Math: Unit Rates & Proportional Reasoning
Ontario Mathematics Proficiency Test — study tool.
Scroll down past the quiz for the complete study guide on this topic.
Scroll down past the quiz for the complete study guide on this topic.
1
François reads 96 pages in 2 hours. 216 pages remain. How much more time does he need?
2
Which toothpaste has the lowest cost per mL?
3
Sam earns $306 for 15 hours. Miko earns $189 for 10 hours. Which statement correctly compares their pay?
4
Max drives 60 km in 50 minutes. He has 90 km remaining. How much longer will the trip take?
5
5 avocados cost $3.70. What is the cost per avocado?
6
12 cookies cost $7.56. How much do 14 cookies cost?
7
Each can of peach paint needs 1/4 red, 5/8 white, and 1/8 yellow. Available: 5 red, 10 white, 1/2 yellow. How many cans can be made?
8
A team raises $1,650 selling 550 pucks. If Harjot collects $24, how many pucks did she sell?
9
530 mL of shampoo costs $7.95. Which bottle has the same cost per mL?
10
12 water bottles cost $6.48. What is the cost per bottle?
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Study Guide: Unit Rates & Proportional Reasoning
WHAT IS A UNIT RATE?
A unit rate is a rate where the denominator is 1.
"3 dollars per avocado" = $3/1 avocado = a unit rate "$7.56 for 12 cookies" = NOT a unit rate (until you divide)
CALCULATING A UNIT RATE
Divide the total by the quantity.
Example: 5 avocados cost $3.70
$3.70 ÷ 5 = $0.74 per avocado
Example: 12 bottles cost $6.48
$6.48 ÷ 12 = $0.54 per bottle
Example: $1,650 for 550 hockey pucks
$1,650 ÷ 550 = $3.00 per puck
COMPARING UNIT RATES
To find the best deal, calculate the unit rate for each option and
compare.
Example: Which toothpaste is cheapest per mL?
A: $1.35 / 100 mL = $0.0135/mL
B: $1.94 / 150 mL = $0.0129/mL
C: $2.40 / 200 mL = $0.0120/mL ← cheapest
D: $3.25 / 250 mL = $0.0130/mL
The lowest unit price wins, even if the total price is higher.
MATCHING UNIT RATES
To find which option has the SAME unit rate, calculate the original
unit rate and then check each option.
Example: 530 mL for $7.95 → $7.95/530 = $0.015/mL
Does 1,280 mL for $19.20 match?
$19.20/1280 = $0.015/mL ✓ Yes, same rate
USING UNIT RATES TO FIND QUANTITIES
Once you have the unit rate, multiply or divide to find what you need.
Example: Pucks cost $3 each. Harjot collected $24. How many pucks?
$24 ÷ $3 = 8 pucks
Example: Cookies cost $0.63 each. How much for 14?
$0.63 × 14 = $8.82
COMPARING TWO WORKERS/RATES
Find each person's unit rate, then compare.
Example: Sam earns $306 for 15 hours. Miko earns $189 for 10 hours.
Sam: $306 ÷ 15 = $20.40/hour
Miko: $189 ÷ 10 = $18.90/hour
Difference: $20.40 - $18.90 = $1.50/hour
For a specific number of hours, multiply the difference by hours:
Over 30 hours: $1.50 × 30 = $45 more
RATE/TIME/DISTANCE PROBLEMS
Rate = Distance / Time
Distance = Rate × Time
Time = Distance / Rate
Example: François reads 96 pages in 2 hours. 216 pages remain.
Rate = 96 / 2 = 48 pages/hour
Time = 216 / 48 = 4.5 hours = 270 minutes
Example: Max drives 60 km in 50 minutes. 90 km remaining.
Rate = 60 / 50 = 1.2 km/min
Time = 90 / 1.2 = 75 minutes
UNITS MATTER: If the rate is in km/min, the time will be in minutes.
If the rate is in pages/hour, the time will be in hours.
LIMITING FACTOR PROBLEMS
When a recipe or mixture needs multiple ingredients, the ingredient
that runs out first determines how much you can make.
Example: Each can of paint needs 1/4 red, 5/8 white, 1/8 yellow.
Available: 5 red, 10 white, 1/2 yellow.
Red allows: 5 ÷ (1/4) = 5 × 4 = 20 cans
White allows: 10 ÷ (5/8) = 10 × 8/5 = 16 cans
Yellow allows: (1/2) ÷ (1/8) = (1/2) × 8 = 4 cans
Yellow runs out first → maximum is 4 cans
The SMALLEST number is the answer — it's the bottleneck.
KEY FORMULAS
Unit rate = total ÷ quantity Total cost = unit rate × quantity Quantity = total ÷ unit rate Speed = distance / time Time = distance / speed Limiting factor = minimum of (supply ÷ requirement) for each ingredient
