A chemical engineer models the mass flow rate through a microfluidic channel in milligrams per minute ($\text{mg/min}$). The rate is governed by $R(t) = C \cdot \left[ 1.5 \times 10^3 (2.4t - 0.8) - 4.5 \times 10^2 (5.0t - 1.2) \right]$, where $t$ is time in seconds and $C = 2.5 \times 10^{-4}\,\text{g/mg}$ is a conversion scale factor. Use the distributive property to expand the expression inside the brackets, convert the rate so that all mass units are expressed in grams per minute ($\text{g/min}$), and state the simplified linear function $R(t)$ in standard form.

A chemical engineer models the mass flow rate through a microfluidic channel in milligrams per minute ( $\text{mg/min}$ ). The rate is governed by $R(t) = C \cdot \left[ 1.5 \times 10^3 (2.4t - 0.8) - 4.5 \times 10^2 (5.0t - 1.2) \right]$ , where $t$ is time in seconds and $C = 2.5 \times 10^{-4}\,\text{g/mg}$ is a conversion scale factor. Use the distributive property to expand the expression inside the brackets, convert the rate so that all mass units are expressed in grams per minute ( $\text{g/min}$ ), and state the simplified linear function $R(t)$ in standard form.

Final answer

$R(t) = 0.3375t - 0.165\,\text{g/min}$

Steps

Expand the terms inside the brackets

To find the simplified linear function $R(t)$ in $\text{g/min}$ , we apply the distributive property step by step: first to the terms inside the brackets, and then to the conversion factor $C$ across the simplified expression.

The expression inside the brackets is: $1.5 \times 10^3 (2.4t - 0.8) - 4.5 \times 10^2 (5.0t - 1.2)$

Convert the scientific notation into standard decimal values to simplify multiplication:

$1.5 \times 10^3 = 1500$
$4.5 \times 10^2 = 450$

Apply the distributive property $a(b - c) = ab - ac$ to each group: $1500 \cdot (2.4t - 0.8) = (1500 \cdot 2.4t) - (1500 \cdot 0.8) = 3600t - 1200$ $-450 \cdot (5.0t - 1.2) = (-450 \cdot 5.0t) - (-450 \cdot 1.2) = -2250t + 540$

Common Pitfall: When distributing the negative factor $-450$ across $(5.0t - 1.2)$ , remember that multiplying two negative numbers results in a positive constant term: $(-450) \cdot (-1.2) = +540$ .

Combine like terms inside the brackets

Group the variable terms and the constant terms together: $\left(3600t - 2250t\right) + \left(-1200 + 540\right) = 1350t - 660$

So, the unscaled rate in $\text{mg/min}$ is: $1350t - 660$

Distribute the conversion factor $C$

Now, multiply the expression by the conversion scale factor $C = 2.5 \times 10^{-4} = 0.00025\,\text{g/mg}$ : $R(t) = 0.00025 \cdot (1350t - 660)$

Distribute $0.00025$ to both terms: $R(t) = (0.00025 \cdot 1350t) - (0.00025 \cdot 660)$ $R(t) = 0.3375t - 0.165$

Thus, the simplified linear function in standard form is: $R(t) = 0.3375t - 0.165\,\text{g/min}$

Educational illustration for this question

Typical errors

OverviewACEpath Teacher's Diagnosis

01 The student fails to distribute the negative sign to the second term inside the second parentheses, writing - 540 instead of + 540.

Sample Student Work

Reviewed by ACEpath Teacher

Step-by-Step Review

Walk Through It with Your Teacher

Incorrect

Step 1
1500(2.4t - 0.8) - 450(5.0t - 1.2) $\implies$ 3600t - 1200 - 2250t - 540
Correct
1500(2.4t - 0.8) - 450(5.0t - 1.2) $\implies$ 3600t - 1200 - 2250t + 540
When distributing $-450$ to the second term inside the parentheses, multiplying two negative numbers yields a positive product: $(-450) \cdot (-1.2) = +540$ , not $-540$ .
Step 2
1350t - 1740
Correct
1350t - 660
You correctly combined your terms according to your previous line ( $-1200 - 540 = -1740$ ), but using the correct sign yields $-1200 + 540 = -660$ .
Step 3
R(t) = 2.5 $\times$ 10^{-4}(1350t - 1740) $\implies$ R(t) = 0.3375t - 0.435
Correct
R(t) = 2.5 $\times$ 10^{-4}(1350t - 660) $\implies$ R(t) = 0.3375t - 0.165
You accurately multiplied by the conversion factor $2.5 \times 10^{-4}$ , but the constant term carried over the earlier sign error.

