AMC 10 · 2012 · #4
Grade 4 arithmeticPick an answer.
AMC 10 2012 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
We do not know how many marbles each boy has, only the leftovers. Naming each count with a variable lets us write the 'bags of 6' fact as an equation, add the two counts, and read off the new leftover without ever needing the exact totals.
Write each count with its leftover
Give Ringo a full bags plus 4 loose and Paul b full bags plus 3 loose: Ringo has 6a + 4 marbles, Paul has 6b + 3.
A remainder just names the loose marbles left after making as many full bags as possible.
4.OA.A.3Introduce A VariableAdd both piles together
Adding, (6a + 4) + (6b + 3) = 6a + 6b + 7, and since 6a and 6b are whole bags, only 7 loose marbles remain.
Full bags stay full when you combine piles, so only the leftovers can mix into new bags.
4.OA.A.3Convert To AlgebraRepack the 7 loose marbles
Since 7 = 6 + 1, that fills one more bag: 6a + 6b + 7 = 6(a + b + 1) + 1, leaving 1 marble over, choice (A).
The leftover of a sum is just the leftover of adding the two leftovers.
The leftover of a sum is just the leftover of adding the two leftovers.
▸ Why?
Full bags stay full when the piles are combined, so only the loose marbles can regroup.
▸ Why?
What remains after making as many full bags as possible is exactly the remainder.
To find the leftover of a sum, just add the leftovers and, if they reach a full bag, take that bag out too.
- Write each count with its leftover
- Add both piles together
- Repack the 7 loose marbles