In an analytical chemistry lab, a student standardizes a sodium hydroxide ($\text{NaOH}$) titrant against primary standard potassium hydrogen phthalate ($\text{KHP}$, molar mass $= 204.22\,\text{g/mol}$). The stoichiometric titration equation yields the delivered titrant volume $V_{\text{titr}}$ (in $\text{mL}$) as a function of the sample mass purity factor $p$: $V_{\text{titr}}(p) = \frac{1}{0.1025} \left[ 1000 \left( \frac{0.5105 p}{204.22} \right) - 0.045(1.20p - 0.85) \right]$. Use the distributive property to expand the expression, collect the terms linear in the purity parameter $p$ and the constant baseline correction, and report $V_{\text{titr}}(p)$ in the form $mp + b$ with coefficients rounded to four significant figures.

In an analytical chemistry lab, a student standardizes a sodium hydroxide ( $\text{NaOH}$ ) titrant against primary standard potassium hydrogen phthalate ( $\text{KHP}$ , molar mass $= 204.22\,\text{g/mol}$ ). The stoichiometric titration equation yields the delivered titrant volume $V_{\text{titr}}$ (in $\text{mL}$ ) as a function of the sample mass purity factor $p$ : $V_{\text{titr}}(p) = \frac{1}{0.1025} \left[ 1000 \left( \frac{0.5105 p}{204.22} \right) - 0.045(1.20p - 0.85) \right]$ . Use the distributive property to expand the expression, collect the terms linear in the purity parameter $p$ and the constant baseline correction, and report $V_{\text{titr}}(p)$ in the form $mp + b$ with coefficients rounded to four significant figures.

Final answer

$V_{\text{titr}}(p) = 23.86p + 0.3732\,\text{mL}$

Steps

Simplify and expand the terms inside the square brackets

To simplify the expression into the linear form $V_{\text{titr}}(p) = mp + b$ , we apply the distributive property step-by-step from the inside out.

The expression inside the brackets is: $1000 \left( \frac{0.5105 p}{204.22} \right) - 0.045(1.20p - 0.85)$

Compute the coefficient of the first term: $1000 \left( \frac{0.5105}{204.22} \right) p = \frac{510.5}{204.22} p \approx 2.499755 p$
Use the distributive property to expand the second term: $-0.045(1.20p - 0.85) = -0.045(1.20p) - 0.045(-0.85)$ $-0.045(1.20p) = -0.054p$ $-0.045(-0.85) = +0.03825$

Common Pitfall: When distributing a negative factor like $-0.045$ across a subtraction, remember that multiplying two negative numbers yields a positive constant term: $-0.045 \cdot (-0.85) = +0.03825$ .

Combine like terms inside the brackets

Group the terms containing the variable $p$ and the constant term: $\left(2.499755 p - 0.054 p\right) + 0.03825 = 2.445755 p + 0.03825$

Distribute the outer scaling factor

Now multiply each term inside the brackets by the outer factor $\frac{1}{0.1025}$ : $V_{\text{titr}}(p) = \frac{1}{0.1025} \left(2.445755 p + 0.03825\right)$ $V_{\text{titr}}(p) = \left(\frac{2.445755}{0.1025}\right) p + \left(\frac{0.03825}{0.1025}\right)$

Evaluating each quotient:

Slope $m = \frac{2.445755}{0.1025} \approx 23.8610 \dots$
Intercept $b = \frac{0.03825}{0.1025} \approx 0.37317 \dots$

Round to four significant figures

Rounding both coefficients to four significant figures:

$m \approx 23.86$
$b \approx 0.3732$

Writing the final equation with units: $V_{\text{titr}}(p) = 23.86p + 0.3732\,\text{mL}$

Educational illustration for this question

Typical errors

OverviewACEpath Teacher's Diagnosis

01 The student forgets to distribute the negative sign to the constant term inside parentheses, writing -0.03825 instead of +0.03825.

Sample Student Work

Reviewed by ACEpath Teacher

Step-by-Step Review

Walk Through It with Your Teacher

Incorrect

Step 1
$1000 * \left(\frac{0.5105}{204.22}\right) = 2.4998$
Step 2
-0.045(1.20p - 0.85) = -0.054p - 0.03825
Correct
-0.045(1.20p - 0.85) = -0.054p + 0.03825
When distributing $-0.045$ across $(1.20p - 0.85)$ , multiplying two negative values yields a positive constant: $(-0.045) \times (-0.85) = +0.03825$ .
Step 3
= $\frac{1}{0.1025}$ * [2.4458p - 0.03825]
Correct
= $\frac{1}{0.1025}$ [2.4458p + 0.03825]
You combined the $p$ -terms correctly, but this expression inherits the negative sign error from the previous step.
Step 4
= 23.86p - 0.3732
Correct
$V_{\text{titr}}(p) = 23.86p + 0.3732\,\text{mL}$
Your arithmetic division by $0.1025$ and significant-figure rounding were accurate, but the sign of the constant term remains negative due to the earlier distribution mistake.

