Algebra Quiz: Using Units In Problem Solving Modeling

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Using Units In Problem Solving ModelingQuestion 1 of 20

A cyclist rides at a constant speed of 18 km/hr for 45 min. Find the distance traveled in kilometers, and show unit tracking (convert minutes to hours).

13.5\text{ km}

810\text{ km}

0.75\text{ km}

13.5\text{ hr}

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Algebra Quiz

Algebra Quiz: Using Units In Problem Solving Modeling

Practice Using Units In Problem Solving Modeling in Algebra with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

What this quiz covers

This quiz focuses on Using Units In Problem Solving Modeling, giving you a quick way to practice the rules, question types, and explanations that matter most for Algebra.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

All questions

Question 1

A cyclist rides at a constant speed of 18 km/hr for 45 min. Find the distance traveled in kilometers, and show unit tracking (convert minutes to hours).

$13.5\text{ km}$ (correct answer)
$810\text{ km}$
$0.75\text{ km}$
$13.5\text{ hr}$

Question 2

A runner completes $3\ \text{miles}$ in $24\ \text{min}$ . Find the runner's average speed in miles per hour (mi/hr), showing unit conversion. (Use $60\ \text{min}=1\ \text{hr}$ .)

$0.125\ \text{mi/hr}$
$1.0\ \text{mi/min}$
$7.5\ \text{mi/hr}$ (correct answer)
$72\ \text{mi/hr}$

Explanation: This question tests your understanding of how units help us solve problems correctly, verify our work, and communicate results clearly—units aren't just labels, they're essential tools for mathematical reasoning. Multi-step problems require careful unit tracking: if you're finding speed in feet per second from miles per hour, you need conversions: start with 60 mph, convert miles to feet (×5280), convert hours to seconds (÷3600), giving 60 × 5280 ÷ 3600 = 88 ft/sec. Solving 'A runner completes 3 miles in 24 min. Find the runner's average speed in mi/hr' with unit tracking: Step 1: Convert time to hours: 24 min × (1 hr/60 min) = 0.4 hr. Step 2: Speed = distance / time: 3 mi / 0.4 hr = 7.5 mi/hr. Final answer: 7.5 mi/hr. Tracking units at each step: (1) ensures we use correct conversion factors, (2) confirms our answer has the right units, (3) catches errors—if we expect feet but get seconds, we know something's wrong! Choice B correctly converts with proper unit cancellation resulting in 7.5 mi/hr. Choice D has the right numerical calculation but the wrong units: the answer should be in mi/hr, not mi/min. This likely means describing what went wrong in conversion or setup. Always state your final answer with units—numbers without units are meaningless in applied problems! Unit conversion strategy: (1) Write the starting value with units, (2) Multiply by conversion factor(s) set up as fractions so unwanted units cancel: (wanted unit)/(starting unit), (3) Cancel units systematically—cross out units that appear in both numerator and denominator, (4) Verify final units match what you want, (5) Calculate the numbers. Example: 3 miles to inches: 3 mi × (5280 ft/mi) × (12 in/ft) = 3 × 5280 × 12 in = 190,080 in. Units guide the whole process! Units help you solve problems even when you're not sure of the formula: think 'what units should my answer have?' If finding distance and you know speed and time, write it with units: ? miles = (60 miles/hour) × (2 hours). Looking at units, what operation makes miles work out? Multiplication! (mi/hr) × hr = mi. The units almost tell you the formula! This is especially helpful when you forget the exact formula but remember what quantities are involved.

Question 3

In the formula $d=rt$ , distance equals rate times time. If $r$ is measured in $\text{miles/hour}$ and $t$ is measured in hours, what are the units of $d$ (using unit cancellation)?

