2010 AMC 10A Problem 3

P1-10~1 minPrint

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

Tyrone had $97$ marbles and Eric had $11$ marbles. Tyrone then gave some of his marbles to Eric so that Tyrone ended with twice as many marbles as Eric. How many marbles did Tyrone give to Eric?

Show solution without answering

Key insight

Passing marbles keeps the total at 108. It ends split 2 to 1, so Tyrone keeps 72 and gave away 25.

Solution

When marbles move from one boy to the other, the total does not change. Use that total.

Together they have $97 + 11 = 108$ marbles, before and after.
At the end Tyrone has twice as many as Eric. Think of it as $3$ equal parts: Eric has $1$ part and Tyrone has $2$ parts.
One part is $108 \div 3 = 36$ . So Eric ends with $36$ and Tyrone ends with $72$ .
Tyrone started with $97$ and ended with $72$ , so he gave away $97 - 72 = 25$ .
Check from Eric's side: $11 + 25 = 36$ . It matches.

The answer is $\boxed{\textbf{(D)}\ 25}$ .

Why this works

When things are only passed around, the total stays the same. Knowing the total and the final ratio tells you everyone's final amount, and the amount passed is just the change. Look for a quantity that stays fixed whenever something is "given" or "moved."

Alternative approach

Let $x$ be the number of marbles Tyrone gives. Afterwards Tyrone has $97 - x$ and Eric has $11 + x$ . Tyrone has twice as many:
$97 - x = 2(11 + x) = 22 + 2x.$
Add $x$ to both sides and subtract $22$ : $75 = 3x$ , so $x = 25$ .

Concise solution

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

Giving marbles back and forth does not change the total, which is $97 + 11 = 108$ .

At the end Tyrone has twice as many as Eric, so the $108$ marbles split in the ratio $2 : 1$ : Eric has $\dfrac{108}{3} = 36$ and Tyrone has $72$ .

Tyrone therefore gave away $97 - 72 = 25$ marbles (equivalently, Eric gained $36 - 11 = 25$ ).

The answer is $\boxed{\textbf{(D)}\ 25}$ .

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)