What quantity is the percentage calculated from?
FNEGE places the Calculation subtest within elementary mathematics. This workshop practices an application of that knowledge without making claims about how often percentages appear on the test. The situations are fictional and explain the underlying mechanisms; these are not presented as past exam questions.
Taking 15% of a quantity means multiplying it by 15/100. But a 15% increase also retains the starting quantity: the result is 115/100 of that quantity, or a factor of 1.15. A 15% decrease instead gives 85/100, or 0.85.
Draw an arrow between values: starting value → factor → final value. With each new change, the previous final value becomes the new base. This makes two questions visible: how large is the change, and how much remains after it?
Do equal percentage increases and decreases cancel out?
No. A 20% increase followed by a 20% decrease multiplies the price by 1.20 × 0.80 = 0.96, for an overall decrease of 4%. The first exercise tracks how the base changes at each step.
Un abonnement coûte 250 €. Son prix augmente de 20 %, puis le nouveau prix diminue de 20 %. Quel est le prix final ?
Choose an answer.
The solution, step by step
€240, an overall decrease of 4%.
The method in action
- After the increase: 250 × 1.20 = €300. The next decrease applies to €300, so it equals €60. The final price is €240.
- In one expression: 250 × 1.20 × 0.80 = 250 × 0.96 = 240. The overall factor of 0.96 corresponds to a 4% decrease.
- The increase adds €50, and the decrease subtracts €60. The rates have the same absolute value, but their bases differ.
How do you work backward from a discounted price to the original price?
Divide the discounted price by 1 − t/100, where t is the discount percentage. This factor must be nonzero: after a 100% discount, a final price of zero is not enough to recover the original price. Adding the discount rate to the reduced price does not reconstruct the original base.
Après une remise de 25 %, un sac coûte 90 €. Quel était son prix avant la remise ?
Choose an answer.
The solution, step by step
120 €.
The method in action
- If P is the original price, 0.75 × P = 90. So P = 90 ÷ 0.75 = 90 × 4/3 = 120.
- Check: one quarter of 120 is 30. Subtracting 30 from 120 gives 90.
- To go from 90 to 120, the increase is 30/90 = 1/3, or 33⅓%. The reverse of a 25% decrease is therefore not a 25% increase.
The pitfall
Confusing percentage points with relative change
A rate rising from 20% to 25% increases by 5 percentage points. Its relative change is 5/20 = 25%. One measures the difference between two rates; the other compares that difference to the starting rate. The question’s wording determines which one to calculate.
Adding rates may be valid when the changes explicitly use the same base. That is generally not true for successive changes. Before choosing an operation, write out “percentage of what?” This question rules out many false shortcuts.
How can you check your result quickly?
After a discount, the final price must be lower than the original. If you work backward to the pre-discount price, the result should be higher. Then check it by applying the original rate to your answer; working forward is often easier than the reverse calculation.
To practice, change one variable at a time: the price, then the rate, then the direction of the question. Keep the same factor-based notation. This helps distinguish a calculation error from a base error, which requires a different correction.
Key takeaways
- Name the base for each percentage.
- Apply successive changes by multiplying their factors.
- To recover the starting value, divide by the nonzero factor; to check, work forward.