Nick and Liz were quoted $15,500, $18,800, $16,900, $19,900, $18,000 and $21,000 for a used car. What must a seventh quote be so that the mean of all seven prices equals the website average of $18,500?

Step-by-step solution

1. 1

   Turn the mean into a total. The mean is the total divided by the count, so the total is the mean times the count. Working with the total is the whole trick here — it converts an averaging question into simple subtraction:

   $\text{required total} = 18{,}500 \times 7 = 129{,}500$

2. 2

   Add the six quotes already in hand.

   $15{,}500 + 18{,}800 + 16{,}900 + 19{,}900 + 18{,}000 + 21{,}000 = 110{,}100$

   Grouping helps: $15{,}500+18{,}800 = 34{,}300$; $16{,}900+19{,}900 = 36{,}800$; $18{,}000+21{,}000 = 39{,}000$; and $34{,}300+36{,}800+39{,}000 = 110{,}100$.

3. 3

   Set up the equation for the missing quote. Let $x$ be the seventh price:

   $110{,}100 + x = 129{,}500$

4. 4

   Solve.

   $x = 129{,}500 - 110{,}100 = 19{,}400$

5. 5

   Check the direction before checking the arithmetic. The six quotes average $110{,}100 \div 6 = 18{,}350$, which is below the target $18{,}500$, so the seventh quote must be above the target to pull the mean up. And $19{,}400 > 18{,}500$ ✓ — the sign of the correction is right.

6. 6

   Verify the mean directly. $110{,}100 + 19{,}400 = 129{,}500$, and $129{,}500 \div 7 = 18{,}500$ exactly ✓.