AMC 8 · 2014 · #10
Grade 4 arithmeticProblem
The first AMC was given in and it has been given annually since that time. Samantha turned years old the year that she took the seventh AMC . In what year was Samantha born?
Pick an answer.
Toolkit + CCSS Solution
Understand
Restated: The first AMC $8$ took place in $1985$ and one has been held every year since. Samantha was $12$ years old in the year she sat the seventh AMC $8$. In what year was she born?
Givens: AMC $8$ #$1$ was held in $1985$; The contest has been held every year since (one per year); Samantha turned $12$ the year of the seventh AMC $8$; Answer choices: (A) $1979$, (B) $1980$, (C) $1981$, (D) $1982$, (E) $1983$
Unknowns: Samantha's birth year
Understand
Restated: The first AMC $8$ took place in $1985$ and one has been held every year since. Samantha was $12$ years old in the year she sat the seventh AMC $8$. In what year was she born?
Givens: AMC $8$ #$1$ was held in $1985$; The contest has been held every year since (one per year); Samantha turned $12$ the year of the seventh AMC $8$; Answer choices: (A) $1979$, (B) $1980$, (C) $1981$, (D) $1982$, (E) $1983$
Plan
Primary tool: #5 Look for a Pattern
Secondary: #7 Identify Subproblems
The years $1985, 1986, 1987, \dots$ form a simple pattern — one contest per year, going up by $1$ each time. Tool #$5$ (Look for a Pattern) turns "the seventh AMC $8$" into a year by counting six steps forward from $1985$. Tool #$7$ (Identify Subproblems) keeps the work tidy by splitting the question into two clean pieces: (a) find the year of the seventh contest, then (b) subtract Samantha's age to get her birth year. Algebra (tool #$13$) is overkill here — a Grade $4$ pattern + subtraction is enough.
Execute — Answer: A
4.OA.C.5 Step 1 List the AMC $8$ years until the seventh one to see the pattern: each year goes up by $1$, so the $n$-th contest is in year $1985 + (n - 1)$.
💡 Writing out the first few terms makes the rule obvious: add $1$ each step. This is the Grade $4$ "generate and analyze patterns" move.
4.NBT.B.4 Step 2 Confirm the pattern with the formula: the seventh contest is $6$ years after the first.
💡 Adding $6$ to a four-digit year is the Grade $4$ "fluently add multi-digit whole numbers" skill.
4.NBT.B.4 Step 3 Now solve the second subproblem: given that Samantha was $12$ in $1991$, subtract to find her birth year.
💡 Splitting the question into "find the year, then find the age" is the Tool #$7$ subproblems move; the subtraction itself is standard Grade $4$ arithmetic.
4.NBT.B.4 Step 4 Match $1979$ to the answer choices.
💡 Reading the choice list is the final "close the subproblem" check.
4.OA.C.5 List the AMC $8$ years until the seventh one to see the pattern: each year goes 4.NBT.B.4 Confirm the pattern with the formula: the seventh contest is $6$ years after the 4.NBT.B.4 Now solve the second subproblem: given that Samantha was $12$ in $1991$, subtrac 4.NBT.B.4 Match $1979$ to the answer choices. Review
Reasonableness: Sanity-check the pattern: the $1$st contest is $1985$, so the $2$nd is $1986$, the $3$rd is $1987$, $\dots$ — six steps later the $7$th must be $1991$. Then a $12$-year-old in $1991$ was born in $1991 - 12 = 1979$, which lines up with choice (A). The other choices ($1980$–$1983$) would mean Samantha was $11, 10, 9,$ or $8$ in $1991$ — none match the "turned $12$" clue.
Alternative: Tool #$11$ (Work Backwards) gives the same answer in one chain: start from "age $12$ at the seventh AMC $8$," rewind $12$ years to the birth year, and substitute the contest year $1991$ — so birth year $= 1991 - 12 = 1979$. Or tool #$6$ (Guess and Check) on the choices: only $1979 + 12 = 1991$ matches the seventh-contest year; $1980 + 12 = 1992$, $1981 + 12 = 1993$, etc., do not.
CCSS standards used (min grade 4)
4.OA.C.5Generate and analyze patterns (Recognizing that AMC $8$ years form the arithmetic pattern $1985, 1986, 1987, \dots$ (step $+ 1$) and using the rule "$n$-th term $= 1985 + (n - 1)$" to land on $1991$ for the seventh contest.)4.NBT.B.4Fluently add and subtract multi-digit whole numbers (Performing the four-digit arithmetic $1985 + 6 = 1991$ and then $1991 - 12 = 1979$ to get Samantha's birth year.)
⭐ This AMC $8$ problem only needs Grade $4$ skills — counting up by $1$ to spot a year pattern, then a single subtraction — that you already know!
⭐ This AMC $8$ problem only needs Grade $4$ skills — counting up by $1$ to spot a year pattern, then a single subtraction — that you already know!
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