The question you opened
Two buys, one average price
You put 120 dollars into a fund at 60 dollars a share, then another 120 at 30 a share.
What did you pay per share on average, in dollars?
$40
The first 120 buys 2 shares and the second buys 4, so 240 dollars bought 6 shares. That is 40 dollars a share.
Count shares before counting prices. The first 120 dollars at 60 a share buys 2 shares. The second 120 at 30 a share buys 4. Altogether 240 dollars bought 6 shares, so the average cost is 240 over 6, which is 40 dollars a share. The tempting 45 is the average of the two prices, and it would be right only if you had bought the same number of shares at each.
You bought a fixed amount of money at each price instead, and a fixed sum buys more of the cheap thing. The low price therefore carries more weight in the average, and the answer is pulled toward it. The general rule is that averaging prices requires knowing the quantities. Total money divided by total quantity is always correct. The plain average of the prices is correct only when the quantities match. When the money matches instead, the result is the harmonic mean, which is always below the plain average.
This is the arithmetic behind buying a fixed sum at regular intervals, often called dollar cost averaging. Spending the same amount every month buys more units when prices are low and fewer when they are high, so the average cost comes out below the average price over the period. That is a genuine property of the method rather than a claim about timing the market.
The same trap appears in fuel, where filling a fixed number of dollars at two different prices per litre gives an average below the midpoint, and in payroll, where averaging two departmental salaries without weighting by headcount produces a figure nobody earns. Whenever an average is quoted, the useful question is what it was weighted by.
Technique: Divide total money by total shares