miles (correct answer)
hours
miles/hour
hour/mile

Explanation: This question tests your understanding of how units help us solve problems correctly, verify our work, and communicate results clearly—units aren't just labels, they're essential tools for mathematical reasoning. Every formula has dimensional consistency: the units on the left must match the units on the right. In distance = rate × time, if rate is in mph and time is in hours, distance MUST be in miles (mph × hr = mi/hr × hr = mi, hours cancel). If your calculation gives distance in hours or rate in miles, something's wrong! Checking units catches setup errors before you even calculate numbers. Checking if d=rt has correct units: r has units mi/hr, t has units hr. Performing the operations: showing unit operations like mi/hr × hr = mi. The result is mi. This matches the expected units for distance, so the formula is dimensionally consistent! Choice A correctly has dimensionally consistent units resulting in miles. Choice C's formula is dimensionally inconsistent: showing the unit mismatch. In a valid formula, both sides must have the same units. Here, the left side has units mi, but the right side has mi/hr if not canceling. This dimensional inconsistency reveals an error in the formula structure! Dimensional analysis for checking formulas: every term added or subtracted must have the SAME units (you can't add apples and oranges). Every multiplication/division produces new units by combining (mi/hr × hr = mi, or mi ÷ hr = mi/hr). Use this to check formulas: if distance = rate × time, check units: mi = (mi/hr) × hr ✓, hours cancel! If a formula is dimensionally wrong, it's mathematically wrong—period. Units help you solve problems even when you're not sure of the formula: think 'what units should my answer have?' If finding distance and you know speed and time, write it with units: ? miles = (60 miles/hour) × (2 hours). Looking at units, what operation makes miles work out? Multiplication! (mi/hr) × hr = mi. The units almost tell you the formula! This is especially helpful when you forget the exact formula but remember what quantities are involved.

Question 4

A recipe uses $2.5\ \text{L}$ of broth. Convert this to milliliters (mL). (Use $1\ \text{L}=1000\ \text{mL}$ .)

$250\ \text{mL}$
$25{,}000\ \text{mL}$
$2.5\ \text{mL}$
$2500\ \text{mL}$ (correct answer)

Explanation: This question tests your understanding of how units help us solve problems correctly, verify our work, and communicate results clearly—units aren't just labels, they're essential tools for mathematical reasoning. Units in calculations work like variables: they multiply, divide, and cancel just like algebraic expressions. When you multiply 60 miles/hour × 2 hours, the 'hours' cancel (like x/x = 1), leaving 120 miles. This dimensional analysis helps you set up conversions correctly: to convert 5 miles to feet, multiply by 5280 ft/1 mile (a fraction equaling 1), so miles cancel and you get 5 × 5280 = 26,400 feet. The unit cancellation guides the calculation! Converting 2.5 L to mL: We need a conversion factor that has mL in the numerator and L in the denominator, so they cancel: 2.5 L × (1000 mL/1 L) = 2.5 × 1000 mL. The units cancel: L × mL/L = mL. Calculation: 2500 mL. The unit cancellation confirms we set up the conversion correctly! Choice B correctly converts with proper unit cancellation resulting in 2500 mL. Choice A uses the conversion factor upside down: it multiplies by 1 L/1000 mL instead of 1000 mL/1 L. To convert L to mL, we need mL/L in the conversion factor so L cancels. Check: L × mL/L = mL ✓. Flipping the fraction gives wrong units! Unit conversion strategy: (1) Write the starting value with units, (2) Multiply by conversion factor(s) set up as fractions so unwanted units cancel: (wanted unit)/(starting unit), (3) Cancel units systematically—cross out units that appear in both numerator and denominator, (4) Verify final units match what you want, (5) Calculate the numbers. Example: 3 miles to inches: 3 mi × (5280 ft/mi) × (12 in/ft) = 3 × 5280 × 12 in = 190,080 in. Units guide the whole process! The golden rule of applied problems: NEVER write a final answer without units! '42' means nothing. '42 meters' means something. '42 mph' means something else. The units complete the answer and show you understand what the number represents. In multi-step problems, carry units through every line of work—this tedious-seeming habit catches errors and makes grading partial credit possible when you make arithmetic mistakes!

Question 5

A rectangle has length $12\ \text{cm}$ and width $0.5\ \text{m}$ . Find the area in $\text{cm}^2$ , showing unit conversion. (Use $1\ \text{m}=100\ \text{cm}$ .)

