Which time unit should you use?
This Upsilar workshop applies the elementary math in the TAGE MAGE Calculation section to travel and production scenarios. It does not predict what will appear on your test. Before calculating, note what each number measures: kilometers, liters, hours, or a fraction of a task.
At a speed measured in kilometers per hour, 45 minutes is 45/60 of an hour, or 0.75 hours. At 48 km/h for 45 minutes, the distance is therefore 48 × 0.75 = 36 km. Writing 48 × 45 would mix the rate’s hour with the time’s minutes. The result may look impressive while being meaningless.
For a flow rate in liters per minute, keep time in minutes if the data allow it. Convert only when doing so makes the model consistent. Also write down assumptions: constant speed, constant flow rate, no losses. If the rate changes, split the problem into periods instead of using one formula.
Why add the speeds when two travelers approach each other?
Diagram: A → ← B. The arrows point toward each other; both movements reduce the gap. In a chase in the same direction, only the difference between the speeds reduces the gap.
Deux cyclistes partent au même instant de deux points distants de 42 km et roulent l’un vers l’autre, à 18 km/h et 24 km/h. Après combien de temps se rencontrent-ils ?
Choose an answer.
The solution, step by step
1 hour.
The method in action
- In one hour, they close the 42 km gap by 18 + 24 = 42 km. The time is 42/42 = 1 hour.
- At the meeting point, they have traveled 18 km and 24 km, respectively; together, that accounts for the original 42 km.
- Seven hours would come from 42/(24 − 18), which models a chase in the same direction. Dividing by one cyclist’s speed would ignore the other cyclist’s movement.
How do you account for an outlet operating at the same time?
Diagram: inlet → tank → outlet. The net flow rate is the inflow minus the outflow. Subtract only during the period when both are flowing at the same time.
Une cuve vide de 180 litres reçoit 8 litres par minute. Une sortie évacue simultanément 2 litres par minute, à débit constant dès le départ. Combien de temps faut-il pour remplir la cuve ?
Choose an answer.
The solution, step by step
30 minutes.
The method in action
- Each minute adds 8 − 2 = 6 liters net. It takes 180/6 = 30 minutes.
- Check: in 30 minutes, 240 liters enter and 60 leave, leaving 180 liters.
- 22.5 minutes ignores the outflow; 18 minutes adds the rates as if both pipes were filling the tank. The direction of the arrows determines the sign.
The pitfall
Adding times instead of rates
A machine that completes a task in 6 hours produces one-sixth of a task per hour. A second machine that completes it in 3 hours produces one-third per hour. Together, they do not take 9 hours: their hourly outputs add, assuming the work is independent and can be divided.
The same principle prevents hasty speed averages. For two trips of equal distance, the travel times differ when the speeds differ. Calculate total distance and total time; their quotient gives the average speed.
Why express work as fractions?
Treat the entire task as one unit. This avoids inventing a volume or number of items not given in the problem.
Une machine réalise seule un lot en 6 heures, une autre en 3 heures. Elles travaillent ensemble à rythme constant, sans se gêner. Combien de temps faut-il pour réaliser un lot ?
Choose an answer.
The solution, step by step
2 hours.
The method in action
- Their combined output is 1/6 + 1/3 = 1/2 of a batch per hour. Producing one batch takes 1 ÷ (1/2) = 2 hours.
- In those two hours, the first machine completes 2/6 of the batch and the second completes 2/3; together they complete 1 whole batch.
- 4.5 hours is the average of the two individual times. It is even longer than the 3 hours needed by the faster machine alone, which contradicts the no-interference model.
Key takeaways
- Make the units consistent before calculating.
- Add or subtract rates according to the actual direction of the activities.
- Check your result by reconstructing the total quantity.