1) Restate & Type
We have fractions 19/36, 5/11, 12/25, 6/11, and 8/18. If they are ordered from least to greatest, which is in the middle (3rd)? Type: arithmetic – comparing fractions.

2) Concept (Big Picture)
To order fractions with different denominators, either:
- convert to decimals, or
- compare two at a time using cross-multiplication.
We just need relative size, not exact values.

3) Long Solution (Step by Step)

Step 1: Simplify where possible
8/18 = 4/9.
So the set is: 19/36, 5/11, 12/25, 6/11, 4/9.

Step 2: Roughly locate each around 1/2 = 0.5
19/36 ≈ 0.53 (a bit more than 1/2)
5/11 ≈ 0.45 (less than 1/2)
12/25 = 0.48 (less than 1/2)
6/11 ≈ 0.55 (more than 1/2)
4/9 ≈ 0.44 (less than 1/2)

So:
- Below 0.5: 4/9, 5/11, 12/25
- Above 0.5: 19/36, 6/11
The middle must be one of 4/9, 5/11, 12/25.

Step 3: Order the three below 0.5 using cross-multiplication

Compare 4/9 and 5/11:
4 * 11 = 44
5 * 9 = 45
Since 44 < 45, we have 4/9 < 5/11.

Compare 5/11 and 12/25:
5 * 25 = 125
12 * 11 = 132
Since 125 < 132, we have 5/11 < 12/25.

Thus:
4/9 < 5/11 < 12/25.

We already know all three are less than 19/36 and 6/11, so the full order is:

4/9 (= 8/18) < 5/11 < 12/25 < 19/36 < 6/11.

The 3rd number (middle) is 12/25.

Answer: 12/25.

4) Common Pitfalls
- Trying to force all fractions to same denominator (slow, error-prone).
- Computing long decimals and rounding badly.
- Forgetting that “middle” of 5 items is the 3rd.

5) Shortcut / Time-Saving Method
Quickly estimate each as a decimal:
8/18 ≈ 0.44
5/11 ≈ 0.45
12/25 = 0.48
19/36 ≈ 0.53
6/11 ≈ 0.55
You can already see the order:
0.44 < 0.45 < 0.48 < 0.53 < 0.55
→ middle is 12/25.


Final answer: 12/25.♦