WHAT YOU DID WELL

Compliments from Your Teacher

You correctly calculated the decimal coefficient for the first term.

You combined the linear terms $2.4998p - 0.054p = 2.4458p$ accurately.

You correctly divided each term by the outer scaling factor $0.1025$ and rounded to four significant figures.

THE ERROR

Where Your ACEpath Teacher Spotted the Mistake

Distributive Sign Error

You forgot that multiplying a negative factor by a subtracted term produces a positive product.

WHY IT HAPPENED

ACEpath Teacher's Diagnosis

Incomplete distribution of the negative sign across parentheses of the form $-a(bx - c)$ , failing to recognize that $-a \cdot (-c) = +ac$ .

REASONING PATH

Your ACEpath teacher’s suggested approach

Distribute Signs with Coefficients: -a(b - c) = -a(b) + (-a)(-c) = -ab + ac
Always circle or explicitly pair the negative sign with the outer factor before multiplying each internal term.

CORRECTED PATH

ACEpath Teacher's Demonstration

-0.045(1.20p - 0.85) = -0.045(1.20p) + (-0.045)(-0.85) = -0.054p + 0.03825
Inside brackets: (2.499755p - 0.054p) + 0.03825 = 2.445755p + 0.03825
Distribute $\frac{1}{0.1025}$ : $\frac{2.445755}{0.1025}$ p + $\frac{0.03825}{0.1025}$ = 23.861p + 0.37317

Final answer

$V_{\text{titr}}(p) = 23.86p + 0.3732\,\text{mL}$

NEXT STEPS

Final Takeaways from Your Teacher

Rewrite expressions of the form $-a(b - c)$ as $-a(b + (-c))$ to keep track of the signs during distribution.
Perform a sanity check on signs: subtracting a negative quantity must increase the baseline value, resulting in a positive intercept.

OverviewACEpath Teacher's Diagnosis

02 The student divides only the first term inside the brackets by 0.1025 and leaves the constant term unscaled.

Sample Student Work

Reviewed by ACEpath Teacher

Step-by-Step Review

Walk Through It with Your Teacher

Partially correct

Step 1
$V_{\text{titr}}(p) = \frac{1}{0.1025} * [2.4998p - 0.054p + 0.03825] \implies = \frac{1}{0.1025} * [2.4458p + 0.03825]$
Step 2
$= \frac{2.4458}{0.1025} p + 0.03825$
Correct
$= \frac{2.4458}{0.1025} p + \frac{0.03825}{0.1025}$
When applying the distributive property $c(A + B) = cA + cB$ , the multiplier $\frac{1}{0.1025}$ must multiply every term inside the brackets. You divided the linear term by $0.1025$ but left the constant $0.03825$ unscaled.
Step 3
$= 23.86p + 0.03825$
Correct
$= 23.86p + 0.3732\,\text{mL}$
Your division of $\frac{2.4458}{0.1025} \approx 23.86$ was accurate, but the final intercept carries forward the unscaled constant from Step 2.

WHAT YOU DID WELL

Compliments from Your Teacher

You correctly computed the quotient and coefficient for the primary standard term inside the brackets.

You handled the sign change correctly when distributing $-0.045$ across $(-0.85)$ to yield $+0.03825$ .

You combined the linear terms inside the bracket accurately to obtain $2.4458p$ .

THE ERROR

Where Your ACEpath Teacher Spotted the Mistake

Incomplete Distribution

You divided only the first term inside the brackets by $0.1025$ , leaving the constant term $+0.03825$ unscaled.

WHY IT HAPPENED

ACEpath Teacher's Diagnosis

Overlooking that an outside scalar multiplier or divisor applies to all grouped terms enclosed within parentheses or brackets, not just the leading variable term.

REASONING PATH

Your ACEpath teacher’s suggested approach

Distribute to All Terms: For any grouping, $k(A + B) = kA + kB$ and $\frac{A + B}{d} = \frac{A}{d} + \frac{B}{d}$ .
Draw arrows from the outer multiplier or divisor to every term inside the brackets before computing values.

CORRECTED PATH

ACEpath Teacher's Demonstration

Distribute $\frac{1}{0.1025}$ to both terms: $V_{\text{titr}}(p) = \frac{2.4458}{0.1025}p + \frac{0.03825}{0.1025}$ .
Calculate each term: $\frac{2.4458}{0.1025} \approx 23.8610$ and $\frac{0.03825}{0.1025} \approx 0.37317$ .
Round each coefficient to four significant figures and attach the volume unit.

