The average height of 22 toddlers increases by 2 inches when two of them leave this group. If the average height of these two toddlers is one-third the average height of the original 22, then the average height, in inches, of the remaining 20 toddlers is

We start with 22 toddlers with some average height, 2 toddlers leave the group, the average height of the remaining 20 toddlers increases by 2 inches, and the 2 who left had an average height that was one-third of the original group's average.

Key insight: When shorter people leave a group, the average height goes up!

---

We define our unknowns clearly:

Let x = average height of the remaining 20 toddlers

Then (x - 2) = average height of the original 22 toddlers

The average height of the 2 who left = $\tfrac{1}{3}(x - 2)$

Why (x - 2)? Because the problem states that the average increases by 2 inches when the 2 toddlers leave. So the original average was 2 inches less than the new average.

---

Here's the crucial concept: The total sum of heights doesn't change when people leave a group.

Sum of heights of 22 toddlers = Sum of heights of 20 toddlers + Sum of heights of 2 toddlers

We write this mathematically:

$22(x - 2) = 20x + 2 \times \tfrac{1}{3}(x - 2)$

Breaking this down:

$22(x - 2)$ = total height of original 22 toddlers

$20x$ = total height of remaining 20 toddlers

$2 \times \tfrac{1}{3}(x - 2)$ = total height of the 2 who left

---

$22(x - 2) = 20x + \tfrac{2}{3}(x - 2)$

$22x - 44 = 20x + \tfrac{2}{3}(x - 2)$

$22x - 44 = 20x + \tfrac{2}{3}x - \tfrac{4}{3}$

$22x - 20x - \tfrac{2}{3}x = 44 - \tfrac{4}{3}$

$2x - \tfrac{2}{3}x = \tfrac{6}{3}x - \tfrac{2}{3}x = \tfrac{4}{3}x$

$44 - \tfrac{4}{3} = \tfrac{132}{3} - \tfrac{4}{3} = \tfrac{128}{3}$

$\tfrac{4}{3}x = \tfrac{128}{3}$

$x = \tfrac{128}{3} \times \tfrac{3}{4} = \tfrac{128}{4} = 32$

---

The average height of the remaining 20 toddlers is 32 inches.

Key Takeaway: When working with averages and groups, remember that the total sum remains constant. This principle helps you set up the right equation every time!