AMC 8 · 2025 · #4

Grade 4 arithmetic

Archetype Arithmetic Expression Decomposition

sequences-arithmeticpattern-recognitionmulti-digit-arithmetic pattern-recognitionformula-substitution ↑ Prerequisites: multi-digit-arithmetic

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Problem

Lucius is counting backward by $7$ s. His first three numbers are $100$ , $93$ , and $86$ . What is his $10$ th number?

Pick an answer.

(A)

(B)

(C)

(D)

(E)

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Toolkit + CCSS Solution

Understand

Restated: Lucius counts backward by $7$, starting from $100$. So the numbers he says are $100$, $93$, $86$, and so on. What is the $10$th number he says?

Givens: The first number is $100$; Each new number is $7$ less than the previous one (the second is $93$, the third is $86$); Answer choices: (A) $30$, (B) $37$, (C) $42$, (D) $44$, (E) $47$

Unknowns: The $10$th number in Lucius's countdown sequence

Understand

Restated: Lucius counts backward by $7$, starting from $100$. So the numbers he says are $100$, $93$, $86$, and so on. What is the $10$th number he says?

Givens: The first number is $100$; Each new number is $7$ less than the previous one (the second is $93$, the third is $86$); Answer choices: (A) $30$, (B) $37$, (C) $42$, (D) $44$, (E) $47$

Plan

Primary tool: #5 Look for a Pattern

Secondary: #3 Eliminate Possibilities

The numbers $100, 93, 86, \dots$ are a clean repeating pattern: each step subtracts $7$. Tool #5 (Look for a Pattern) is the natural fit — list a few more terms to confirm the rule, then jump to the $10$th term. Because we are counting from the $1$st term, the $10$th term is $9$ steps away, so we subtract $7$ a total of $9$ times (equivalently subtract $9 \times 7 = 63$). Tool #3 (Eliminate) is a fast sanity check on the multiple-choice list — only one of the five choices can be $100 - 63$.

Execute — Answer: B

#5 Look for a Pattern 4.OA.C.5 Step 1

Confirm the pattern by listing a few more terms.
The rule "subtract $7$" gives $100, 93, 86, 79, 72, \dots$ — the differences $100-93$, $93-86$, $86-79$, $79-72$ are all $7$, so the rule is correct.

$$100,\;93,\;86,\;79,\;72,\;\dots$$

💡 Generating the next few numbers from a stated rule is exactly the Grade 4 "pattern from a rule" skill.

#5 Look for a Pattern 3.OA.D.9 Step 2

Count how many steps of $-7$ separate the $1$st number from the $10$th.
From term $1$ to term $10$ there are $10 - 1 = 9$ jumps, and each jump subtracts $7$.

$$\text{jumps} = 10 - 1 = 9$$

💡 Noticing that the $n$-th term is $n-1$ jumps from the start is a Grade 3 arithmetic-pattern observation.

#5 Look for a Pattern 3.OA.C.7 Step 3

Find the total amount subtracted over $9$ jumps.
Each jump removes $7$, so over $9$ jumps we remove $9 \times 7 = 63$.

$$9 \times 7 = 63$$

💡 The product $9 \times 7$ is a Grade 3 "multiply within $100$" basic fact.

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

Subtract that total from the starting number $100$ to land on the $10$th term.

$$a_{10} = 100 - 63 = 37$$

💡 Subtracting a $2$-digit number from $100$ is a Grade 4 fluent subtraction with multi-digit whole numbers.

#3 Eliminate Possibilities 4.NBT.B.4 Step 5

Check against the choices.
Only one choice equals $37$, which is choice (B); every other option fails the rule "$100$ minus $9$ sevens".

$$\text{Match: } 37 \;\Rightarrow\; \textbf{(B)}$$

💡 Comparing the computed value to the five listed numbers is straightforward Grade 4 whole-number reasoning.

[1] #5 4.OA.C.5 Confirm the pattern by listing a few more terms. The rule "subtract $7$" gives $

[2] #5 3.OA.D.9 Count how many steps of $-7$ separate the $1$st number from the $10$th. From ter

[3] #5 3.OA.C.7 Find the total amount subtracted over $9$ jumps. Each jump removes $7$, so over

[4] #5 4.NBT.B.4 Subtract that total from the starting number $100$ to land on the $10$th term.

[5] #3 4.NBT.B.4 Check against the choices. Only one choice equals $37$, which is choice (B); eve

Review

Reasonableness: Each step shrinks the number by $7$, so after $9$ steps we should be well below $100$ but still positive (since $9 \times 7 = 63 < 100$). The answer $37$ lies between $30$ and $44$ — exactly where the choices cluster — and a direct hand-count $100, 93, 86, 79, 72, 65, 58, 51, 44, 37$ confirms the $10$th number is $37$.

Alternative: Tool #2 (Systematic List): just write all ten terms in order — $100, 93, 86, 79, 72, 65, 58, 51, 44, 37$ — and read off the last one. That avoids any multiplication and is a great backup for a learner who prefers counting over formulas.

CCSS standards used (min grade 4)

3.OA.C.7 Fluently multiply and divide within 100 (Computing $9 \times 7 = 63$, the total amount subtracted over the nine jumps from term 1 to term 10.)
3.OA.D.9 Identify arithmetic patterns and explain using properties of operations (Recognizing that reaching the $10$th term from the $1$st takes $9$ equal jumps in the "$-7$" pattern.)
4.OA.C.5 Generate a number or shape pattern following a given rule (Extending the sequence $100, 93, 86, \dots$ using the stated "subtract $7$" rule to confirm the pattern.)
4.NBT.B.4 Fluently add and subtract multi-digit whole numbers (Performing the subtraction $100 - 63 = 37$ to land on the $10$th term and matching it to choice (B).)

⭐ This AMC 8 problem only needs Grade 4 number-pattern skills — "subtract the same amount over and over" — that you already know!

Problem

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: B

Review

CCSS standards used (min grade 4)

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