Round each number using the Huntington-Hill rounding rules. (Hint:
is all the information you need. You won’t need a calculator or to look up any tables.) (a) 1.5 (b) 1.4 (c) 1.41 (d) 1.42
The Correct Answer and Explanation is:
Answers (Using the Huntington-Hill rounding rules):
(a) 2
(b) 1
(c) 1
(d) 1
Explanation (Like in a textbook):
The Huntington-Hill rounding method, also known as the method of equal proportions, is used in contexts like apportioning seats in government or decision-making bodies. This method uses a special rounding rule based not just on standard decimal rules, but on a geometric mean between two integers.
Here’s how it works:
Let a number xxx lie between two integers nnn and n+1n+1n+1. Instead of rounding at the usual 0.5 mark, the Huntington-Hill method uses the geometric mean of nnn and n+1n+1n+1 as the cutoff point. The geometric mean is calculated as: Geometric Mean=n(n+1)\text{Geometric Mean} = \sqrt{n(n+1)}Geometric Mean=n(n+1)
The rules are:
- If x<n(n+1)x < \sqrt{n(n+1)}x<n(n+1), round down to nnn
- If x>n(n+1)x > \sqrt{n(n+1)}x>n(n+1), round up to n+1n+1n+1
- If x=n(n+1)x = \sqrt{n(n+1)}x=n(n+1), round up to n+1n+1n+1
Now let’s apply this to each case:
(a) 1.5
Here, n=1n = 1n=1, so geometric mean = 1⋅2=2≈1.414\sqrt{1 \cdot 2} = \sqrt{2} \approx 1.4141⋅2=2≈1.414
Since 1.5 > 1.414, we round up to 2
(b) 1.4
Again, geometric mean = 2≈1.414\sqrt{2} \approx 1.4142≈1.414
Since 1.4 < 1.414, we round down to 1
(c) 1.41
1.41 < 1.414 ⇒ round down to 1
(d) 1.42
1.42 > 1.414 ⇒ round up to 2
But wait!
Actually, 1.42 is only slightly above 1.414 — we must check carefully. Let’s break this down:
- 1.41 < 1.414 ⇒ round down
- 1.42 > 1.414 ⇒ round up
BUT the exact value of 1.42 is closer to 1.4142, so the true comparison is:
- Is 1.42 < or > 2≈1.4142\sqrt{2} \approx 1.41422≈1.4142?
Since 1.42 > 1.4142, we round up to 2
So the correct answer to (d) is 2, not 1.
Final corrected answers:
(a) 2
(b) 1
(c) 1
(d) 2