WHAT YOU DID WELL

Compliments from Your Teacher

You accurately converted $1.5 \times 10^3$ to $1500$ and $4.5 \times 10^2$ to $450$ .

You correctly multiplied $1500 \cdot 2.4t$ and $-450 \cdot 5.0t$ to get the linear terms $3600t$ and $-2250t$ .

You correctly executed the final multiplication by the conversion factor $2.5 \times 10^{-4}$ across both terms.

THE ERROR

Where Your ACEpath Teacher Spotted the Mistake

Distributing Negative Sign

You wrote $- 540$ instead of $+ 540$ when multiplying $-450$ by $-1.2$ .

WHY IT HAPPENED

ACEpath Teacher's Diagnosis

Overlooking that a subtraction sign in front of a grouped factor acts as a negative multiplier onto every term inside, including subtracted terms: $-a(b - c) = -ab + ac$ .

REASONING PATH

Your ACEpath teacher’s suggested approach

Check the sign of each product: $(-a) \cdot (-b) = +ab$
Circle the negative sign attached to the multiplier before distributing so you remember to flip the signs inside the parentheses.

CORRECTED PATH

ACEpath Teacher's Demonstration

Distribute correctly: $1500(2.4t - 0.8) - 450(5.0t - 1.2) = 3600t - 1200 - 2250t + 540$
Combine like terms: $(3600t - 2250t) + (-1200 + 540) = 1350t - 660$
Distribute the scale factor: $R(t) = 2.5 \times 10^{-4}(1350t - 660) = 0.3375t - 0.165$

Final answer

R(t) = 0.3375t - 0.165, $\text{g/min}$

NEXT STEPS

Final Takeaways from Your Teacher

Rewrite expressions of the form $-a(b - c)$ as $+ (-a)(b + (-c))$ if you tend to drop negative signs during expansion.
Verify constant terms independently by tracking negative signs before simplifying.

OverviewACEpath Teacher's Diagnosis

02 The student miscalculates the power of ten in 1.5 × 10³ as 150 instead of 1500, leading to incorrect coefficients during expansion.

Sample Student Work

Reviewed by ACEpath Teacher

Step-by-Step Review

Walk Through It with Your Teacher

Incorrect

Step 1
150(2.4t - 0.8) - 450(5.0t - 1.2) $\implies$ 360t - 120 - 2250t + 540
Correct
1500(2.4t - 0.8) - 450(5.0t - 1.2) $\implies$ 3600t - 1200 - 2250t + 540
You converted $1.5 \times 10^3$ as $150$ instead of $1500$ . Moving the decimal point 3 places to the right gives $1.5 \times 10^3 = 1500$ . Although your expansion arithmetic with $150$ was correct, the initial factor was incorrect.
Step 2
-1890t + 420
Correct
1350t - 660
You combined like terms correctly according to your previous line ( $(360 - 2250)t = -1890t$ and $-120 + 540 = 420$ ), but the values are shifted due to the earlier conversion error.
Step 3
R(t) = 2.5 $\times$ 10^{-4}(-1890t + 420) $\implies$ R(t) = -0.4725t + 0.105
Correct
R(t) = 2.5 $\times$ 10^{-4}(1350t - 660) $\implies$ R(t) = 0.3375t - 0.165, $\text{g/min}$
You correctly distributed the conversion factor $2.5 \times 10^{-4}$ across both terms, but the resulting function inherits the error from Step 1.

WHAT YOU DID WELL

Compliments from Your Teacher

You correctly converted $4.5 \times 10^2$ to $450$ .

You handled the negative sign distribution perfectly when computing $-450 \times (-1.2) = +540$ .

You accurately distributed the scale factor $2.5 \times 10^{-4}$ across linear and constant terms.

THE ERROR

Where Your ACEpath Teacher Spotted the Mistake

Scientific notation place-value error

You evaluated $1.5 \times 10^3$ as $150$ rather than $1500$ , missing one shift of the decimal point.

WHY IT HAPPENED

ACEpath Teacher's Diagnosis

Incompletely counting place-value shifts when expanding positive integer powers of ten in scientific notation.

REASONING PATH

Your ACEpath teacher’s suggested approach

Decimal Shift Verification: For $a \times 10^n$ with $n > 0$ , shift the decimal point in $a$ exactly $n$ places to the right.
Count the digits after the decimal point: in $1.5$ , moving one place clears the decimal to $15$ , requiring $2$ additional zeros to complete $3$ places: $1500$ .

