AMC 10 · 2010 · #10

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
A run of numbers plus one extra has an average equal to a large multiple of that extra. Find the extra number.

Pick an answer.

(A)
$\dfrac{49}{101}$
(B)
$\dfrac{50}{101}$
(C)
$\dfrac{1}{2}$
(D)
$\dfrac{51}{101}$
(E)
$\dfrac{50}{99}$
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

Pairing adds the run to 4950.

1 + 2 + … + 99 = (99 · 100)/2 = 4950
2STEP 2

Write the average as an equation

The average becomes one plain equation.

(4950 + x)/100 = 100x
3STEP 3

Solve for x

Gathering the unknown gives a single fraction.

4950 + x = 10000x → 4950 = 9999x → x = 4950/9999
4STEP 4

Reduce the fraction

Reducing it gives 50/101, choice (B).

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

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