AMC 10 · 2024 · #5
Grade 7 arithmeticalgebraPick an answer.
The twenty numbers are never revealed, so no amount of listing or guessing at entries can help. What a mean does give is a total: Tool #11 (Work Backwards) runs the mean formula in reverse to recover the sum 20 · 45 = 900 before anything is deleted. Once totals are the currency, Tool #13 (Convert to Algebra) does the real work — name the number of 6s k, write the surviving count as 20 - k and the surviving sum as 900 - 6k, and the second mean becomes a single linear equation in k. Tool #15 (Organize Information in More Ways) closes the loop: regrouping the same twenty numbers into "the 6s" and "the rest" and balancing their distances from 45 reproduces k from a completely different direction, which is the check that the setup was right.
Turn the mean into a total
Twenty times forty-five gives a total of 900.
A mean is a total that has been shared out evenly, so multiplying it back by the number of shares rebuilds the total.
A mean is a total that has been shared out evenly, so multiplying it back by the count rebuilds the total.
▸ Why?
An average is a total divided by how many there are, so the two are two readings of one thing.
▸ Why?
That total is the individual values added together, so each group contributes its own share of it.
Name the count of 6s
With k sixes the surviving sum is 900 minus 6k.
One letter absorbs the only thing the problem hides, and every other quantity becomes an expression in that letter.
6.EE.A.2Convert To AlgebraWrite the second mean as an equation
The surviving sum over the surviving count is 66.
The second mean is a second fact about the same twenty numbers, and written in totals it becomes one equation in one unknown.
7.EE.B.4Convert To AlgebraSolve for k
It reduces to sixty k equals 420, so k is 7.
Gathering the unknown on one side is just undoing, in reverse order, the operations that built the equation.
7.EE.A.1Convert To AlgebraConfirm with the balance view
Sixes sit 39 below, survivors 21 above — again 7.
The mean is the balance point of the data, so how far a group sits from it, times how big that group is, must cancel across the two sides.
6.SP.A.3Organize Information In More WaysThis AMC 12 problem only needs Grade 6 "mean equals total divided by count" and a Grade 7 linear equation — stop thinking about the twenty hidden numbers and think about their total, and the number of 6s falls out in one line.
- Turn the mean into a total
- Name the count of 6s
- Write the second mean as an equation
- Solve for k
- Confirm with the balance view