What exactly does “two to three” mean?
FNEGE emphasizes translating a prompt into mathematical relationships. Ratios are a useful way to practice this skill. The situations below illustrate the models; they do not represent an official distribution of test topics.
If A and B are in a 2-to-3 ratio, write A = 2k and B = 3k. The total is 5k. So A is 2/5 of the total, not 2/3. The fraction A/B is indeed 2/3, but its denominator is B, not the sum A + B.
For a scale, first make the units match. At a scale of 1:50,000, one centimeter on the map represents 50,000 real centimeters, or 500 meters. A distance of 3 cm therefore represents 1.5 km. Area does not use the same factor: both dimensions are multiplied, so the factor is squared.
How do you divide a quantity according to a ratio?
Find the value of one share after counting all the shares. This step produces a clearer solution than cross-multiplying before defining the quantities.
Deux ateliers se partagent 84 litres de peinture dans le ratio 3 pour 4. Quelle quantité reçoit l’atelier correspondant aux 3 parts ?
Choose an answer.
The solution, step by step
36 liters.
The method in action
- There are 3 + 4 = 7 shares. One share is 84 ÷ 7 = 12 liters.
- The first workshop receives 3 × 12 = 36 liters; the second receives 4 × 12 = 48 liters.
- Both checks work: 36 + 48 = 84 and 36/48 = 3/4. Calculating 84 × 3/4 would give 63 liters by using the wrong base.
Why do concentrations need to be weighted?
In this exercise, assume the volumes are additive. This lets you divide the total amount of syrup by the total volume of the mixture.
On mélange 2 litres d’une boisson contenant 30 % de sirop et 3 litres d’une boisson contenant 10 % de sirop. Quelle est la proportion de sirop obtenue ?
Choose an answer.
The solution, step by step
18%.
The method in action
- The first drink contributes 2 × 0.30 = 0.60 liters of syrup. The second contributes 3 × 0.10 = 0.30 liters.
- The mixture contains 0.90 liters of syrup in 5 liters total. The proportion is 0.90/5 = 0.18, or 18%.
- The result is between 10% and 30%, and closer to 10% because the larger drink contains 10% syrup.
The pitfall
Averaging rates without considering their weights
The simple average of 30% and 10% is 20%, but that would apply here only to equal volumes. With 2 liters and 3 liters, it treats different contributions as if they were equal. First add the amount of syrup, then the drink volumes.
Likewise, two quantities are not automatically proportional just because they increase together. A price made up of a fixed subscription and a per-unit charge is not proportional to consumption. The factor must remain constant, and in a proportional model a zero quantity must correspond to a zero total.
What checks should you keep in your scratch work?
For a division, check the sum and the ratio. For a scale, check the units and direction of scaling. For a mixture with positive volumes, make sure the final rate stays between the starting rates.
If the result is wrong, return to the names of the quantities before redoing the arithmetic. Writing “liters of syrup” and “liters of drink” prevents you from dividing two values that do not play the intended roles. This translation discipline also helps when a problem adds a step or hides the total.
Key takeaways
- In a 3-to-4 ratio, the total has 7 shares.
- Make the units consistent before setting up a proportion.
- Weight a mixture by the amounts actually contributed.