3.4 Translating Word Problems to Algebraic Equations

The Importance of Algebraic Translation on the TEAS 7

Translating word problems into algebraic equations is one of the most heavily tested skills on the ATI TEAS 7 Mathematics section. Rather than presenting equations pre-packaged for solving, the exam frequently presents real-world scenarios—such as fluid concentration mixing, patient intake/output tracking, age-weight dosage relationships, or nursing staff ratios—written entirely in conversational English.

To succeed, you must act as a mathematical translator: converting English words into variables, operation symbols, and algebraic equations. Once translated accurately, solving the problem relies on the exact linear equation algorithms mastered in prior sections.

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The English-to-Algebra Translation Dictionary

Every algebraic operation has key indicator words in English. Memorize this translation dictionary to recognize operations instantly under exam conditions:

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Handling "Turn-Around" Phrases

The single most common error on TEAS translation questions occurs with turn-around phrases in subtraction and division. In English, phrases such as "less than" or "subtracted from" place the subtracted quantity first, but algebraically it must be written last.

Exam Tip: Notice the difference between "less than" and "is less than". "8 less than $x$" is an expression ($x - 8$). "8 is less than $x$" is an inequality ($8 < x$).

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Universal 4-Step Word Problem Translation Protocol

When faced with any word problem on test day, apply this structured protocol:

1. Identify the Target Unknown: Read the final question sentence first to identify exactly what quantity you are asked to find.
2. Assign a Variable: Let $x$ (or another letter) represent the primary unknown quantity. Write down its units (e.g., $x = \text{mL of 50% solution}$).
3. Build the Equation: Translate the problem statement sentence by sentence, substituting your variable expressions.
4. Solve and Verify: Solve the linear equation and check that your answer makes physical and clinical sense.

Step-by-Step Worked Example 1: Translation of a Number Relationship

Problem: Six less than four times a number is equal to twice the sum of the number and three. Find the number.

• Step 1 (Assign Variable): Let $n =$ the unknown number.
• Step 2 (Translate Left Side): "Four times a number" $= 4n$. "Six less than four times a number" $= 4n - 6$.
• Step 3 (Translate Equals Sign): "is equal to" $= =$.
• Step 4 (Translate Right Side): "the sum of the number and three" $= (n + 3)$. "twice the sum..." $= 2(n + 3)$.
• Step 5 (Assemble Equation): $4n - 6 = 2(n + 3)$
• Step 6 (Solve):
  • Distribute: $4n - 6 = 2n + 6$
  • Subtract $2n$: $2n - 6 = 6$
  • Add $6$: $2n = 12 \implies n = 6$

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Healthcare Application: Solution & Medication Mixture Problems

In clinical pharmacy and IV therapy, nurses must often mix two solutions of different concentrations to create a desired intermediate strength. Mixture problems are built on the Solute Conservation Principle:

$\text{Solute from Solution 1} + \text{Solute from Solution 2} = \text{Total Solute in Final Mixture}$ $\left(C_1 \cdot V_1\right) + \left(C_2 \cdot V_2\right) = C_f \cdot \left(V_1 + V_2\right)$

where $C$ represents concentration percentage (as a decimal) and $V$ represents volume.

Clinical Problem: Dextrose IV Mixture

A nurse needs to prepare a $10%$ Dextrose solution. How many milliliters of a $50%$ Dextrose solution must be added to $400\text{ mL}$ of pure water ($0%$ Dextrose) to yield a $10%$ Dextrose mixture?

• Step 1 (Assign Variable): Let $x = \text{mL of 50% Dextrose solution}$.
• Step 2 (Determine Volumes & Concentrations):
  • Solution 1 (50% Dextrose): Volume $= x\text{ mL}$, Concentration $= 0.50$
  • Solution 2 (Pure Water): Volume $= 400\text{ mL}$, Concentration $= 0.00$
  • Final Mixture (10% Dextrose): Volume $= (x + 400)\text{ mL}$, Concentration $= 0.10$
• Step 3 (Set up the Solute Equation): $0.50(x) + 0.00(400) = 0.10(x + 400)$
• Step 4 (Simplify and Solve):
  • $0.50x + 0 = 0.10x + 40$
  • Subtract $0.10x$ from both sides: $0.40x = 40$
  • Divide by $0.40$: $x = \frac{40}{0.40} = 100\text{ mL}$
• Clinical Result: The nurse must add $100\text{ mL}$ of $50%$ Dextrose solution to $400\text{ mL}$ of water.

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Healthcare Application: 24-Hour Patient Fluid Balance (Intake & Output)

Tracking patient fluid balance over a shift requires translating relationships between total intake ($I$) and output ($O$).

Clinical Problem: Fluid Output Relationship

Over a 24-hour period, a surgical patient's total fluid output was $150\text{ mL}$ less than twice their total fluid intake. If the patient's combined fluid intake and output totaled $3,750\text{ mL}$, what was the patient's total fluid intake?

• Step 1 (Define Variables): Let $I = \text{fluid intake in mL}$.
• Step 2 (Express Output in terms of Intake): Output is "$150\text{ mL}$ less than twice intake" $\implies O = 2I - 150$.
• Step 3 (Set up Total Volume Equation): $\text{Intake} + \text{Output} = 3750 \implies I + (2I - 150) = 3750$
• Step 4 (Combine like terms and solve):
  • $3I - 150 = 3750$
  • Add $150$: $3I = 3900$
  • Divide by $3$: $I = 1,300\text{ mL}$
• Verification: Intake $= 1,300\text{ mL}$. Output $= 2(1300) - 150 = 2450\text{ mL}$. Total $= 1300 + 2450 = 3,750\text{ mL}$ (VERIFIED!).

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TEAS 4-Function Calculator Strategy

1. Decimal Conversions for Percentages: Always convert percentage strengths to decimals before performing calculator entry ($50% = 0.50$, $10% = 0.10$, $4% = 0.04$).
2. Calculating Fractional Divisors: When dividing by decimals such as $\frac{40}{0.40}$, enter 40 / 0.40 = directly to get 100 without rounding intermediate steps.

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Common Student Traps & How to Avoid Them

1. Direct Order Subtraction Trap: Translating '5 less than x' as $5 - x$. Remedy: Re-read 'less than' as 'subtracted from' to remind yourself to flip the order: $x - 5$.
2. Forgetting Final Combined Volume in Mixtures: Writing $0.50x = 0.10(400)$ without adding $x$ to the final volume. Remedy: Remember the final mixture volume is always the sum of the component volumes: $(V_1 + V_2)$.
3. Misidentifying Final Variable Request: Solving for $x = 8$ and choosing 8, when the question asked for Day Shift nurses ($2x + 4 = 20$). Remedy: Reread the exact question prompt before selecting your final quiz option.