$300\ \text{cm}^2$
$600\ \text{cm}^2$ (correct answer)
$6\ \text{cm}^2$
$0.6\ \text{m}^2$

Explanation: This question tests your understanding of how units help us solve problems correctly, verify our work, and communicate results clearly—units aren't just labels, they're essential tools for mathematical reasoning. Multi-step problems require careful unit tracking: if you're finding speed in feet per second from miles per hour, you need conversions: start with 60 mph, convert miles to feet (×5280), convert hours to seconds (÷3600), giving 60 × 5280 ÷ 3600 = 88 ft/sec. Solving 'A rectangle has length 12 cm and width 0.5 m. Find the area in cm²' with unit tracking: Step 1: Convert width to cm: 0.5 m × (100 cm/1 m) = 50 cm. Step 2: Area = length × width: 12 cm × 50 cm = 600 cm². Final answer: 600 cm². Tracking units at each step: (1) ensures we use correct conversion factors, (2) confirms our answer has the right units, (3) catches errors—if we expect feet but get seconds, we know something's wrong! Choice B correctly converts with proper unit cancellation resulting in 600 cm². Choice D makes a unit conversion error in the multi-step process: converting area to m² instead of cm², likely forgetting to square the conversion factor for area units. When chaining conversions, write out each step with units and verify they cancel correctly. One wrong conversion factor throws off the entire result! Unit conversion strategy: (1) Write the starting value with units, (2) Multiply by conversion factor(s) set up as fractions so unwanted units cancel: (wanted unit)/(starting unit), (3) Cancel units systematically—cross out units that appear in both numerator and denominator, (4) Verify final units match what you want, (5) Calculate the numbers. Example: 3 miles to inches: 3 mi × (5280 ft/mi) × (12 in/ft) = 3 × 5280 × 12 in = 190,080 in. Units guide the whole process! Units help you solve problems even when you're not sure of the formula: think 'what units should my answer have?' If finding distance and you know speed and time, write it with units: ? miles = (60 miles/hour) × (2 hours). Looking at units, what operation makes miles work out? Multiplication! (mi/hr) × hr = mi. The units almost tell you the formula! This is especially helpful when you forget the exact formula but remember what quantities are involved.

Question 6

Based on the graph shown, what are the most appropriate units for the slope of the line?

dollars per month (correct answer)
months per dollar
total dollars
total months

Explanation: The slope represents the change in y-values divided by the change in x-values. From the graph, y-axis shows dollars and x-axis shows months, so slope = change in dollars √∑ change in months = dollars per month. This represents the rate of savings per month. Choice B inverts the units, while choices C and D represent total quantities rather than rates.

Question 7

In the formula for pressure, $P=\dfrac{F}{A}$ , force $F$ is measured in newtons (N) and area $A$ is measured in square meters ( $\text{m}^2$ ). What are the units of $P$ ?

N·m $^2$
N/m $^2$ (correct answer)
m$^2$/N
N/m

Explanation: This question tests your understanding of how units help us solve problems correctly, verify our work, and communicate results clearly—units aren't just labels, they're essential tools for mathematical reasoning. Every formula has dimensional consistency: the units on the left must match the units on the right. In P = F/A, if force F is in newtons (N) and area A is in square meters (m²), pressure P must be in N/m² (newtons divided by square meters = N/m²). The unit division guides the calculation! Checking if P = F/A has correct units: Force F has units N (newtons), area A has units m² (square meters). Performing the division: N ÷ m² = N/m². The result is N/m² (newtons per square meter). This is the standard unit for pressure (also called a Pascal), so the formula is dimensionally consistent! Units verify the formula structure! Choice B correctly shows that pressure units are N/m² when force is divided by area, following the rules of unit division. Choice A has units N·m², which would come from multiplying force by area, not dividing. This would give a quantity with different physical meaning (like work or torque). Division and multiplication of units give completely different results—track operations carefully! Dimensional analysis for checking formulas: every term added or subtracted must have the SAME units (you can't add apples and oranges). Every multiplication/division produces new units by combining (N × m = N·m for work, or N ÷ m² = N/m² for pressure). Use this to check formulas: if pressure = force/area, check units: N/m² = N ÷ m² ✓, units work out! If a formula is dimensionally wrong, it's mathematically wrong—period.

Question 8

Which expression has correct units for a distance $d$ if speed $v$ is in meters/sec and time $t$ is in seconds?

