AMC 10 · 2010 · #14

Grade 8 algebra

Archetype Coordinate Plane Figure Computation └ Sub-archetype Figure Constraint Equations for Unknown Quantity

mean-median-mode-rangesequences-arithmeticlinear-equations-one-var convert-to-algebra ↑ Prerequisites: mean-median-mode-range

📏 Medium solution 💡 2 insights

Problem

The list holds the whole numbers 1 through 99 plus one extra number x, so 100 numbers in all. The average of these 100 numbers equals 100 times x. Find x.

Pick an answer.

(A)

$\dfrac{49}{101}$

(B)

$\dfrac{50}{101}$

(C)

$\dfrac{1}{2}$

(D)

$\dfrac{51}{101}$

(E)

$\dfrac{50}{99}$

AMC 10 2010 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Convert to Algebra

The word 'average' hides an equation. Turn the sentence into algebra: (sum of all 100 numbers) / 100 = 100x. First handle the fixed part as a subproblem (add 1 through 99), then solve the equation for x. A pattern in the numbers 9999 and 4950 makes the final fraction easy to reduce.

1STEP 1

Add 1 through 99

Pair the fixed numbers from the ends: 1+99, 2+98, and so on. Each pair makes 100 and 50 sits alone, so the sum is 4950.

1 + 2 + … + 99 = (99 · 100)/2 = 4950

2STEP 2

Write the average as an equation

Counting x there are 100 numbers summing to 4950 + x, so the average $\frac{4950 + x}{100}$ must equal 100x.

(4950 + x)/100 = 100x

3STEP 3

Solve for x

Multiply both sides by 100 to clear the fraction, then subtract x: 4950 + x = 10000x becomes 4950 = 9999x.

4950 + x = 10000x → 4950 = 9999x → x = 4950/9999

4STEP 4

Reduce the fraction

Because 9999 = 99 times 101 and 4950 = 99 times 50, cancelling the shared 99 leaves x = $\frac{50}{101}$ .

x = 4950/9999 = (99 · 50)/(99 · 101) = 50/101

Answer

50/101

x = 50/101 is a little less than 1/2, so 100x is a little less than 50. The 100 numbers run from 1 up to 99 plus a tiny x, so their average should be a bit under 50 as well. Both sides land just under 50, which is consistent, and 50/101 matches choice (B).

💡Key takeaway

The word 'average' is really an equation: add everything up, divide by how many, and set it equal to what you were told.

Add 1 through 99
Write the average as an equation
Solve for x
Reduce the fraction

Add 1 through 99

Write the average as an equation

Solve for x

Reduce the fraction

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