Word problems (motion, work, mixture)

Motion problems

Basic relation: S = v · t. If two of distance, speed and time are known, the third can be found.

Situation	Closing (or separating) speed
Towards each other	v₁ + v₂
In the same direction (chase)	v₁ − v₂
Downstream	v + u
Upstream	v − u

Here v is the speed in still water, u is the speed of the current. Meeting time = distance / closing speed.

• Average speed = total distance / total time.
• Average speed is not the arithmetic mean: if the speeds are 15 and 10 km/h, the average speed is 12, not 12.5.

Work problems

The whole job is taken as 1. If it is finished in t days, the work rate per day is 1/t.

• Working together: 1/t₁ + 1/t₂ = 1/t.
• If two workers finish in 6 and 3 days, together they do 1/6 + 1/3 = 1/2, that is, they finish in 2 days.
• Add the rates, not the times.

Mixture problems

If a solution has mass m and concentration p%, the amount of pure substance is m · p / 100.

1. Write the amount of pure substance in each solution
2. When mixed, the pure substances add up
3. Total pure substance / total mass = new concentration

• Pure substance and mass are conserved: m₁p₁ + m₂p₂ = (m₁ + m₂) · p.
• When water is added, the amount of pure substance does not change, the mass increases.

Examples

• 200 g of a 10% solution and 300 g of a 20% solution were mixed. Pure substance: 20 + 60 = 80 g, mass 500 g, concentration 80/500 = 16%.
• The distance between cities A and B is 120 km. Two cars set out towards each other at 50 and 70 km/h. They meet after 120 / 120 = 1 hour.