$d=\dfrac{v}{t}$
$d=v+t$
$d=v\,t$ (correct answer)
$d=\dfrac{t}{v}$

Explanation: This question tests your understanding of how units help us solve problems correctly, verify our work, and communicate results clearly—units aren't just labels, they're essential tools for mathematical reasoning. Every formula has dimensional consistency: the units on the left must match the units on the right. For distance d with speed v in m/s and time t in s, we need an expression where the units work out to meters. Let's check each option by tracking units! Checking dimensional consistency for each option: Option A: d = v/t gives (m/s) ÷ s = m/s² (acceleration units, not distance!). Option B: d = v + t gives (m/s) + s, but you can't add different units—dimensionally invalid! Option C: d = vt gives (m/s) × s = m (seconds cancel, leaving meters—correct for distance!). Option D: d = t/v gives s ÷ (m/s) = s²/m (strange units, not distance!). Only option C gives units of meters for distance. Choice C correctly multiplies speed by time (vt), where the units (m/s) × s = m give the proper distance units—this is the familiar distance = rate × time formula! Choice A divides speed by time, giving m/s² (acceleration units), not meters. This formula structure doesn't match the physical relationship between distance, speed, and time. Dimensional analysis reveals the error before any calculation! Units help you solve problems even when you're not sure of the formula: think 'what units should my answer have?' If finding distance and you know speed (m/s) and time (s), write it with units: ? m = (? m/s) × (? s). Looking at units, what operation makes meters work out? Multiplication! (m/s) × s = m. The units almost tell you the formula! This is especially helpful when you forget the exact formula but remember what quantities are involved.

Question 9

A gasoline container holds 3.5 gallons. Convert this to liters. (Use $1\text{ gal}=3.785\text{ L}$ .)

$13.25\text{ gal}$
$13.25\text{ L}$ (correct answer)
$3.5\text{ L}$
$0.925\text{ L}$

Explanation: This question tests your understanding of how units help us solve problems correctly, verify our work, and communicate results clearly—units aren't just labels, they're essential tools for mathematical reasoning. Units in calculations work like variables: they multiply, divide, and cancel just like algebraic expressions. When you multiply 60 miles/hour × 2 hours, the 'hours' cancel (like x/x = 1), leaving 120 miles. This dimensional analysis helps you set up conversions correctly: to convert 5 miles to feet, multiply by 5280 ft/1 mile (a fraction equaling 1), so miles cancel and you get 5 × 5280 = 26,400 feet. The unit cancellation guides the calculation! Converting 3.5 gal to L: We need a conversion factor that has L in the numerator and gal in the denominator, so they cancel: 3.5 gal × (3.785 L/1 gal) = 3.5 × 3.785 L. The units cancel: gal × L/gal = L. Calculation: 3.5 × 3.785 = 13.2475 ≈ 13.25 L. The unit cancellation confirms we set up the conversion correctly! Choice B correctly converts with proper unit cancellation resulting in 13.25 L. Choice D has the right numerical calculation but the wrong units: the answer should be in L, not gal. This likely means forgetting to multiply by the conversion factor, leaving it in gallons. Always state your final answer with units—numbers without units are meaningless in applied problems! Unit conversion strategy: (1) Write the starting value with units, (2) Multiply by conversion factor(s) set up as fractions so unwanted units cancel: (wanted unit)/(starting unit), (3) Cancel units systematically—cross out units that appear in both numerator and denominator, (4) Verify final units match what you want, (5) Calculate the numbers. Example: 3 miles to inches: 3 mi × (5280 ft/mi) × (12 in/ft) = 3 × 5280 × 12 in = 190,080 in. Units guide the whole process!

Question 10

A rectangular room is 12 ft long and 9 ft wide. Find the area and include correct units. Use the formula $A=\ell\times w$ and check that the units are consistent.

$108\text{ ft}^2$ (correct answer)
$108\text{ ft}$
$21\text{ ft}^2$
$1{,}296\text{ ft}^2$

Explanation: This question tests your understanding of how units help us solve problems correctly, verify our work, and communicate results clearly—units aren't just labels, they're essential tools for mathematical reasoning. Every formula has dimensional consistency: the units on the left must match the units on the right. In distance = rate × time, if rate is in mph and time is in hours, distance MUST be in miles (mph × hr = mi/hr × hr = mi, hours cancel). If your calculation gives distance in hours or rate in miles, something's wrong! Checking units catches setup errors before you even calculate numbers. Checking if A=ℓ×w has correct units: ℓ has units ft, w has units ft. Performing the operations: ft × ft = ft². The result is ft². This matches the expected units for area, so the formula is dimensionally consistent! Choice A correctly has dimensionally consistent units resulting in 108 ft². Choice B uses the formula upside down: it might add instead of multiply, giving ft + ft = ft, not ft². To get area in ft², we need ft × ft = ft². Check: 12 ft × 9 ft = 108 ft² ✓. Using addition gives wrong units! Dimensional analysis for checking formulas: every term added or subtracted must have the SAME units (you can't add apples and oranges). Every multiplication/division produces new units by combining (mi/hr × hr = mi, or mi ÷ hr = mi/hr). Use this to check formulas: if distance = rate × time, check units: mi = (mi/hr) × hr ✓, hours cancel! If a formula is dimensionally wrong, it's mathematically wrong—period.

