AMC 8 · 2014 · #10

Grade 4 arithmetic

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

sequences-arithmeticmulti-digit-arithmetic identify-subproblems ↑ Prerequisites: sequences-arithmeticmulti-digit-arithmetic

📏 Short solution 💡 2 insights

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Problem

The first AMC $8$ was given in $1985$ and it has been given annually since that time. Samantha turned $12$ years old the year that she took the seventh AMC $8$ . In what year was Samantha born?

Pick an answer.

(A)

1979

(B)

1980

(C)

1981

(D)

1982

(E)

1983

AMC 8 2014 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.

View mode:

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?

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

#5 Look for a Pattern 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 $1985 + (n - 1)$.

$$1985, 1986, 1987, 1988, 1989, 1990, \underline{1991}$$

💡 Writing out the first few terms makes the rule obvious: add $1$ each step. This is the Grade $4$ "generate and analyze patterns" move.

#5 Look for a Pattern 4.NBT.B.4 Step 2

Confirm the pattern with the formula: the seventh contest is $6$ years after the first.

$$1985 + (7 - 1) = 1985 + 6 = 1991$$

💡 Adding $6$ to a four-digit year is the Grade $4$ "fluently add multi-digit whole numbers" skill.

#7 Identify Subproblems 4.NBT.B.4 Step 3

Now solve the second subproblem: given that Samantha was $12$ in $1991$, subtract to find her birth year.

$$1991 - 12 = 1979$$

💡 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.

#7 Identify Subproblems 4.NBT.B.4 Step 4

Match $1979$ to the answer choices.

$$1979 \;\Rightarrow\; \textbf{(A)}$$

💡 Reading the choice list is the final "close the subproblem" check.

[1] #5 4.OA.C.5 List the AMC $8$ years until the seventh one to see the pattern: each year goes

[2] #5 4.NBT.B.4 Confirm the pattern with the formula: the seventh contest is $6$ years after the

[3] #7 4.NBT.B.4 Now solve the second subproblem: given that Samantha was $12$ in $1991$, subtrac

[4] #7 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.5 Generate 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.4 Fluently 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!

Problem

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: A

Review

CCSS standards used (min grade 4)

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