Final answer

$V_{\text{titr}}(p) = 23.86p + 0.3732\,\text{mL}$

NEXT STEPS

Final Takeaways from Your Teacher

Practice drawing explicit distribution arcs from outer factors to every term inside brackets.
Check that the units and physical scale of the constant term match expectations after division by small decimal factors.

OverviewACEpath Teacher's Diagnosis

03 The student multiplies by 0.1025 instead of dividing by it when distributing the outer factor.

Sample Student Work

Reviewed by ACEpath Teacher

Step-by-Step Review

Walk Through It with Your Teacher

Incorrect

Step 1
$V_{\text{titr}}(p) = 0.1025 * [2.4998p - 0.054p + 0.03825]$
Correct
$V_{\text{titr}}(p) = \frac{1}{0.1025} [2.4998p - 0.054p + 0.03825]$
You correctly evaluated $1000 \left(\frac{0.5105}{204.22}\right)p \approx 2.4998p$ and distributed $-0.045(1.20p - 0.85) = -0.054p + 0.03825$ . However, the leading factor in the original problem is $\frac{1}{0.1025}$ , which means dividing by $0.1025$ , not multiplying by $0.1025$ .
Step 2
= 0.1025 * [2.4458p + 0.03825]
Correct
= $\frac{1}{0.1025}$ [2.4458p + 0.03825]
You correctly combined the $p$ -terms ( $2.4998p - 0.054p = 2.4458p$ ), but this line is affected by the incorrect outer factor.
Step 3
= 0.1025 * 2.4458p + 0.1025 * 0.03825 $\implies$ 0.2507p + 0.003921
Correct
$= \left(\frac{2.4458}{0.1025}\right)p + \frac{0.03825}{0.1025} \implies 23.86p + 0.3732$
You applied the distributive property structure correctly to your expression, but distributing the incorrect factor produced coefficients that are too small by a factor of $0.1025^2$ .

WHAT YOU DID WELL

Compliments from Your Teacher

You handled the negative sign distribution $-0.045(-0.85) = +0.03825$ with complete accuracy.

You combined the linear terms in $p$ correctly inside the brackets.

You demonstrated a solid understanding of how to distribute an outer factor across a sum.

THE ERROR

Where Your ACEpath Teacher Spotted the Mistake

Reciprocal Factor Inversion

You replaced the fraction factor $\frac{1}{0.1025}$ with multiplication by $0.1025$ .

WHY IT HAPPENED

ACEpath Teacher's Diagnosis

Confusing a unit-fraction scaling factor $\frac{1}{a}$ with multiplication by $a$ , leading to multiplying the grouped terms by $0.1025$ rather than dividing by $0.1025$ .

REASONING PATH

Your ACEpath teacher’s suggested approach

Fraction Factor vs. Direct Multiplier: Multiplying by $\frac{1}{c}$ is equivalent to dividing by $c$ , not multiplying by $c$ : $\frac{1}{c}(u + v) = \frac{u}{c} + \frac{v}{c}$ .
Always copy fraction coefficients exactly as given before distributing or evaluating.
Order of Magnitude Sanity Check: Dividing by a small positive number less than $1$ (like $0.1025 \approx \frac{1}{10}$ ) makes quantities roughly $10$ times larger, whereas multiplying by it makes them $10$ times smaller.
Check whether distributing your outer scalar should scale terms up or down before computing the final numbers.

CORRECTED PATH

ACEpath Teacher's Demonstration

Retain the outer fraction factor: $V_{\text{titr}}(p) = \frac{1}{0.1025} (2.4458p + 0.03825)$ .
Distribute by dividing each coefficient by $0.1025$ : $V_{\text{titr}}(p) = \left(\frac{2.4458}{0.1025}\right)p + \left(\frac{0.03825}{0.1025}\right)$ .
Compute the values: $\frac{2.4458}{0.1025} \approx 23.8615$ and $\frac{0.03825}{0.1025} \approx 0.37317$ .
Round each coefficient to four significant figures and include units.

Final answer

$V_{\text{titr}}(p) = 23.86p + 0.3732\,\text{mL}$

NEXT STEPS

Final Takeaways from Your Teacher

Review how reciprocal factors $\frac{1}{k}$ distribute over sums.
Practice distributing both $\frac{1}{k}$ and $k$ across binomial expressions to make the distinction automatic.

Lock it in

Question

What is the result of expanding $-a(b - c)$ using the distributive property?

Answer

$-ab + ac$ . A negative multiplied by a negative yields a positive term.

Resources

Answer and steps

Final answer

Steps

Typical errors

01 The student forgets to distribute the negative sign to the constant term inside parentheses, writing -0.03825 instead of +0.03825.

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 divides only the first term inside the brackets by 0.1025 and leaves the constant term unscaled.

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 multiplies by 0.1025 instead of dividing by it when distributing the outer factor.

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

Similar question