Question 11

A runner's average speed is computed by $v=\dfrac{d}{t}$ . Which expression has correct units for speed if distance is in meters (m) and time is in seconds (sec)?

$\text{m}\cdot\text{sec}$
$\dfrac{\text{sec}}{\text{m}}$
$\dfrac{\text{m}}{\text{sec}}$ (correct answer)
$\text{m}+\text{sec}$

Explanation: This question tests your understanding of how units help us solve problems correctly, verify our work, and communicate results clearly—units aren't just labels, they're essential tools for mathematical reasoning. Every formula has dimensional consistency: the units on the left must match the units on the right. For speed = distance/time, if distance is in meters and time is in seconds, speed must be in meters per second (m/sec). The units guide the formula structure! Checking units for v = d/t: Distance d has units m (meters), time t has units sec (seconds). Performing the division: v = d/t means speed units = m/sec. This reads as 'meters per second'—it tells us how many meters are traveled in each second. The fraction bar in the formula becomes the fraction bar in the units! Choice A correctly shows m/sec as the units for speed when distance is in meters and time is in seconds. Choice B has inverted units (sec/m)—this would mean 'seconds per meter' or how much time it takes to travel one meter, which is slowness, not speed! The position of units in the fraction matches their position in the formula. Dimensional analysis for checking formulas: if someone claims speed = distance × time, check units: m × sec = m·sec. But we know speed should be distance per time (like 60 miles per hour), not distance times time! The dimensional check immediately reveals the formula error. Units aren't just labels—they're formula validators!

Question 12

In the formula for density, $\rho=\dfrac{m}{V}$ , mass $m$ is measured in grams (g) and volume $V$ is measured in cubic centimeters (cm $^3$ ). What are the units of density $\rho$ ?

$\text{cm}^3/\text{g}$
$\text{g}\cdot\text{cm}^3$
$\text{g}/\text{cm}^3$ (correct answer)
$\text{g}/\text{cm}$

Explanation: This question tests your understanding of how units help us solve problems correctly, verify our work, and communicate results clearly—units aren't just labels, they're essential tools for mathematical reasoning. Every formula has dimensional consistency: the units on the left must match the units on the right. For density = mass/volume, if mass is in grams and volume is in cm³, we can determine density's units by performing the division with units included. Checking units for ρ = m/V: Mass m has units g (grams), volume V has units cm³ (cubic centimeters). Performing the division: ρ = m/V means density units = g/cm³. This reads as 'grams per cubic centimeter'—it tells us how many grams of material fit in each cubic centimeter of space. The division of units works just like division of numbers! Choice C correctly shows g/cm³ as the units of density when mass is in grams and volume is in cubic centimeters. Choice A has the units inverted (cm³/g)—this would be 'volume per mass' or specific volume, not density! Remember: in a fraction, the numerator unit goes on top, denominator unit goes on bottom. Units help you solve problems even when you're not sure of the formula: think 'what units should density have?' Density measures how much mass fits in a given volume, so it should be mass per volume: mass/volume. This reasoning leads directly to g/cm³. The units almost tell you the formula!

Question 13

A recipe uses 750 mL of water. Convert this amount to liters. (Use $1000\text{ mL}=1\text{ L}$ .)

