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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.5Step 1
List a few more terms
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.9Step 2
Count the jumps
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.7Step 3
Find the total decrease
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.4Step 4
Subtract from 100
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.4Step 5
Match against the choices
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.
[1]
#5 4.OA.C.5Confirm the pattern by listing a few more terms. The rule "subtract $7$" gives $
[2]
#5 3.OA.D.9Count how many steps of $-7$ separate the $1$st number from the $10$th. From ter
[3]
#5 3.OA.C.7Find the total amount subtracted over $9$ jumps. Each jump removes $7$, so over
[4]
#5 4.NBT.B.4Subtract that total from the starting number $100$ to land on the $10$th term.
[5]
#3 4.NBT.B.4Check 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!
⭐ 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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