Read the receipt · worked lesson
Does 25% more SC mean a 25% lower unit cost?
No. At a 25% higher coin rate, cost per coin falls by 20%. The rate and its reciprocal use different starting quantities, so their percentage changes are not equal.
Written · Formula v1 · No operator prices
Made-up inputs. Real arithmetic.
More units per dollar and fewer dollars per unit describe the same receipts from opposite directions.
Comparison package
- Purchase price
- $20.00
- Premium coins (SC)
- 25
- SC per dollar
- 1.250
- Cost per SC
- $0.800
Reference package
- Purchase price
- $20.00
- Premium coins (SC)
- 20
- SC per dollar
- 1.000
- Cost per SC
- $1.000
Rate uplift against this reference: 25.0%. Teaching inputs, not a verified baseline or an offer recommendation.
Use the same two receipts
Suppose a normal package contains 20 SC for $20 and a comparison package contains 25 SC for $20. These are made-up inputs, not operator prices. The normal rate is 1.000 SC per dollar and the comparison rate is 1.250. Dividing 1.250 by 1.000 and subtracting one gives a 25.0% rate uplift.
Now turn both fractions around. Normal cost per SC is $20 divided by 20, or $1.000. The comparison cost is $20 divided by 25, or $0.800. The cost falls by $0.200 from a $1.000 starting point: a 20% reduction. Both descriptions are correct. They refer to different units and different denominators.
The reciprocal is the reason
A rate of 1.25 times as many coins per dollar has a reciprocal cost of 1 ÷ 1.25, or 0.8 times as many dollars per coin. The reduction is 1 − 0.8 = 0.2. In general, if rate uplift is u written as a decimal, the unit-cost reduction is u ÷ (1 + u), for a positive uplift.
For example, 50% more coins per dollar gives a unit cost two thirds as large: a reduction of about 33.3%, not 50%. Doubling the rate gives half the unit cost, a 50% reduction. These are mathematical transformations of the same inputs, not extra benefits that can be added together.
Name the denominator when you share a percentage
‘25% better’ is incomplete. ‘25% more SC per dollar than the entered normal package’ identifies what changed and what it was compared with. ‘20% lower effective cost per SC than that package’ states the inverse relationship. Keep the package prices and amounts with either statement so somebody else can reproduce it.
Do not combine a rate uplift and a unit-cost reduction into a total percentage. In this example, adding 25 and 20 to claim a 45% improvement counts one price relationship twice. Nor does either percentage describe the share of your purchase price that will ultimately be redeemed.
Check rounding before calling it a discrepancy
The checker calculates with fixed-point decimal arithmetic and rounds for display. A displayed rate is a shortened representation, so dividing two rounded rates by hand can differ from an uplift calculated before display rounding. Reproduce a comparison from the original prices and coin amounts, not from a copied rounded headline.
This is especially useful when comparing small prices or fractional premium coins. Record the actual cents and coin precision instead of rounding them to make the inputs convenient. If the input is ambiguous, resolve it from the source rather than treating a formatting guess as a price.
Lower purchase cost does not price the outcome
Effective cost per SC describes the purchase package. It does not establish a redemption value for that SC, subtract the cost of meeting playthrough, or predict what a session returns. A lower unit cost can coexist with conditions you would not accept. The checker leaves those conditions visible as context rather than inventing a combined value.
Try both receipts in the checker and read the rate, inverted cost and comparison together. Then inspect the current purchase terms separately. If no supported normal package exists, describe only the raw purchase rate. A transparent partial answer is more useful than a broad value judgment the inputs do not support.