2.3 Word Problems & Multi-Step Reasoning

Many NAPT math items are not presented as bare equations — they are short scenarios you must translate into arithmetic, algebra, or geometry before you can solve anything. This section builds a repeatable method for reading a word problem, works through multi-step reasoning chains that combine several math skills in one scenario, and covers basic probability and combinatorics (counting) word problems.

Translating Word Problems into Equations

The core skill is converting an English sentence into a mathematical expression. Four steps make this repeatable:

1. Read the problem twice and identify exactly what is being asked — the unknown quantity.
2. Assign a variable to the unknown (commonly x, but any letter works).
3. Translate keywords into operations.
4. Write the equation, solve it, and check the answer against the original scenario — does the number make sense?

Keyword Translation Table

Worked Example

A recruit's monthly base pay is $50 more than twice his housing allowance. His total monthly pay (base pay plus allowance) is $2,150. What is his housing allowance?

Let a = the allowance. Then base pay = 2a + 50 (the phrase twice means × 2, and more than means + 50).

The word total tells us to add base pay and allowance:

a + (2a + 50) = 2,150 3a + 50 = 2,150 3a = 2,100 a = $700

Checking the answer: base pay = 2(700) + 50 = $1,450, and 700 + 1,450 = $2,150, which matches the total given.

Multi-Step Reasoning Chains

Many NAPT problems require two or more separate calculations, where the result of one step becomes the input to the next. The key is to solve one step completely, write down the intermediate result, and only then move to the next step — trying to do it all in your head increases the chance of an error.

Worked Example: Rate + Arithmetic

A ship travels at 24 knots for 3.5 hours, then reduces speed to 18 knots for 2 hours while entering a channel. How many total nautical miles did it travel?

• Step 1 (distance = speed × time): 24 × 3.5 = 84 nautical miles
• Step 2: 18 × 2 = 36 nautical miles
• Step 3 (combine): 84 + 36 = 120 nautical miles

Worked Example: Combined Work Rate (Algebra)

Pump A can fill a tank alone in 6 hours; Pump B can fill the same tank alone in 3 hours. Working together, how long will it take to fill the tank?

Each pump's rate is 1 tank divided by its time alone:

• Pump A's rate = 1/6 tank per hour
• Pump B's rate = 1/3 tank per hour
• Combined rate = 1/6 + 1/3 = 1/6 + 2/6 = 3/6 = 1/2 tank per hour
• Time = 1 ÷ (1/2) = 2 hours

Worked Example: Geometry + Algebra Combined

A rectangular storage room has a perimeter of 46 feet. Its length is 5 feet more than its width. What is the room's area?

Let w = width, so length l = w + 5. The rectangle perimeter formula is P = 2(l + w):

2[(w + 5) + w] = 46 (w + 5) + w = 23 2w + 5 = 23 2w = 18 w = 9, so l = 9 + 5 = 14 feet

Area = l × w = 14 × 9 = 126 square feet

This example chains three skills in a single problem: translating the phrase 5 feet more than into algebra, solving a linear equation for the width, and then applying the rectangle area formula from Section 2.1.

Basic Probability & Combinatorics Word Problems

Probability measures how likely an event is to occur, expressed as a fraction, decimal, or percent between 0 (impossible) and 1 (certain):

Probability = (favorable outcomes) / (total possible outcomes)

Worked Example: Simple Probability

A box contains 5 red pens and 3 blue pens. If one pen is drawn at random, what is the probability it is blue?

Probability = 3 / (5 + 3) = 3/8 = 0.375 (37.5%)

Worked Example: Independent Events

Independent events are events where one outcome does not affect the other — for independent events, multiply the individual probabilities together. If two fair coins are flipped, what is the probability both land heads?

P(heads) × P(heads) = (1/2) × (1/2) = 1/4 (25%)

Combinatorics: The Fundamental Counting Principle

Combinatorics is the branch of math that counts possible arrangements or selections. The fundamental counting principle says: if one choice can be made in m ways and a second, independent choice can be made in n ways, the two choices together can be made in m × n ways.

A locker combination uses 3 digits, each digit 0–9, and digits may repeat. How many possible combinations exist?

10 × 10 × 10 = 1,000 combinations

Combinatorics: Choosing a Group (Combinations)

When the order of a selection does not matter (choosing a team, not ranking it), use the combination formula:

C(n, r) = n! / [r!(n − r)!]

where n! (n factorial) means n × (n − 1) × (n − 2) × ... × 1.

How many different 2-person teams can be formed from a group of 5 sailors?

C(5, 2) = 5! / (2! × 3!) = (5 × 4 × 3 × 2 × 1) / [(2 × 1) × (3 × 2 × 1)] = 120 / 12 = 10 possible teams