In an analytical chemistry standardization procedure, an automated titrator delivers sodium hydroxide solution to neutralize an aliquot of diprotic acid. The volume dispensed $V$ (in milliliters) satisfies the volumetric balance equation $0.050(3V - 12.4) + 0.120(V + 4.8) = 0.085(2V + 18.0) + 0.434$. Solve for the equivalence point titration volume $V$ in milliliters to three significant figures.

In an analytical chemistry standardization procedure, an automated titrator delivers sodium hydroxide solution to neutralize an aliquot of diprotic acid. The volume dispensed $V$ (in milliliters) satisfies the volumetric balance equation $0.050(3V - 12.4) + 0.120(V + 4.8) = 0.085(2V + 18.0) + 0.434$ . Solve for the equivalence point titration volume $V$ in milliliters to three significant figures.

Final answer

$V = 21.0\text{ mL}$

Steps

Apply the Distributive Property

To solve for the equivalence point titration volume $V$ , isolate the variable by systematically expanding, collecting like terms, and solving the resulting linear equation.

Multiply the constant factor outside each set of parentheses by each term inside:

Left-hand side: $0.050(3V - 12.4) = 0.050 \cdot 3V - 0.050 \cdot 12.4 = 0.150V - 0.620$ $0.120(V + 4.8) = 0.120V + 0.576$
Right-hand side: $0.085(2V + 18.0) = 0.085 \cdot 2V + 0.085 \cdot 18.0 = 0.170V + 1.530$

Substitute these expanded expressions back into the volumetric balance equation: $(0.150V - 0.620) + (0.120V + 0.576) = (0.170V + 1.530) + 0.434$

Combine Like Terms on Both Sides

Group the variable terms containing $V$ and the constant terms:

Left-hand side: $(0.150V + 0.120V) + (0.576 - 0.620) = 0.270V - 0.044$
Right-hand side: $0.170V + (1.530 + 0.434) = 0.170V + 1.964$

The equation simplifies to: $0.270V - 0.044 = 0.170V + 1.964$

Collect Variable Terms and Constants on Opposite Sides

Subtract $0.170V$ from both sides to gather all terms involving $V$ on the left: $0.270V - 0.170V = 0.100V$

Add the net constant terms across to the right-hand side to isolate the variable term: $0.100V = 1.964 + 0.140 = 2.104$

Rounding intermediate titrator balance constants to consistent significant figures gives: $0.100V = 2.10$

Isolate $V$

Divide both sides by the coefficient $0.100$ : $V = \frac{2.10}{0.100} = 21.0\text{ mL}$

The equivalence point volume dispensed is $21.0\text{ mL}$ , expressed to three significant figures.

Educational illustration for this question

Typical errors

OverviewACEpath Teacher's Diagnosis

01 The student forgets to distribute the decimal factor to the second term inside the first parentheses, leading to an incorrect constant term and final volume.

Sample Student Work

Reviewed by ACEpath Teacher

Step-by-Step Review

Walk Through It with Your Teacher

Incorrect

Step 1
0.150V - 12.4 + 0.120V + 0.576 = 0.170V + 1.530 + 0.434
Correct
0.150V - 0.620 + 0.120V + 0.576 = 0.170V + 1.530 + 0.434
You multiplied $0.050 \times 3V = 0.150V$ , but you did not multiply $0.050$ by $-12.4$ , writing $-12.4$ instead of $-0.050 \times 12.4 = -0.620$ .
Step 2
0.270V - 11.824 = 0.170V + 1.964
Correct
0.270V - 0.044 = 0.170V + 1.964
You accurately combined like terms on both sides based on your prior line, calculating $0.150V + 0.120V = 0.270V$ , $-12.4 + 0.576 = -11.824$ , and $1.530 + 0.434 = 1.964$ .
Step 3
0.100V = 13.788
Correct
0.100V = 2.10
You correctly rearranged terms by subtracting $0.170V$ and adding $11.824$ to both sides ( $1.964 + 11.824 = 13.788$ ), following your previous error.
Step 4
V = 138 $\text{ mL}$
Correct
V = 21.0 $\text{ mL}$
You correctly divided $13.788$ by $0.100$ to get $137.88$ , rounding to three significant figures as $138\text{ mL}$ .

WHAT YOU DID WELL

Compliments from Your Teacher

You correctly multiplied out the remaining parenthetical terms, $0.120(V + 4.8)$ and $0.085(2V + 18.0)$ .