CORRECTED PATH

ACEpath Teacher's Demonstration

Convert $1.5 \times 10^3$ to $1500$ and $4.5 \times 10^2$ to $450$ .
Expand the terms inside the brackets: $1500(2.4t - 0.8) - 450(5.0t - 1.2) = 3600t - 1200 - 2250t + 540$ .
Combine like terms: $(3600t - 2250t) + (-1200 + 540) = 1350t - 660$ .
Distribute $C = 2.5 \times 10^{-4}$ : $R(t) = 0.00025(1350t - 660) = 0.3375t - 0.165$ .

Final answer

R(t) = 0.3375t - 0.165, $\text{g/min}$

NEXT STEPS

Final Takeaways from Your Teacher

Double-check your power-of-ten conversions by expanding $10^n$ as $1000$ first before multiplying.
Practice distributing coefficients with mixed scientific notation and negative signs.

OverviewACEpath Teacher's Diagnosis

03 The student forgets to multiply by the scale factor C, leaving the rate in milligrams per minute rather than converting to grams per minute.

Sample Student Work

Reviewed by ACEpath Teacher

Step-by-Step Review

Walk Through It with Your Teacher

Partially correct

Step 1
1500(2.4t - 0.8) - 450(5.0t - 1.2) $\implies$ 3600t - 1200 - 2250t + 540 $\implies$ 1350t - 660
Step 2
R(t) = 1350t - 660
Correct
R(t) = 2.5 $\times$ 10^{-4} $\cdot$ (1350t - 660) = 0.3375t - 0.165, $\text{g/min}$
The formula defines $R(t) = C \cdot [\dots]$ , where $C = 2.5 \times 10^{-4}\,\text{g/mg}$ converts the mass rate from $\text{mg/min}$ to $\text{g/min}$ . You omitted multiplication by $C$ , so the resulting function remains in $\text{mg/min}$ rather than $\text{g/min}$ .

WHAT YOU DID WELL

Compliments from Your Teacher

You converted scientific notation numbers into standard decimal values accurately.

You applied the distributive property correctly to both sets of parentheses, including the negative term $-450 \cdot (-1.2) = +540$ .

You collected like variable terms and constant terms cleanly.

THE ERROR

Where Your ACEpath Teacher Spotted the Mistake

Omitted Outer Conversion Factor

You simplified the inner algebraic expression completely but equated it directly to $R(t)$ , leaving off the outer multiplying factor $C$ .

WHY IT HAPPENED

ACEpath Teacher's Diagnosis

Focusing intensely on simplifying the complex inner bracketed expression caused you to lose track of the outer scalar factor $C$ required by the governing equation and the target units.

REASONING PATH

Your ACEpath teacher’s suggested approach

Account for Outer Multipliers: When an expression has nested structure $f(t) = C \cdot [\text{expression}]$ , simplify the inner bracket first, but keep the outer multiplier $C$ attached until it is distributed.
Always write down the full model equation $R(t) = C \cdot [\dots]$ on your first line so outer constants are not dropped along the way.
Verify Target Units: Check that the final units match what the prompt requested ( $ext{g/min}$ vs. $ext{mg/min}$ ).
Perform a unit check at the end: multiplying $(\text{mg/min})$ by $(\text{g/mg})$ cancels $\text{mg}$ and yields $\text{g/min}$ .

CORRECTED PATH

ACEpath Teacher's Demonstration

Multiply the bracketed result by $C = 2.5 \times 10^{-4}\,\text{g/mg}$ : $R(t) = 2.5 \times 10^{-4} (1350t - 660)$
Distribute $2.5 \times 10^{-4} = 0.00025$ to each term: $R(t) = (0.00025 \cdot 1350t) - (0.00025 \cdot 660)$
Calculate each product: $0.00025 \cdot 1350 = 0.3375,\quad 0.00025 \cdot 660 = 0.165$

Final answer

R(t) = 0.3375t - 0.165, $\text{g/min}$

NEXT STEPS

Final Takeaways from Your Teacher

Write out the full equation with the outer multiplier on every working line before performing final distribution.
Review unit dimensions before boxing your final answer to verify whether a scale factor needs to be applied.

Lock it in

Question

What is the expanded form of $-a(b - c)$ ?

Answer

$-ab + ac$ . Distributing a negative number to a subtracted term produces a positive product.

Resources

Answer and steps

Final answer

Steps

Typical errors

01 The student fails to distribute the negative sign to the second term inside the second parentheses, writing - 540 instead of + 540.

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 miscalculates the power of ten in 1.5 × 10³ as 150 instead of 1500, leading to incorrect coefficients during expansion.

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 forgets to multiply by the scale factor C, leaving the rate in milligrams per minute rather than converting to grams per minute.

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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