AMC 10 · 2016 · #20
Grade 8 algebraPick an answer.
The operation is never told to us directly, so treat the result a◆ b as an unknown expression and feed clever inputs into the two rules until the expression is forced. Substituting equal arguments collapses Rule 1 using Rule 2, which pins the operation down as ordinary division. After that the equation is routine: plug in, simplify, and solve a linear equation.
Make two arguments equal
Making two arguments equal simplifies the first rule.
Feeding equal inputs lets Rule 2 erase the messy inner term and expose a simpler relationship.
6.EE.B.6Introduce A VariablePin down a ◇ 1
That pins down one special value.
A second smart substitution turns one unknown into a number we actually know.
6.EE.A.2Introduce A VariableIdentify the operation
The operation is then recognisable.
Once a quantity times b equals a, that quantity has to be a divided by b.
Once a quantity multiplied by one number gives another, that quantity has to be the second divided by the first.
▸ Why?
Dividing undoes multiplying exactly, so the unknown factor is recovered in one move.
▸ Why?
Two expressions naming the same operation name the same rule, so the identification is complete.
Check division fits both rules
It really does satisfy both rules.
A candidate rule is only safe once it survives every condition it must satisfy.
6.EE.A.2Guess And CheckPlug into the equation
Substituting leaves a linear equation.
Dividing by a fraction flips it, so the x climbs into the numerator and the problem becomes linear.
6.NS.A.1Introduce A VariableSolve and reduce
Reducing gives 109, choice (A).
Isolate x, then strip the common factor so p and q are truly relatively prime.
8.EE.C.7Work BackwardsWhen an operation is defined only by rules, feed it equal inputs to collapse the rules until you recognize a familiar operation hiding underneath.
- Make two arguments equal
- Pin down a ◇ 1
- Identify the operation
- Check division fits both rules
- Plug into the equation
- Solve and reduce