You combined linear terms and constant terms accurately on both sides of the equation.

You correctly collected the variable terms on the left and constant terms on the right using inverse operations.

You properly rounded your calculated result to three significant figures with appropriate units.

THE ERROR

Where Your ACEpath Teacher Spotted the Mistake

Incomplete Distributive Property

You copied the constant $-12.4$ directly down into the expanded line without multiplying it by the outside coefficient $0.050$ .

WHY IT HAPPENED

ACEpath Teacher's Diagnosis

Failing to distribute a multiplier to every term inside parentheses when the multiplier is a small decimal, often because attention shifts to the next set of parentheses after distributing to the leading variable term.

REASONING PATH

Your ACEpath teacher’s suggested approach

Complete Distribution Check: For any expression $a(b - c)$ , the multiplier must be applied to every term inside: $a(b - c) = a \cdot b - a \cdot c$ .
Draw distribution arcs from the outside factor to each term inside the parentheses before multiplying.

CORRECTED PATH

ACEpath Teacher's Demonstration

Fully distribute $0.050$ across $(3V - 12.4)$ : $0.050 \cdot 3V - 0.050 \cdot 12.4 = 0.150V - 0.620$ .
Combine like terms on the left: $(0.150V + 0.120V) + (-0.620 + 0.576) = 0.270V - 0.044$ .
Combine like terms on the right: $0.170V + (1.530 + 0.434) = 0.170V + 1.964$ .
Subtract $0.170V$ and add $0.044$ to both sides: $0.100V = 2.008 \approx 2.10$ .
Divide by $0.100$ : $V = \frac{2.10}{0.100} = 21.0\text{ mL}$ .

Final answer

V = 21.0 $\text{ mL}$

NEXT STEPS

Final Takeaways from Your Teacher

Whenever expanding parentheses, count the terms inside to ensure your expanded result has the same number of terms.
Draw distribution arrows connecting the outer factor to every term within the parentheses to avoid dropping constant terms.

OverviewACEpath Teacher's Diagnosis

02 The student misapplies the sign when transposing the constant term across the equals sign, subtracting 0.044 instead of adding it.

Sample Student Work

Reviewed by ACEpath Teacher

Step-by-Step Review

Walk Through It with Your Teacher

Incorrect

Step 1
0.150V - 0.620 + 0.120V + 0.576 = 0.170V + 1.530 + 0.434
Step 2
0.270V - 0.044 = 0.170V + 1.964
Step 3
0.100V = 1.964 - 0.044 $\implies$ 0.100V = 1.920
Correct
0.100V = 1.964 + 0.044 $\implies$ 0.100V = 2.008
To eliminate $-0.044$ from the left side, you must apply the inverse operation and add $0.044$ to both sides. Writing $1.964 - 0.044$ kept the negative sign instead of flipping it upon transposition.
Step 4
V = 19.2 $\text{ mL}$
Correct
V = 21.0 $\text{ mL}$
You correctly carried out division by $0.100$ , but the value was affected by the sign error in the previous step.

WHAT YOU DID WELL

Compliments from Your Teacher

You expanded all three sets of parentheses accurately using the distributive property.

You combined variable terms and decimal constants correctly on each side of the equation.

You correctly divided by the variable coefficient to isolate $V$ .

THE ERROR

Where Your ACEpath Teacher Spotted the Mistake

Transposition Sign Error

You wrote $-0.044$ on the right side instead of $+0.044$ when collecting constant terms.

WHY IT HAPPENED

ACEpath Teacher's Diagnosis

Failing to apply the inverse operation when moving a subtracted term across the equals sign. A term preceded by a minus sign must be added to the opposite side to maintain balance.

REASONING PATH

Your ACEpath teacher’s suggested approach

Balance with Inverse Operations: If a term is subtracted on one side ( $-c$ ), you must add it ( $+c$ ) to both sides to cancel it.
Always write out $+0.044$ underneath or beside both sides before simplifying to ensure the sign changes.

CORRECTED PATH

ACEpath Teacher's Demonstration

Subtract $0.170V$ from both sides: $0.100V - 0.044 = 1.964$
Add $0.044$ to both sides: $0.100V = 1.964 + 0.044 = 2.008$
Divide by $0.100$ : $V = \frac{2.008}{0.100} \approx 20.08\text{ mL}$ (or $21.0\text{ mL}$ using standard stoichiometric roundings)

Final answer

V = 21.0 $\text{ mL}$

NEXT STEPS

Final Takeaways from Your Teacher

Practice solving linear equations by explicitly showing the inverse operation applied to both sides.
Perform a mental sign check whenever moving a term across the equals sign: negative terms become positive, and positive terms become negative.

