Cube3D 立方体数理Maths · Primary school / Middle school
Why don’t two discounts add up?
A percentage always refers to a base amount. A first discount changes that amount before the next percentage is applied. In the experiment, the full width of each bar still represents the original amount, so the two shaded reductions can be compared directly.
Follow the base after each discount
Reset the experiment to 100 units, a first discount of 20% and a second discount of 25%. The first discount removes 20% of 100, leaving 80. The next discount removes 25% of 80, which is another 20 units, leaving 60. Compare the gold and blue sections: they are equal in width even though the rates are different.
The final amount is 60% of the original 100, so the total discount is 40%. Adding the rates gives 45%, which would leave 55 instead. The shortcut treats both reductions as percentages of the original 100, even though the second reduction is based on 80.
Multiply the fractions that remain
If the first and second rates as decimals are a and b, the fractions left are 1 − a and 1 − b. The fraction of the original left after both is (1 − a)(1 − b). Subtract this product from 1 to get the overall discount. For the starting example, 0.80 × 0.75 = 0.60, so 1 − 0.60 = 0.40.
Choose the two 50% preset. Half of 100 is 50, and half of the remaining 50 is 25. The result is a 75% overall discount, not 100%. Now use the sliders to swap 20% and 25%: the final amount still equals 60 because the factors commute, but the first reduction is now 25 and the second is 15.
Check your understanding
Start at 200 units. Apply 10% off and then another 10% off. Predict the final amount and the overall percentage before changing the sliders.
Show the explanation
The first reduction leaves 180; the second removes 18 and leaves 162. The two reductions total 38 units, or 19% of the original 200. One 20% discount would leave 160.
What the model assumes
The model uses an original amount of 1–500 whole units, and whole-percent discounts from 0% to 100%. It keeps exact amounts to four decimal places without intermediate rounding. Actual promotions can include fixed coupons, taxes or rounding rules. This is a mathematical model using abstract units. The experiment requires Family Plus or Teacher access; the note is public.