$0.75\text{ L}$ (correct answer)
$0.75\text{ mL}$
$750\text{ L}$
$7.5\text{ L}$

Explanation: This question tests your understanding of how units help us solve problems correctly, verify our work, and communicate results clearly—units aren't just labels, they're essential tools for mathematical reasoning. Units in calculations work like variables: they multiply, divide, and cancel just like algebraic expressions. When you convert 750 mL to liters, divide by 1000 (since 1000 mL = 1 L). This dimensional analysis helps you set up conversions correctly: 750 mL × (1 L/1000 mL) = 750/1000 L = 0.75 L. The unit cancellation guides the calculation! Converting 750 mL to L: We need a conversion factor that has L in the numerator and mL in the denominator, so they cancel: 750 mL × (1 L/1000 mL) = 750/1000 L. The units cancel: mL × L/mL = L. Calculation: 750 ÷ 1000 = 0.75 L. The unit cancellation confirms we set up the conversion correctly! Choice B correctly converts milliliters to liters with proper unit cancellation resulting in 0.75 L. Choice A multiplies by 10 instead of dividing by 1000: this would be correct if converting from centiliters (cL) to liters, but not from milliliters. The prefix 'milli-' means 1/1000, so 1000 mL = 1 L, which means we divide by 1000 to convert mL to L. Always pay attention to metric prefixes! Unit conversion strategy: (1) Write the starting value with units, (2) Multiply by conversion factor(s) set up as fractions so unwanted units cancel: (wanted unit)/(starting unit), (3) Cancel units systematically—cross out units that appear in both numerator and denominator, (4) Verify final units match what you want, (5) Calculate the numbers. For metric conversions, remember: kilo- = 1000, centi- = 1/100, milli- = 1/1000!

Question 14

A car travels 150 miles on 5 gallons of gas. What is the fuel efficiency in miles per gallon (mi/gal)? (Track units: miles ÷ gallons.)

30 mi/gal (correct answer)
750 mi/gal
0.033 mi/gal
30 gal/mi

Explanation: This question tests your understanding of how units help us solve problems correctly, verify our work, and communicate results clearly—units aren't just labels, they're essential tools for mathematical reasoning. Every formula has dimensional consistency: the units on the left must match the units on the right. For fuel efficiency in miles per gallon, we divide miles by gallons: 150 miles ÷ 5 gallons = 30 mi/gal. The units tell us the operation: mi ÷ gal = mi/gal. Checking units catches setup errors before you even calculate numbers. Solving 'fuel efficiency' with unit tracking: We want miles per gallon (mi/gal), so we divide total miles by total gallons: 150 mi ÷ 5 gal = 30 mi/gal. The units work out: mi ÷ gal = mi/gal ✓. This tells us how many miles the car travels on one gallon of gas. Calculation: 150 ÷ 5 = 30. Final answer: 30 mi/gal. Tracking units confirms we set up the division correctly—if we had divided gallons by miles, we'd get gal/mi, which is fuel consumption, not efficiency! Choice A correctly divides miles by gallons with proper unit tracking, resulting in 30 mi/gal—the standard way to express fuel efficiency. Choice D has the right number but inverted units: 30 gal/mi would mean the car uses 30 gallons per mile—that would be terrible efficiency! This shows why tracking units matters: the same numbers with different unit arrangements mean completely different things. Always verify your units match what the problem asks for! The golden rule of applied problems: NEVER write a final answer without units! '30' means nothing. '30 mi/gal' means good fuel efficiency. '30 gal/mi' means you're driving a rocket ship! The units complete the answer and show you understand what the number represents. In rate problems, pay special attention to which quantity goes in the numerator versus denominator—this determines the meaning!

Question 15

A container has density $2.4\ \text{g/mL}$ and volume $750\ \text{mL}$ . Find the mass in kilograms. Use $1000\ \text{g}=1\ \text{kg}$ . (Track units through the calculation.)

1.8 kg (correct answer)
1800 kg
0.0018 kg
1.8 g

Explanation: This question tests your understanding of how units help us solve problems correctly, verify our work, and communicate results clearly—units aren't just labels, they're essential tools for mathematical reasoning. Multi-step problems require careful unit tracking: if you're finding mass from density and volume, then converting to different mass units, you need to track carefully: start with density × volume = 2.4 g/mL × 750 mL = 1800 g, then convert grams to kilograms (÷1000), giving 1800 g ÷ 1000 = 1.8 kg. At each step, track units: g/mL × mL = g, then g × (1 kg/1000 g) = kg. Units guide which conversions to use! Solving 'mass from density and volume' with unit tracking: Step 1: Mass = density × volume: 2.4 g/mL × 750 mL = 1800 g (mL cancels). Step 2: Convert grams to kilograms: 1800 g × (1 kg/1000 g) = 1800/1000 kg. Step 3: Calculate: 1800 ÷ 1000 = 1.8 kg. Final answer: 1.8 kg. Tracking units at each step: (1) ensures we use correct conversion factors, (2) confirms our answer has the right units, (3) catches errors—if we expect kg but get mL, we know something's wrong! Choice A correctly multiplies density by volume to get mass in grams, then converts to kilograms with proper unit cancellation, resulting in 1.8 kg. Choice D has the right numerical calculation but the wrong units: the answer should be in kg (as requested), not g. This shows they calculated mass correctly but forgot the final conversion. Always read what units the problem asks for—getting the right number with wrong units is still wrong! Units help you solve problems even when you're not sure of the formula: think 'what units should my answer have?' If finding mass and you know density (g/mL) and volume (mL), write it with units: ? g = (2.4 g/mL) × (750 mL). Looking at units, what operation makes g work out? Multiplication! (g/mL) × mL = g. The units almost tell you the formula! This is especially helpful when you forget the exact formula but remember what quantities are involved.

