AMC 10 · 2013 · #8

Grade 8 algebra
Archetype Arithmetic Expression Decomposition └ Sub-archetype Fraction / Expression Sum Evaluation
exponentsfraction-arithmetic convert-to-algebraeasier-related-problem ↑ Prerequisites: exponents
📏 Short solution 💡 1 insight
Problem
A fraction has 2²⁰¹⁴+2²⁰¹² on top and 2²⁰¹⁴-2²⁰¹² on the bottom. Find its exact value.

Pick an answer.

(A)
-1
(B)
1
(C)
$\frac{5}{3}$
(D)
2013
(E)
$2^{4024}$

AMC 10 2013 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Solve an Easier Related Problem

The exponents 2014 and 2012 look like the difficulty, but they are decoys: the only fact used anywhere is that they differ by 2. That points at Tool #9 (Solve an Easier Related Problem) — replace the unwritable powers by the same fraction built from one small block. Tool #4 (Introduce a Variable) does the replacing: name the shared power x=2²⁰¹², so the top and bottom become 4x+x and 4x-x. The move the whole problem turns on is cancelling the common factor x, and that step is legal only because x ≠ 0, so it deserves a check that does not use it at all: Tool #11 (Work Backwards) treats the value itself as the unknown t and recovers it from a linear equation, never factoring anything. Tool #3 (Eliminate Possibilities) then bounds the value between 1 and 2, which by itself leaves only one choice standing.

1STEP 1

The exponents differ by two

2014=2012+2, so 2²⁰¹⁴=2² · 2²⁰¹²=4 · 2²⁰¹²: the big term is just four copies of the small one.

2²⁰¹⁴=2²⁰¹²⁺²=2² · 2²⁰¹²=4 · 2²⁰¹²
2STEP 2

Name the shared power x

Let x=2²⁰¹², a product of 2012 twos, so x ≠ 0. Then 2²⁰¹⁴=4x and the whole fraction is (4x+x)/(4x-x).

x=2²⁰¹² ＞ 0, (2²⁰¹⁴+2²⁰¹²)/(2²⁰¹⁴-2²⁰¹²)=(4x+x)/(4x-x)
3STEP 3

Factor out x and cancel it

Factor: 4x+x=5x and 4x-x=3x, so the fraction is 5x/3x; since x ≠ 0 that x cancels, leaving 5/3.

(4x+x)/(4x-x)=5x/3x=5/3
4STEP 4

Confirm without cancelling anything

With a=2²⁰¹⁴, b=2²⁰¹² and t=(a+b)/(a-b), the quotient rule gives a/b=4, so (t+1)/(t-1)=4 and t=5/3 again.

a/b=(t+1)/(t-1) and a/b=2²⁰¹⁴⁻²⁰¹²=4 → t+1=4t-4 → t=5/3
Answer
5/3
The answer choices can be narrowed by size alone, which is an independent confirmation rather than a feeling. With b=2²⁰¹² ＞ 0 the numerator is 5b and the denominator is 3b, both positive, so the value is positive and (A) -1 is out. Since 5b ＞ 3b, the value exceeds 1, so (B) 1 is out. Since 5b ＜ 6b=2 · 3b, the value is less than 2, so (D) 2013 and (E) 2⁴⁰²⁴ are out — both are far above 2. Exactly one choice, 5/3, lies strictly between 1 and 2, so the bounds alone force (C), and 5/3≈ 1.67 indeed sits there. A numerical spot check agrees: with exponents 4 and 2 in place of 2014 and 2012, (16+4)/(16-4)=20/12=5/3. That small case is a confirmation of the identity already proved, not the reason for believing it — the proof works for any exponents two apart. The bait choices are explained too: (E) is what wrongly adding the exponents would give, and (D) is just the year.
💡Key takeaway

When powers of the same base are added or subtracted, pull the smaller power out as a common factor: it cancels, and all that is left is the gap between the exponents.

  • The exponents differ by two
  • Name the shared power x
  • Factor out x and cancel it
  • Confirm without cancelling anything

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