AMC 10 · 2016 · #4

Grade 7 number-theory

Archetype Arithmetic Expression Decomposition └ Sub-archetype Fraction / Expression Sum Evaluation

floor-functionfraction-arithmetic formula-substitution ↑ Prerequisites: fraction-arithmetic

📏 Medium solution 💡 3 insights

Problem

A function called rem is defined for real numbers x and y ≠ 0 by rem(x,y)=x-y⌊ x/y⌋, where ⌊ t⌋ is the greatest integer that is less than or equal to t. Compute rem(3/8,-2/5).

Pick an answer.

(A)

$-\frac{3}{8}$

(B)

$-\frac{1}{40}$

(C)

(D)

$\frac{3}{8}$

(E)

$\frac{31}{40}$

AMC 10 2016 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Identify Subproblems

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.

1STEP 1

Plug the numbers into the formula

Put x=3/8 and y=-2/5 into the formula to get 3/8-(-2/5)⌊ (3/8)/(-2/5)⌋; the only tricky piece is the floor.

rem(3/8,-2/5)=3/8-(-2/5)⌊ (3/8)/(-2/5)⌋

2STEP 2

Divide the two fractions

Dividing by a fraction flips it: 3/8÷(-2/5)=3/8·(-5/2), and one minus sign makes the quotient -15/16.

(3/8)/(-2/5)=3/8·(-5/2)=-15/16

3STEP 3

Take the floor of a negative number

-15/16=-0.9375 sits between -1 and 0, and floor steps left to the greatest integer below it, so ⌊ -15/16⌋=-1, not 0.

⌊ -15/16⌋=⌊ -0.9375⌋=-1

4STEP 4

Substitute back and subtract

With the floor known, 3/8-(-2/5)(-1)=3/8-2/5=15/40-16/40=-1/40, which is (B).

3/8-(-2/5)(-1)=15/40-16/40=-1/40=(B)

Answer

-1/40

The result -1/40 passes two quick tests. Sign: 3/8=15/40 is just barely smaller than 2/5=16/40, so the difference is negative, as expected. Denominator: combining eighths and fifths forces denominator 40. Only choice (B) is both negative and over 40, matching exactly.

💡Key takeaway

Floor 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

Plug the numbers into the formula

Divide the two fractions

Take the floor of a negative number

Substitute back and subtract

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