AMC 10 · 2016 · #7

Grade 6 arithmetic
Archetype Coordinate Plane Figure Computation └ Sub-archetype Figure Constraint Equations for Unknown Quantity
mean-median-mode-range convert-to-algebra ↑ Prerequisites: mean-median-mode-range
📏 Medium solution 💡 2 insights
Problem
Seven data values are 60, 100, x, 40, 50, 200, 90. The mean, the median, and the mode of these seven values are all equal to the same number x. Find x.

Pick an answer.

(A)
50
(B)
60
(C)
75
(D)
90
(E)
100

AMC 10 2016 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Convert to Algebra

The mean condition is the strongest one: it is a single equation that pins down x exactly. Tool #13 (Convert to Algebra) turns the sentence "the mean equals x" into the equation (540+x)/7 = x, which solves in one move. The median and mode conditions are then used with Tool #2 (Make a Systematic List) — sort the seven numbers — and Tool #3 (Eliminate Possibilities) to confirm the same x works, so we know the answer is consistent rather than a coincidence.

1STEP 1

Add the six known numbers

Six values are plain numbers; add them: 60+100+40+50+200+90 = 540, so all seven sum to 540 + x.

60+100+40+50+200+90 = 540
2STEP 2

Turn "mean equals x" into an equation

The mean is (540+x)/7 = x; clearing the fraction gives 540 = 6x, so x = 90.

(540+x)/7 = x → 540+x = 7x → 540 = 6x → x = 90
3STEP 3

Check the median by sorting

Put x=90 back and sort: 40, 50, 60, 90, 90, 100, 200. The 4th of seven values — the median — is 90, matching x.

40, 50, 60, 90, 90, 100, 200 → median = 90
4STEP 4

Check the mode and read the answer

In 40, 50, 60, 90, 90, 100, 200 only 90 repeats, so the mode is 90 too — mean, median, mode all agree, giving (D).

mode = 90 = median = mean → x = 90 → (D)
Answer
90
All three measures land on 90, exactly as the problem demands, and 90 is one of the answer choices (D). A quick sanity check on the mean: ignore x for a moment and average the other six numbers, 540/6 = 90. When the mean of a set already sits in the set, adding that same value again leaves the mean unchanged — so the full seven-value mean is also 90, matching x. The value 90 also sits comfortably in the middle of the data, which is what a median and mean should do.
💡Key takeaway

When a problem says the mean equals some unknown, set the total equal to that many copies of the unknown — then sort the list to double-check the median and mode.

  • Add the six known numbers
  • Turn "mean equals x" into an equation
  • Check the median by sorting
  • Check the mode and read the answer

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