AMC 10 · 2014 · #1
Grade 8 arithmeticPick an answer.
The expression stacks three separate operations, so the clean move is to split it into subproblems (Tool #7): first add the fractions, then handle the -1 exponent, then multiply by 10. The -1 is the one trap here — Tool #16 (Change Focus) reframes "raise to the power -1" as the simpler idea "flip the fraction over." Because it is multiple-choice, Tool #3 lets us confirm the final value lands exactly on a listed choice.
Add the three fractions
The three fractions add to 4/5.
A common denominator turns three different-sized pieces into one kind of piece you can just count up.
5.NF.A.1Identify SubproblemsFlip for the -1 exponent
The exponent flips that fraction.
A power of -1 just means "turn it over": numerator and denominator swap places.
A power of minus one just means turn it over: the top and the bottom swap places.
▸ Why?
A number multiplied by its reciprocal gives one, so the reciprocal is exactly what undoes it.
▸ Why?
Renaming a fraction over a common denominator first keeps its value, so the flip acts on a single number.
Multiply by 10
Multiplying gives 25/2, choice (C).
Ten copies of 5/4 is 50/4, and halving top and bottom by 2 gives 25/2.
4.NF.B.4Identify SubproblemsA power of -1 just means "flip the fraction," and the rest is add-then-multiply.
- Add the three fractions
- Flip for the -1 exponent
- Multiply by 10