Unit Pricing & Constant Speed - Sixth Grade Mathematics

Comparing unit rates allows consumers and analysts to evaluate commercial offers, determine the "better buy" among packaged goods, and analyze relative motion.

Unit Pricing: The Consumer's Decision Tool

In grocery stores, goods are sold in containers of varying sizes and costs. Comparing the total sticker price alone is misleading: an 18-ounce box of cereal costing $4.50 might appear cheaper than a 30-ounce box costing $6.00, but calculating the unit price reveals which option gives more product per dollar.

Unit Price Formula:
               Unit Price = Total Cost / Total Units (Ounces, Pounds, etc.)

Brand Alpha:  $4.50 / 18 oz  =  $0.25 per ounce
Brand Beta:   $6.00 / 30 oz  =  $0.20 per ounce
Conclusion:   Brand Beta is the better buy (costs $0.05 less per ounce)!

The Rule for Unit Pricing

When calculating unit price, always divide the MONEY (dollars and cents) by the QUANTITY (ounces, grams, or count). Putting the quantity in the numerator produces units per dollar rather than cost per unit, which can cause confusion.

Comparing Constant Speeds

When evaluating vehicles, runners, or moving objects, speed is the unit rate of distance divided by time.

Speed Formula:
               Speed = Distance / Time

Runner A: Runs 5 miles in 40 minutes  --> Speed = 5 / 40 = 0.125 miles per min
Runner B: Runs 3 miles in 21 minutes  --> Speed = 3 / 21 = 0.143 miles per min
Conclusion: Runner B is faster because Runner B covers more distance each minute!

Comparing Using Time per Distance

Alternatively, athletic races can be compared by pacing (minutes per mile): Runner A: 40 minutes / 5 miles = 8.0 minutes per mile. Runner B: 21 minutes / 3 miles = 7.0 minutes per mile. In pacing, a SMALLER unit rate indicates a faster speed (less time taken per mile).

Multi-Step Comparison Scenarios

Real-world comparisons often involve multiple items and package options.

Case Study: The School Art Supply Order
A school district is purchasing marker sets.
Vendor 1: Offers a 24-pack of markers for $15.60.
Vendor 2: Offers a 36-pack of markers for $21.60.
Which vendor offers the lower price per marker, and how much would the
school save on an order of 720 markers by choosing the better vendor?

1. Calculate unit price for Vendor 1:
   $15.60 / 24 = $0.65 per marker.
2. Calculate unit price for Vendor 2:
   $21.60 / 36 = $0.60 per marker.
   Vendor 2 offers the lower price (saves $0.05 per marker).
3. Calculate savings for 720 markers:
   720 markers * $0.05 savings/marker = $36.00 saved.

Exercises

Exercise 1. Determine the unit price for each option and state which is the better buy: Option A: 16 ounces of peanut butter for $3.84; Option B: 28 ounces of peanut butter for $6.16.

Exercise 2. Car X travels 275 miles on 11 gallons of fuel. Car Y travels 336 miles on 14 gallons of fuel. Which vehicle has the higher fuel efficiency in miles per gallon?

Exercise 3. Cyclist 1 rides 18 miles in 1.5 hours. Cyclist 2 rides 26 miles in 2 hours. Who is traveling at a greater speed, and by how many miles per hour?

Solutions

Solution to Exercise 1. For Option A: divide cost by ounces: $3.84 / 16 = $0.24 per ounce. For Option B: divide cost by ounces: $6.16 / 28. Set up division: 28 into 61 goes 2 times (2 * 28 = 56; remainder 5). Bring down 6 to make 56: 28 into 56 goes 2 times (remainder 0). Option B costs $0.22 per ounce. Comparing $0.24 and $0.22, Option B is the better buy because it is $0.02 cheaper per ounce.

Solution to Exercise 2. Calculate miles per gallon for each vehicle: Car X: 275 miles / 11 gallons = 25 miles per gallon. Car Y: 336 miles / 14 gallons. Divide: 14 into 33 goes 2 times (2 * 14 = 28; remainder 5). Bring down 6 to make 56: 14 into 56 goes 4 times (4 * 14 = 56; remainder 0). Car Y achieves 24 miles per gallon. Comparing 25 mpg to 24 mpg, Car X has the higher fuel efficiency by 1 mile per gallon.

Solution to Exercise 3. Find speed in miles per hour: Cyclist 1: 18 / 1.5 = 180 / 15 = 12 miles per hour. Cyclist 2: 26 / 2 = 13 miles per hour. Comparing 13 mph and 12 mph, Cyclist 2 is traveling at a greater speed, moving 1 mile per hour faster than Cyclist 1.