AMC 10 · 2021 · #3
Grade 7 algebraPick an answer.
The value of the whole stack is given and the starting number x is hidden at the very bottom, which is exactly the setup for Tool #11 (Work Backwards): undo the operations in reverse order. Reading the left side from the outside in, the operations are add 2, take a reciprocal, add 1, take a reciprocal, add 2, take a reciprocal (of the 2 on top), add 3. Tool #7 (Identify Subproblems) makes each layer its own tiny task, and after each undo the equation that remains is a shorter stack of the same shape — Tool #9 (Solve an Easier Related Problem) in action. No expanding of the giant fraction is needed.
Undo the outer plus two
Undo the outer addition first.
Strip the outermost layer first, the same way you take off a coat before a sweater.
7.NS.A.1Work BackwardsFlip to undo the reciprocal
Flip to undo the reciprocal.
If 1/A=p/q then A=q/p, because flipping a fraction twice returns it unchanged.
Flipping a fraction twice returns it unchanged, so one flip undoes a reciprocal.
▸ Why?
A number multiplied by its reciprocal gives one, so the reciprocal is what undoes it.
▸ Why?
Turning a fraction over swaps its top and bottom without changing what it counts.
Subtract one, then flip again
Subtract, then flip again.
Each level is the same tiny puzzle, so the same pair of moves clears it every time.
7.NS.A.1Identify SubproblemsPeel down to three plus x
Peel down to the bottom layer.
A numerator of 2 instead of 1 just means the flip leaves a factor of 2 to carry along.
6.NS.A.1Work BackwardsSolve for x
Solving gives three quarters.
The last wrapper is the simplest one, so the final undo is a single subtraction.
7.EE.B.4Work BackwardsA tall stacked fraction unwraps from the outside in: subtract whatever was added, flip whatever sits under the 1, and repeat until only x is left.
- Undo the outer plus two
- Flip to undo the reciprocal
- Subtract one, then flip again
- Peel down to three plus x
- Solve for x