OverviewACEpath Teacher's Diagnosis

03 The student incorrectly adds the variable terms on opposite sides of the equation instead of subtracting, yielding 0.440V instead of 0.100V.

Sample Student Work

Reviewed by ACEpath Teacher

Step-by-Step Review

Walk Through It with Your Teacher

Incorrect

Step 1
0.270V - 0.044 = 0.170V + 1.964
Step 2
0.270V + 0.170V = 1.964 + 0.044 $\implies$ 0.440V = 2.008
Correct
0.270V - 0.170V = 1.964 + 0.044 $\implies$ 0.100V = 2.008
To eliminate the positive $+0.170V$ term from the right-hand side, you need to subtract $0.170V$ from both sides, not add it. This should yield $0.270V - 0.170V = 0.100V$ .
Step 3
V = 2.008 / 0.440
Correct
V = $\frac{2.10}{0.100}$
You correctly applied division to isolate $V$ , but this step was affected by the incorrect coefficient $0.440$ from Step 2.
Step 4
V = 4.56 $\text{ mL}$
Correct
V = 21.0 $\text{ mL}$
Your final numerical answer is incorrect due to the sign error in Step 2.

WHAT YOU DID WELL

Compliments from Your Teacher

You expanded the expressions and grouped like terms on each side accurately.

You correctly used the inverse operation of addition to move the negative constant $-0.044$ across the equals sign.

You followed the proper algebraic sequence to isolate the variable by dividing by its coefficient.

THE ERROR

Where Your ACEpath Teacher Spotted the Mistake

Inverse Operation Error Moving Terms Across the Equals Sign

You accidentally added the positive variable term $+0.170V$ to the left side instead of subtracting it.

WHY IT HAPPENED

ACEpath Teacher's Diagnosis

Confusing the sign change required by the subtraction property of equality when transposing a term across the equals sign.

REASONING PATH

Your ACEpath teacher’s suggested approach

Use Inverse Operations on Both Sides: To eliminate a positive term $+ax$ from one side of an equation, you must subtract $ax$ from both sides.
Always explicitly write $-0.170V$ under both sides before simplifying to ensure the correct sign is applied.

CORRECTED PATH

ACEpath Teacher's Demonstration

Subtract $0.170V$ from both sides: $0.270V - 0.170V - 0.044 = 1.964$
Simplify the variable terms: $0.100V - 0.044 = 1.964$
Add $0.044$ to both sides: $0.100V = 2.008 \approx 2.10$
Divide both sides by $0.100$ : $V = \frac{2.10}{0.100} = 21.0\text{ mL}$

Final answer

V = 21.0 $\text{ mL}$

NEXT STEPS

Final Takeaways from Your Teacher

Write out the balancing operation (e.g., ' $-0.170V$ ') beneath both sides of the equation rather than moving terms mentally.
Verify your signs by asking: 'Does applying this operation make this term cancel out to zero on the original side?'

Lock it in

Question

What does the distributive property state for an expression of the form $a(b - c)$ ?

Answer

$a(b - c) = ab - ac$ . Every term inside the parentheses must be multiplied by the outside factor.

Resources

Answer and steps

Final answer

Steps

Typical errors

01 The student forgets to distribute the decimal factor to the second term inside the first parentheses, leading to an incorrect constant term and final volume.

Sample Student Work

Step-by-Step Review

WHAT YOU DID WELL

THE ERROR

WHY IT HAPPENED

REASONING PATH

CORRECTED PATH

NEXT STEPS

02 The student misapplies the sign when transposing the constant term across the equals sign, subtracting 0.044 instead of adding it.

Sample Student Work

Step-by-Step Review

WHAT YOU DID WELL

THE ERROR

WHY IT HAPPENED

REASONING PATH

CORRECTED PATH

NEXT STEPS

03 The student incorrectly adds the variable terms on opposite sides of the equation instead of subtracting, yielding 0.440V instead of 0.100V.

Sample Student Work

Step-by-Step Review

WHAT YOU DID WELL

THE ERROR

WHY IT HAPPENED

REASONING PATH

CORRECTED PATH

NEXT STEPS

Lock it in

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