Check successive percentage changes: source rows, changing bases and recovery rates

Supply records rather than asking for a slogan

All three records and the wrong explanation below are original editorial examples, not captured model output, actual products, investments or invoices. A starts 100, falls 20%, then rises 20%. B starts 250, falls 10%, then rises 10%. C starts 80, rises 25%, then falls 20%. Ask for intermediate values, final values, changes relative to the originals and recovery rates from the final values.

The OpenStax discount explanation supplies the underlying percentage relationship; this article develops a chained-change checking exercise. Do not average row percentages when unequal original values determine a group total.

Inputs: A=100, then-20%, then+20%; B=250, then-10%, then+10%; C=80, then+25%, then-20%. For each row show every base, factor, intermediate and final value, net change from the original, and rate needed to recover the original from the final value. Then calculate the group-total change from original values. Do not average row percentages.

Separate the source, calculations and comparison questions

The ZIP contains source.csv, reference-results.csv, comparison-ledger.csv, wrong-added-rates.csv and bilingual instructions. Every row is fictional editorial material. Rates in source.csv are decimal fractions: −0.20 means −20%, not −0.20%. A, B and C are assumed to use the same additive unit for the total exercise.

item,initial,first_rate,second_rate
A,100,-0.20,0.20
B,250,-0.10,0.10
C,80,0.25,-0.20

Keep every intermediate value in the reference

A’s second rise uses 80 as its base, so it adds 16 and ends 96. Recovering 100 then requires 4÷96≈4.1667%. A different question asks how to recover immediately after the first drop to 80: that requires 20÷80=25%. These recoveries have different starting points.

B gives 250×0.9=225, then 225×1.1=247.5, a 1% net fall. C rises to 100 and then loses 20, returning to 80. Its rates are not symmetric:1.25×0.8=1.

Locate the first false step in a deliberately wrong draft

Suppose an invented draft says: “A subtracts 20 and adds 20, returning 100; B similarly returns 250; C’s rates add to+5%, giving 84.” Each claim treats the second change as though it used the original base. Repair the explanation by requiring that base at each step, rather than replacing only the last number.

Use the two inputs in the direction of the question

Open the Percentage calculator. Value A is the comparison value and Base value B is the denominator. The result includes both A as a share of B and the change from B to A. This tool checks one pair at a time; calculate the sequential values from the source first and keep the comparison ledger yourself.

1. 1
   For A’s net change put 96 in Value A and 100 in Base value B, then Generate result. The share is 96.00%; the change is −4.00%. Record the change line for the net-change question.
2. 2
   For recovery from the final 96, use A=100 and B=96. The change is 4.17% on the two-decimal display. For recovery immediately after the first fall, use A=100 and B=80: 25.00%.
3. 3
   Check B’s net pair 247.5/250 for −1.00% and C’s 80/80 for 0.00%. For the whole group use A=423.5 and B=430: −1.51% on display.
4. 4
   Retain the higher-precision reference calculations and write the displayed value into comparison-ledger.csv or your own copy. This panel shows the comparison result; it does not export the complete three-row working table.

These comparisons use positive nonzero bases. Display rounding does not create a new source value or validate zero, missing or mixed-unit denominators.

Reconstruct the group total from values

Original values total 430; final values total 423.5. The decrease 6.5 gives-6.5÷430≈-1.5116%. Averaging-4%,-1% and 0% gives about-1.6667%, which is not the actual combined change.

This assumes additive values in the same unit. Different units, populations or non-additive scores cannot simply be summed. Continue with percent versus percentage-point checks for report denominators and the weighted-average exercise for reconstructing totals.

Reverse the inputs once to expose a plausible wrong answer

For the question “What is A’s net change?”, deliberately enter 100 as A and 96 as B. The calculator correctly reports 4.17%, but it answers the recovery question. Reject it as the net-change answer, restore A=96 and B=100 and verify −4.00%. A working calculator cannot choose the intended denominator for you.

wrong-added-rates.csv is a separate editorial fault: it gives A 100, B 250 and C 84 by adding the rate labels. Recompute from source.csv for 96,247.5,80 and total 423.5. Do not repair it by merely changing the group percentage; all three source IDs and their intermediate bases must remain auditable.

Change the conditions to test understanding

Reverse A’s percentage order: rising 20% and then falling 20% still ends 96, because these factors commute. But “subtract a fixed 20, then add 20%” ends 96, whereas “add 20%, then subtract a fixed 20” ends 100. Mixing fixed amounts with percentages removes that order-independence.

For any positive starting value, a decrease by proportion r followed by an increase by r, with 0<r<1, has factor(1-r)(1+r)=1-r². This explains the failure to cancel without relying on the special starting value 100. It does not cover floors, caps or piecewise rules.

Accept the AI explanation by checking rows, changed starting points, recovery questions and the aggregate. Retain sufficient precision for checking and round only for the required display. A rounded display value is not a new original measurement.