Rates, Work & Speed • Topic 2 of 4

Average Rate

The single most tested trap in this chapter: average speed is total distance divided by total time, never the plain average of the individual speeds — unless the times spent are equal. If you drive 120 miles at 40 mph and the same 120 miles back at 60 mph, you spend 3 hours out and 2 hours back, so the average is 240 miles / 5 hours = 48 mph, not 50. When the two distances are equal, there is a clean shortcut: the average is the harmonic mean 2ab/(a + b) of the two speeds. When instead the two times are equal, the plain average does work. The GRE loves to reward the test-taker who spots which case they are in. The same logic covers average price when you buy different quantities, or average production rate across shifts — always rebuild it as total amount over total time or total units.

✅ Solved examples

1. A cyclist rides 30 km at 15 km/h, then 30 km at 10 km/h. Find the average speed.
Time = 30/15 + 30/10 = 2 + 3 = 5 h for 60 km. Average = 60/5 = 12 km/h. (Harmonic mean: 2·15·10/25 = 12.)
2. Quantitative Comparison. A trip covers equal distances at 40 mph and 60 mph. Quantity A: the average speed. Quantity B: 50 mph. Which is greater?
Equal distances ⇒ average = 2·40·60/100 = 48 mph. So Quantity B (50) is greater. The plain-average 50 is the trap.
3. A driver travels for 2 hours at 50 km/h and 3 hours at 80 km/h. Find the average speed.
Distance = 100 + 240 = 340 km over 5 h. Average = 340/5 = 68 km/h.
4. A shopper buys 2 kg of apples at $3/kg and 3 kg at $4/kg. Find the average price per kg.
Total cost = 6 + 12 = $18 for 5 kg. Average = 18/5 = $3.60 per kg.

✏️ Practice — try these, take hints as needed

1. A car goes 60 km at 30 km/h and 60 km at 20 km/h. Find the average speed.
Find each time.
60/30 + 60/20 = 2 + 3 = 5 h.
120 km / 5 h.
24 km/h
2. Equal distances are covered at 20 mph and 30 mph. Find the average speed.
Equal distance ⇒ harmonic mean.
2·20·30/(20+30).
1200/50.
24 mph
3. A runner jogs 1 hour at 8 km/h and 2 hours at 5 km/h. Average speed?
Distances: 8 and 10.
Total 18 km over 3 h.
18/3.
6 km/h
4. A trip is half distance at 40 km/h and half at 60 km/h. Average speed?
Equal halves ⇒ harmonic mean.
2·40·60/100.
4800/100.
48 km/h
5. A factory makes 200 units in the first 4 hours and 360 units in the next 6 hours. Average rate per hour?
Total units = 560.
Total time = 10 h.
560/10.
56 units per hour

📝 Topic test — 8 questions

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