AMC 10 · 2002 · #1
Grade 8 rate-ratioThe ratio 102001+102001102000+102002 is closest to which of the following numbers?
Pick an answer.
AMC 10 2002 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Try it yourself first — the explanation is most useful after you’ve attempted it.
Toolkit + CCSS Solution
Understand
Restated: Evaluate $\dfrac{10^{2000}+10^{2002}}{10^{2001}+10^{2001}}$ and decide which of the listed numbers it is closest to.
Givens: The fraction $\dfrac{10^{2000}+10^{2002}}{10^{2001}+10^{2001}}$; Numerator: two powers of $10$ added, with exponents $2000$ and $2002$; Denominator: the same power $10^{2001}$ added to itself; Answer choices: (A) $0.1$, (B) $0.2$, (C) $1$, (D) $5$, (E) $10$
Unknowns: Which single choice the fraction is closest to
Understand
Restated: Evaluate $\dfrac{10^{2000}+10^{2002}}{10^{2001}+10^{2001}}$ and decide which of the listed numbers it is closest to.
Givens: The fraction $\dfrac{10^{2000}+10^{2002}}{10^{2001}+10^{2001}}$; Numerator: two powers of $10$ added, with exponents $2000$ and $2002$; Denominator: the same power $10^{2001}$ added to itself; Answer choices: (A) $0.1$, (B) $0.2$, (C) $1$, (D) $5$, (E) $10$
Plan
Primary tool: #7 Identify Subproblems
Secondary: #9 Solve an Easier Related Problem, #3 Eliminate Possibilities
The fraction looks frightening only because of the giant exponents. Tool #7 (Identify Subproblems) splits the job into three clean pieces: factor the numerator, collapse the denominator, then cancel. Tool #9 (Solve an Easier Related Problem) is what makes the exponents harmless — pulling the common factor $10^{2000}$ out of the top and bottom removes every huge power at once, leaving a tiny fraction of small whole numbers. Tool #3 (Eliminate Possibilities) then handles 'closest to': once the value is $5.05$, the choices $0.1$, $0.2$, $1$, and $10$ are all far away, so only $5$ survives.
Execute — Answer: D
8.EE.A.1 Step 1 Factor the numerator
- Both terms on top are powers of $10$, so pull out the smaller one, $10^{2000}$.
- Since $10^{2002}=10^{2000}\cdot 10^{2}$, the numerator becomes $10^{2000}+10^{2002}=10^{2000}(1+10^{2})=10^{2000}\cdot 101$.
- The scary exponent $2002$ is now just a factor of $101$.
💡 Adding powers of the same base lets you factor out the smallest power, shrinking the exponents into an ordinary number.
6.EE.A.3 Step 2 Collapse the denominator
- The bottom is one quantity added to itself: $10^{2001}+10^{2001}=2\cdot 10^{2001}$.
- To match the numerator's factor of $10^{2000}$, write $10^{2001}=10\cdot 10^{2000}$, so the denominator is $2\cdot 10\cdot 10^{2000}=20\cdot 10^{2000}$.
💡 Anything plus itself is just twice that thing, and one extra factor of $10$ turns $10^{2001}$ into $10\cdot 10^{2000}$.
8.EE.A.1 Step 3 Cancel the common power of 10
- Now the fraction is $\dfrac{101\cdot 10^{2000}}{20\cdot 10^{2000}}$.
- The factor $10^{2000}$ appears top and bottom and cancels completely, leaving $\dfrac{101}{20}$.
- Every giant exponent is gone.
💡 A factor shared by numerator and denominator divides out to $1$, no matter how enormous it is.
5.NBT.B.7 Step 4 Divide and pick the closest choice
- Divide: $\dfrac{101}{20}=5.05$.
- Compare with the choices — $0.1$, $0.2$, $1$, and $10$ are all far off, while $5$ is only $0.05$ away.
- So the fraction is closest to $5$, which is choice (D).
💡 Once the monster fraction reduces to $101/20$, ordinary division settles which choice is nearest.
8.EE.A.1 Both terms on top are powers of $10$, so pull out the smaller one, $10^{2000}$. 6.EE.A.3 The bottom is one quantity added to itself: $10^{2001}+10^{2001}=2\cdot 10^{2001 8.EE.A.1 Now the fraction is $\dfrac{101\cdot 10^{2000}}{20\cdot 10^{2000}}$. The factor 5.NBT.B.7 Divide: $\dfrac{101}{20}=5.05$. Compare with the choices — $0.1$, $0.2$, $1$, an Review
Reasonableness: A quick estimate confirms it: the top is dominated by $10^{2002}$, which is $100$ times $10^{2000}$, and the bottom is $2\cdot 10^{2001}=20\cdot 10^{2000}$, so the ratio is roughly $\tfrac{100}{20}=5$. The small $+10^{2000}$ on top only nudges it up to $5.05$, so a value near $5$ is exactly what to expect — and it rules out every other choice.
Alternative: Divide each term by $10^{2000}$ first: the fraction becomes $\dfrac{1+10^{2}}{10+10}=\dfrac{1+100}{20}=\dfrac{101}{20}=5.05$, landing on (D) without ever factoring.
CCSS standards used (min grade 8)
8.EE.A.1Know and apply the properties of integer exponents (Rewriting $10^{2002}=10^{2000}\cdot 10^{2}$ and $10^{2001}=10\cdot 10^{2000}$ to factor out and then cancel the common power $10^{2000}$.)6.EE.A.3Apply the properties of operations to generate equivalent expressions (Combining the identical denominator terms $10^{2001}+10^{2001}$ into $2\cdot 10^{2001}$.)5.NBT.B.7Add, subtract, multiply, and divide decimals to hundredths (Dividing $\tfrac{101}{20}=5.05$ and comparing it to the answer choices to find the closest one.)
⭐ When every term is a power of the same base, factor out the smallest power and cancel it — the giant exponents disappear and a tiny fraction is left.
⭐ When every term is a power of the same base, factor out the smallest power and cancel it — the giant exponents disappear and a tiny fraction is left.
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