AMC 8 · 2025 · #4
Easy mode Grade 4Problem
Picture Lucius counting numbers out loud, but going backward. He starts at 100 and each time he says the next number, he goes down by 7.
So the first number he says is 100. The second is 93. The third is 86. He keeps going like this, dropping by 7 every step.
What is the 10th number that he says?
Pick an answer.
AMC 8 2025 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: 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
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.
💡 Generating the next few numbers from a stated rule is exactly the Grade 4 "pattern from a rule" skill.
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 $7$.
💡 Noticing that the $n$-th term is $n-1$ jumps from the start is a Grade 3 arithmetic-pattern observation.
3.OA.C.7 Step 3 - Find the total amount subtracted over $9$ jumps.
- Each jump removes $7$, so over $9$ jumps we remove $9 \times 7 = 63$.
💡 The product $9 \times 7$ is a Grade 3 "multiply within $100$" basic fact.
4.NBT.B.4 Step 4 Subtract that total from the starting number $100$ to land on the $10$th term.
💡 Subtracting a $2$-digit number from $100$ is a Grade 4 fluent subtraction with multi-digit whole numbers.
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".
💡 Comparing the computed value to the five listed numbers is straightforward Grade 4 whole-number reasoning.
4.OA.C.5 Confirm the pattern by listing a few more terms. The rule "subtract $7$" gives $ 3.OA.D.9 Count how many steps of $-7$ separate the $1$st number from the $10$th. From ter 3.OA.C.7 Find the total amount subtracted over $9$ jumps. Each jump removes $7$, so over 4.NBT.B.4 Subtract that total from the starting number $100$ to land on the $10$th term. 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.7Fluently 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.9Identify 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.5Generate 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.4Fluently 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!
⭐ This AMC 8 problem only needs Grade 4 number-pattern skills — "subtract the same amount over and over" — that you already know!
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