2013 AMC 10B Problem 3

P1-10~2 minPrint

Linear Equations & Word Problems Set up the equation/formula and compute; no special trick needed

On a particular January day, the high temperature in Lincoln, Nebraska, was $16$ degrees higher than the low temperature, and the average of the high and the low temperatures was $3\,^\circ$ . In degrees, what was the low temperature in Lincoln that day?

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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 $L$ . The high is $16$ degrees higher, so it is $L + 16$ .
2. The average of two numbers is their sum divided by $2$ . The average is $3$ , so the sum is $2 \times 3 = 6$ :
$L + (L + 16) = 6.$
3. That says $2L + 16 = 6$ . Take $16$ away from both sides: $2L = -10$ . Divide by $2$ : $L = -5$ .
4. Check: the high is $-5 + 16 = 11$ , and the average is $\frac{11 + (-5)}{2} = \frac{6}{2} = 3$ . That matches.

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

Why this works

"The average of two numbers is $3$ " means exactly the same thing as "the two numbers add up to $6$ ". 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 $16$ degrees apart, so each one is half of that, $8$ degrees, away from the average. The low is $3 - 8 = -5$ and the high is $3 + 8 = 11$ .

Concise solution

A shorter write-up, the way contest solutions are usually written. It may take a different route from the solution above.

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

The average of the two is $3$ , so their sum is $6$ :
$L + (L + 16) = 6 \quad\Longrightarrow\quad 2L = -10 \quad\Longrightarrow\quad L = -5.$

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

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

Review in the AoPS books

Introduction to Algebra, Chapter 3: One-Variable Linear Equations (§3.3, §3.4)
Introduction to Algebra, Chapter 7: Proportion (§7.4)