The Unit Rate - Math 6

1. Fundamental Concepts

Definition: A unit rate is a special type of rate that has a denominator of 1 (when expressed as a fraction). It represents "how much of the first quantity corresponds to 1 unit of the second quantity".
Essence: It is a simplified form of a proportional relationship. By converting the ratio of two quantities into a "value corresponding to 1 unit", it facilitates subsequent calculations.
Distinction: The denominator of a regular rate can be any number (e.g., 6 veggie bowls for 18 dollars), while a unit rate must have the denominator simplified to 1 (e.g., 1 veggie bowl for 3 dollars).

Calculation Principle: Whether comparing "part-to-whole" or "part-to-part", divide the first quantity by the second quantity to convert the second quantity (denominator) to 1. The result is the unit rate.
Flexibility: The result of a unit rate can be a decimal, not limited to integers (e.g., the unit rate for avocados might be 0.75 dollars per avocado).
Practicality: Once the unit rate is known, the value corresponding to any quantity can be quickly calculated using the formula: Unit Rate × Target Quantity (e.g., 1 bowl costs 3 dollars, so 7 bowls cost 3 × 7 = 21 dollars).

Problem 1: 6 veggie bowls cost 18 dollars. Find the price of 1 veggie bowl (unit rate: dollars per bowl).
- Solution: 18 ÷ 6 = 3. The unit rate is 3 dollars per bowl.
Problem 2: There are 36 triangles for 12 squares. Find the number of triangles per square (unit rate: triangles per square).
- Solution: 36 ÷ 12 = 3. The unit rate is 3 triangles per square.

Problem 1: 8 bicycles correspond to 16 skateboards. Find the number of bicycles per skateboard (unit rate: bicycles per skateboard).
- Solution: 8 ÷ 16 = 0.5. The unit rate is 0.5 bicycles per skateboard.
Problem 2: 10 carrots are for 20 veggie bowls. Find the number of carrots per veggie bowl (unit rate: carrots per veggie bowl).
- Solution: 10 ÷ 20 = 0.5. The unit rate is 0.5 carrots per veggie bowl.

Problem 1: 6 dollars buys 8 avocados. Find the price of 1 avocado (unit rate: dollars per avocado), and calculate the total cost for 10 avocados.
- Solution: First, calculate the unit rate: 6 ÷ 8 = 0.75 dollars per avocado. Then, calculate the total cost for 10 avocados: 0.75 × 10 = 7.5 dollars.
Problem 2: 3 fish are in 6 houses. Find the number of fish per house (unit rate: fish per house), and calculate the number of fish for 15 houses.
- Solution: First, calculate the unit rate: 3 ÷ 6 = 0.5 fish per house. Then, calculate the number of fish for 15 houses: 0.5 × 15 = 7.5 fish.

Clarify the Goal: First, identify the "type of unit rate" to be calculated, i.e., "Unit of the First Quantity / Unit of the Second Quantity" (e.g., dollars per piece, pieces per vehicle).
Calculate the Unit Rate: Divide the "amount of the first quantity" by the "amount of the second quantity" to ensure the denominator of the result is 1. If the result is a decimal, keep it as is.
Extended Calculation: If you need to find the value corresponding to any quantity, use Unit Rate × Target Quantity to get the result quickly.
Verification Check: After calculation, verify by reversing the process (e.g., Result ÷ Target Quantity) to confirm if it equals the previously calculated unit rate, avoiding errors.