3.4 Multi-Step Problem Solving & Algebraic Word Problems

3.4 Multi-Step Problem Solving & Algebraic Word Problems

Multi-step algebraic word problems evaluate your ability to breakdown complex real-world scenarios into structured mathematical equations. Success on these higher-level AFCT Arithmetic Reasoning questions requires a systematic translation framework, precise variable definitions, and careful avoidance of distractor trap answers.

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1. The AFCT 5-Step Problem-Solving Framework

When approaching multi-step word problems, follow this disciplined 5-step process:

1. Identify the Target Question: Read the final sentence first to determine exactly what quantity you are solving for (e.g., "the larger integer", "Ross's present age", or "the total cost per budget").
2. Define Variables: Choose clear variable names representing unknown quantities. Express secondary unknowns in terms of a single primary variable whenever possible.
3. Formulate the Equation: Translate the verbal relationships into a mathematical equation using key algebraic operational keywords.
4. Solve the Equation: Perform algebraic manipulations (combining like terms, isolating the variable).
5. Verify and Check: Plug the result back into the original word problem context to confirm it makes physical and numerical sense.

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2. Consecutive Integer Problems

Consecutive integers are integers that follow each other in order without gaps.

Algebraic Representations

• Consecutive Integers: $n, n+1, n+2, n+3, \dots$
• Consecutive Even Integers: $n, n+2, n+4, n+6, \dots$ (where $n$ is even)
• Consecutive Odd Integers: $n, n+2, n+4, n+6, \dots$ (where $n$ is odd)

AFCT Shortcut: The average (mean) of a set of consecutive integers equals the middle term. If the sum of 3 consecutive integers is $S$, the middle integer is $\frac{S}{3}$.

Step-by-Step Worked Example 1: The sum of three consecutive even integers is 78. What is the value of the largest integer?

• Step 1: Let integers be $n$, $n+2$, and $n+4$.
• Step 2: Set up sum equation: $n + (n+2) + (n+4) = 78$.
• Step 3: Combine like terms: $3n + 6 = 78$.
• Step 4: Solve for $n$: $3n = 72 \implies n = 24$.
• Step 5: Determine largest integer ($n+4$): $24 + 4 = 28$.

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3. Age Word Problems & Timeline Matrix Technique

Age word problems compare the ages of two or more individuals across different time frames (past, present, future).

Constructing the Age Matrix

Always construct a 2D timeline matrix to keep relationships clear:

Step-by-Step Worked Example 2: Captain Vance is currently 4 times as old as Lieutenant Ross. In 10 years, Captain Vance will be only 2 times as old as Lieutenant Ross. How old is Lieutenant Ross right now?

• Step 1: Define present ages: Let Ross's present age $= R$. Vance's present age $= 4R$.
• Step 2: Express future ages in 10 years: Ross $= R + 10$, Vance $= 4R + 10$.
• Step 3: Set up future equation: $\text{Vance}{10} = 2 \times (\text{Ross}{10})$.
• Step 4: Substitute expressions: $4R + 10 = 2(R + 10)$.
• Step 5: Expand and solve: $4R + 10 = 2R + 20 \implies 2R = 10 \implies R = 5\text{ years old}$.

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4. Multi-Step Budgeting & Supply Allocation Math

Procurement and supply problems require tracking multiple item quantities, tier pricing structures, flat fees, and distribution rules.

Step-by-Step Worked Example 3: A military unit purchases 15 cases of equipment at $120 per case, 8 cases at $150 per case, and pays a flat shipping fee of $100. If the total cost is divided equally among 5 logistics budgets, how much is charged to each budget?

• Step 1: Calculate cost of first equipment tier: $15 \times $120 = $1,800$.
• Step 2: Calculate cost of second equipment tier: $8 \times $150 = $1,200$.
• Step 3: Sum sub-costs and shipping: $$1,800 + $1,200 + $100 = $3,100\text{ total cost}$.
• Step 4: Divide total cost by 5 budgets: $\frac{$3,100}{5} = $620\text{ per budget}$.

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5. Common AFCT Arithmetic Reasoning Traps