AMC 10 · 2016 · #3
Grade 7 number-theoryPick an answer.
The formula looks heavy, but it is really three small jobs done in order, which is exactly what Tool #7 (Identify Subproblems) is for: first divide x by y, then take the floor of that quotient, then substitute back and subtract. Tool #4 (Introduce a Variable) keeps x=3/8 and y=-2/5 straight so each piece drops into the right slot. Tool #3 (Eliminate Possibilities) gives a fast sanity check: the result must be negative with denominator 40, which already points at one choice.
Plug the numbers into the formula
The formula is applied literally.
A formula is just a fill-in-the-blank: put each given number where its letter sits.
6.EE.A.2Introduce A VariableDivide the two fractions
The quotient comes out negative.
Flip-and-multiply turns a messy fraction-over-fraction into one clean fraction, and one minus sign makes it negative.
Flipping and multiplying turns a fraction over a fraction into one clean fraction.
▸ Why?
Dividing by a number is multiplying by its reciprocal, since the two undo each other.
▸ Why?
Multiplying top and bottom by the same amount renames a fraction without changing its value.
Take the floor of a negative number
Rounding it down gives -1, not zero.
Floor always steps left to the nearest integer, so a number just below zero lands on -1.
6.NS.C.7Identify SubproblemsSubstitute back and subtract
Substituting back gives -1/40, choice (B).
With the floor pinned down, the whole formula collapses to one ordinary fraction subtraction.
7.NS.A.1Identify SubproblemsFloor always rounds down toward smaller numbers, so a value just below zero like -0.9375 drops to -1, not 0.
- Plug the numbers into the formula
- Divide the two fractions
- Take the floor of a negative number
- Substitute back and subtract