Question 16

A water tank is being filled at a constant rate of 2.5 gallons/min. How much water is added in 18 minutes? Show unit tracking so minutes cancel correctly.

$45\text{ gallons}$ (correct answer)
$45\text{ min}$
$7.2\text{ gallons}$
$2.5\text{ gallons}$

Question 17

A storage bin has dimensions 40 cm by 25 cm by 30 cm. Find its volume in cubic centimeters ( $\text{cm}^3$ ). Show unit tracking.

$30,000\text{ cm}^2$
$95\text{ cm}^3$
$30,000\text{ cm}^3$ (correct answer)
$3,000\text{ cm}^3$

Explanation: This question tests your understanding of how units help us solve problems correctly, verify our work, and communicate results clearly—units aren't just labels, they're essential tools for mathematical reasoning. Units in calculations work like variables: they multiply, divide, and cancel just like algebraic expressions. When finding volume of a rectangular box, multiply length × width × height: 40 cm × 25 cm × 30 cm. The units multiply too: cm × cm × cm = cm³. This dimensional analysis confirms we're calculating volume (3D space)! Calculating volume with unit tracking: Volume = length × width × height = 40 cm × 25 cm × 30 cm. The units multiply: cm × cm × cm = cm³ (cubic centimeters). Calculation: 40 × 25 × 30 = 1,000 × 30 = 30,000 cm³. The unit tracking confirms we're finding volume—if we got cm² (area) or cm (length), we'd know something was wrong! Choice B correctly multiplies all three dimensions with proper unit tracking, resulting in 30,000 cm³. Choice C has the right numerical value but wrong units: the answer should be in cm³ (volume), not cm² (area). This shows a fundamental misunderstanding—area is 2D (length × width), while volume is 3D (length × width × height). The units reveal the dimensionality of what you're measuring! Every formula has dimensional consistency: the units on the left must match the units on the right. For geometric formulas: perimeter has units of length (cm), area has units of length² (cm²), volume has units of length³ (cm³). If your calculation for volume gives cm² or cm, you've made an error—the units themselves catch the mistake!

Question 18

A recipe uses 750 mL of broth. Convert 750 mL to liters. Use $1000\text{ mL}=1\text{ L}$ .

$7.5\text{ L}$
$0.75\text{ L}$ (correct answer)
$750{,}000\text{ L}$
$0.75\text{ mL}$

Explanation: This question tests your understanding of how units help us solve problems correctly, verify our work, and communicate results clearly—units aren't just labels, they're essential tools for mathematical reasoning. Units in calculations work like variables: they multiply, divide, and cancel just like algebraic expressions. To convert 750 mL to liters, we use the fact that 1000 mL = 1 L, so we multiply by (1 L)/(1000 mL)—a fraction equaling 1. The mL units cancel, leaving L! Converting 750 mL to L: We need a conversion factor with L in the numerator and mL in the denominator: 750 mL × (1 L)/(1000 mL) = 750/1000 L = 0.75 L. The units cancel: mL × (L/mL) = L ✓. This unit cancellation confirms we set up the conversion correctly—if we had used (1000 mL)/(1 L), we'd get mL²/L, which makes no sense! Choice B correctly divides 750 by 1000 (since 1000 mL = 1 L) to get 0.75 L with proper unit conversion. Choice A multiplies by 10 instead of dividing by 1000—this is the conversion factor upside down! To go from a smaller unit (mL) to a larger unit (L), we expect a smaller number, not larger. Unit tracking catches this error immediately. The golden rule of unit conversion: when converting from smaller to larger units, your number gets smaller (750 mL → 0.75 L). When converting from larger to smaller units, your number gets bigger (0.75 L → 750 mL). This quick check helps verify you've set up the conversion correctly!

