distinction changes the annual commitment.
Carry the new base into the next period
Compounding applies each period's proportional change to the updated amount. The SEC's Investor.gov glossary explains compound interest as interest on both the principal and accumulated interest. [1] The same multiplication can describe an assumed sequence of reductions. It does not tell us whether a factory can achieve them.
For four reductions of 10%, multiply by 0.9 four times. Starting at 10%, the faulty-unit rate becomes 9%, 8.1%, 7.29%, and finally 6.561%. The reduction relative to the starting rate is 34.39%. The change in the rate itself is 3.439 percentage points. Those are different quantities.
OpenStax distinguishes repeated multiplication from adding the same amount each period. [2] A promise to remove one percentage point per quarter would take 10% to 6% after four quarters. That is the 40% relative reduction in the slide, but it requires progressively larger proportional improvements as the remaining rate falls. It is a different plan.
Four-period view. Each quarter starts from the preceding quarter's rate: 10% → 9% → 8.1% → 7.29% → 6.561%. At a constant 10,000 units per quarter, the final projected count is 656.1 faulty units.
Hypothetical example: a 10% relative reduction each quarter. At 10,000 inspected units in the final quarter, a 6.561% rate corresponds to 656.1 modeled faulty units. The fractional count is an expectation, not an observed unit count.
Keep the denominator attached
The quantity being reduced needs a precise name. Here it is the share of units with at least one fault, not the number of separate faults. NIST defines the proportion nonconforming as nonconforming items divided by total items. [3] A unit with several faults still counts once in that proportion.
At 10,000 units, a projected rate of 6.561% corresponds to 656.1 faulty units. The fraction is a modeled expected count, not a claim that someone will find a tenth of a unit. Actual counts are whole numbers and vary; rounding each intermediate calculation would also change the arithmetic.
If production reaches 20,000 units in the final quarter, the same rate corresponds to 1,312.2 faulty units. The rate has improved while the projected count exceeds the starting 1,000. A repair team may care about that count; a process comparison may care about the rate. Record both before promising less downstream work.
Comparability matters beyond batch size. A change in product mix, inspection coverage, or the definition of a fault could change the observed rate. The calculator assumes those definitions remain consistent. It cannot establish that the underlying process improved.
Let the quarterly assumptions differ
There is no need to repeat a rate just because a spreadsheet makes it convenient. Suppose the relative reductions are 10%, 5%, zero, and 20%. The successive rates are 9%, 8.55%, 8.55%, and 6.84%. The overall relative reduction is 31.6%, rather than the 35% obtained by adding the four reductions.
An increase and an equal percentage decrease also fail to cancel. A 20% increase from 10% gives 12%; a subsequent 20% decrease gives 9.6%. The two changes use different bases. Their combined result is a 4% relative decrease from the starting rate.
Try these cases in the projection playground. Enter a starting rate and a signed relative change for each quarter. Negative changes reduce the rate; positive changes increase it. Change production volume separately to see its effect on the projected count. Each row retains the preceding value so you can inspect the multiplication.
A zero starting rate stays zero under purely proportional changes. A 100% reduction also reaches zero, which later multiplications cannot raise. A real process could develop new faults after that; the formula would then need a mechanism for adding them. The tool also rejects a projected rate above 100%, because a share of faulty units cannot exceed all units inspected.
Ask what will produce each improvement
The arithmetic can be correct while the plan is weak. The first quarter might remove a recurring setup error, while later quarters face different causes. A constant reduction assumes the team can keep finding changes that deliver it. Neither the interest definition nor the algebra text supplies evidence for that assumption.
This is where the earlier discussion of second-order consequences becomes useful. Fewer faulty units might release repair time, while higher production adds work elsewhere. Trace those mechanisms separately; a clean projection does not establish their effects.
Take ten minutes with the proposed improvement slide. Write the starting numerator and denominator, then calculate all four periods using the stated rates. Label the relative change and percentage-point change separately. Before approving the annual target, ask the owner of each quarter's change to name the work, the evidence, and the review date that support it.
Sources
[1] U.S. Securities and Exchange Commission, Investor.gov, “Compound Interest”. Definition of interest on principal and accumulated interest. The quality example is an original mathematical application, with no investment advice or return prediction.
[2] OpenStax, College Algebra 2e, section 6.1, “Exponential Functions”. Explains multiplicative versus additive change. Varying quarterly multipliers here are evaluated one period at a time; no constant real-world improvement law is claimed.
[3] NIST/SEMATECH, e-Handbook of Statistical Methods, section 6.3.3.2, “Proportions Control Charts”. Used for the numerator and denominator of the proportion nonconforming. The playground does not implement a control chart or statistical test.
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