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2013 AMC 10B Problem 3

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2013 AMC 10B Problem 3AMC 10 Step by Step
Name:                             Date: October 28, 2026
Linear Equations & Word Problems · difficulty 1 of 5 · about 2 min · position P1-10

On a particular January day, the high temperature in Lincoln, Nebraska, was 1616 degrees higher than the low temperature, and the average of the high and the low temperatures was 3 ∘3\,^\circ . In degrees, what was the low temperature in Lincoln that day?

  1. A)−13-13
  2. B)−8-8
  3. C)−5-5
  4. D)−3-3
  5. E)1111

Answer

C

Key insight

If two numbers average 3, they add up to 6. The high is 16 more than the low, so the low is 8 below the average.

Solution

1. Call the low temperature LL. The high is 1616 degrees higher, so it is L+16L + 16.
2. The average of two numbers is their sum divided by 22. The average is 33, so the sum is 2×3=62 \times 3 = 6:
L+(L+16)=6.L + (L + 16) = 6.
3. That says 2L+16=62L + 16 = 6. Take 1616 away from both sides: 2L=−102L = -10. Divide by 22: L=−5L = -5.
4. Check: the high is −5+16=11-5 + 16 = 11, and the average is 11+(−5)2=62=3\frac{11 + (-5)}{2} = \frac{6}{2} = 3. That matches.

The answer is (C) −5\boxed{\textbf{(C)}\ -5}.

Why this works

"The average of two numbers is 33" means exactly the same thing as "the two numbers add up to 66". Once you know the sum and the difference of two numbers, both numbers are fixed. Turning an average into a sum is a move you will use again and again.

Alternative approach

The average of two numbers sits exactly halfway between them. The high and low are 1616 degrees apart, so each one is half of that, 88 degrees, away from the average. The low is 3−8=−53 - 8 = -5 and the high is 3+8=113 + 8 = 11.

The trap

Using 3 as the sum of the two temperatures instead of 6, or picking 11, which is the high temperature, not the low.

Concise solution

Let the low temperature be LL, so the high is L+16L + 16.

The average of the two is 33, so their sum is 66:
L+(L+16)=6⟹2L=−10⟹L=−5.L + (L + 16) = 6 \quad\Longrightarrow\quad 2L = -10 \quad\Longrightarrow\quad L = -5.

Check: the high is 1111 and 11+(−5)2=3\frac{11 + (-5)}{2} = 3.

The answer is (C) −5\boxed{\textbf{(C)}\ -5}.

Review in the AoPS books

Techniques

Set up the equation/formula and compute; no special trick needed