Question 19

A runner's pace is 8 min/mi. How long will it take to run 6 miles? Show unit tracking. (Answer in minutes.)

$0.75\text{ min}$
$48\text{ min}$ (correct answer)
$14\text{ min}$
$48\text{ mi}$

time = pace \times distance = 8 \text{ min/mi} \times 6 \text{ mi} = 48 \text{ min}

. At each step, track units:

\text{min/mi} \times \text{ mi} = \text{ min}

(miles cancel). Units guide the calculation! Solving 'time to run 6 miles at 8 min/mi pace' with unit tracking: Step 1: Identify what we're finding: time in minutes. Step 2: Set up calculation with units:

time = pace \times distance = 8 \text{ min/mi} \times 6 \text{ mi}

. Step 3: Cancel units:

\text{min/mi} \times \text{ mi} = \text{ min}

✓. Step 4: Calculate:

8 \times 6 = 48 \text{ min}

. Final answer: 48 min. Tracking units at each step ensures we multiply (not divide) pace by distance—the units tell us the operation! Choice B correctly multiplies pace by distance with proper unit cancellation resulting in 48 minutes. Choice A divides distance by pace (

6 \div 8 = 0.75

) instead of multiplying. This is a common error when working with rates! The units show why multiplication is correct:

\text{min/mi} \times \text{ mi} = \text{ min}

, but

\text{mi} \div (\text{min/mi}) = \text{mi} \times (\text{mi/min}) = \text{mi}^2/\text{min}

, which makes no sense for time. Always let units guide your operations! Units help you solve problems even when you're not sure of the formula: think 'what units should my answer have?' If finding time and you know pace (min/mi) and distance (mi), write it with units:

? \text{ min} = (8 \text{ min/mi}) \times (6 \text{ mi})

. Looking at units, what operation makes minutes work out? Multiplication!

( \text{min/mi} ) \times \text{ mi} = \text{ min}

. The units almost tell you the formula!

Question 20

Convert 72 miles/hr to feet/sec. Use $1\text{ mi}=5280\text{ ft}$ and $1\text{ hr}=3600\text{ sec}$ .

105.6 ft/sec (correct answer)
105.6 ft/hr
19.8 ft/sec
380,160 ft/sec

Explanation: This question tests your understanding of how units help us solve problems correctly, verify our work, and communicate results clearly—units aren't just labels, they're essential tools for mathematical reasoning. Units in calculations work like variables: they multiply, divide, and cancel just like algebraic expressions. When you convert 72 miles/hour to feet/second, you need two conversions: miles to feet (×5280) and hours to seconds (÷3600). This gives 72 × 5280 ÷ 3600 = 105.6 ft/sec. The unit cancellation guides the calculation! Converting 72 mi/hr to ft/sec: We need conversion factors that change miles to feet and hours to seconds: 72 mi/hr × (5280 ft/1 mi) × (1 hr/3600 sec) = 72 × 5280 ÷ 3600 ft/sec. The units cancel: (mi/hr) × (ft/mi) × (hr/sec) = ft/sec. Calculation: 72 × 5280 = 380,160, then 380,160 ÷ 3600 = 105.6 ft/sec. The unit cancellation confirms we set up the conversion correctly! Choice A correctly uses both conversion factors with proper unit cancellation, resulting in 105.6 ft/sec with correct units for speed. Choice D makes a unit conversion error: they multiplied by 5280 but forgot to divide by 3600, getting 380,160 ft/sec. When chaining conversions, write out each step with units and verify they cancel correctly. One wrong conversion factor throws off the entire result! Unit conversion strategy: (1) Write the starting value with units, (2) Multiply by conversion factor(s) set up as fractions so unwanted units cancel: (wanted unit)/(starting unit), (3) Cancel units systematically—cross out units that appear in both numerator and denominator, (4) Verify final units match what you want, (5) Calculate the numbers. Example: 72 mi/hr to ft/sec: 72 mi/hr × (5280 ft/mi) × (1 hr/3600 sec) = 105.6 ft/sec. Units guide the whole process!