A set of numbers has an average of 18. When the number 46 is added to the set, the average becomes 20. How many numbers were in the set to begin with?

With $n$ numbers to start, $18n+46=20(n+1)$, so $18n+46=20n+20$ and $2n=26$, giving $n=13$. Check: $13\times 18=234$, and $(234+46)/14=280/14=20$. A neater route: 46 sits 26 above the new average of 20, and that 26 is exactly what lifts the other $n$ numbers by 2 each, so $n=13$. 14 is the count AFTER the new number joined. 26 is the intermediate value $2n$. 23 is 46/2, which mixes up the new value with a count.