Ratios and rates (ratio tables, unit rate) Knowledge-Based Study

Study notes

1. What is a Ratio?

   A ratio is a mathematical comparison of two quantities by division. It describes the relative sizes of two or more values.

   For example, if a fruit bowl contains 3 apples and 5 oranges, the ratio of apples to oranges can be written in three equivalent ways:

   • With a colon: $3 : 5$
   • As a fraction: $\frac{3}{5}$
   • In words: 3 to 5

   What is a Rate?

   A rate is a specific type of ratio that compares two quantities measured in different units.

   Common real-world examples of rates include:

   • Distance and time: 150 miles in 3 hours
   • Cost and quantity: $12 for 4 notebooks
   • Heartbeats and time: 140 beats in 2 minutes

   What is a Unit Rate?

   A unit rate is a rate simplified so that the second quantity (the denominator) is equal to 1 unit. Unit rates make it easy to compare quantities and analyze patterns.

   To find a unit rate, divide the first quantity by the second quantity:

   $$\text{Unit Rate} = \frac{\text{Quantity A}}{\text{Quantity B}}$$

   Worked Example

   A car travels 240 miles in 4 hours. What is its unit rate of speed?

   1. Write the given rate as a fraction: $\frac{240\text{ miles}}{4\text{ hours}}$
   2. Divide both the numerator and denominator by 4: $\frac{240 \div 4}{4 \div 4} = \frac{60\text{ miles}}{1\text{ hour}}$
   3. State the unit rate: 60 miles per hour (or 60 mph).

   Unit Price

   When a unit rate expresses cost per single item or unit of measurement, it is called a unit price. For example, if 5 pounds of flour cost 7.50, the unit price is \(\frac{7.50}{5} = \1.50\) per pound.

   What is a Ratio Table?

   A ratio table is a structured tool used to organize pairs of numbers that form equivalent ratios. Every column or row in a ratio table maintains the exact same proportional relationship.

   Techniques for Completing Ratio Tables

   • Scaling Up (Multiplication): Multiply both quantities in a ratio by the same non-zero constant to generate an equivalent ratio.
   • Scaling Down (Division): Divide both quantities by a common factor to simplify the ratio or find smaller benchmark values.
   • Repeated Addition or Subtraction: Combine columns to find values that do not share an obvious single integer factor.

   Example Table

   Consider a recipe that uses 2 cups of sugar for every 3 cups of flour:

   • Sugar (cups): 2 | 4 | 6 | 10
   • Flour (cups): 3 | 6 | 9 | 15

   Notice that multiplying the first column by 2 gives $4 : 6$, multiplying by 3 gives $6 : 9$, and multiplying by 5 gives $10 : 15$. Each pair simplifies to $\frac{2}{3}$.

   Connecting Unit Rates to Ratio Tables

   The unit rate is often the most efficient bridge for finding missing values in a ratio table.

   Step-by-Step Application

   A printer produces 72 pages in 3 minutes. How many pages does it produce in 8 minutes?

   Method 1: Using the Unit Rate

   1. Find the unit rate: $\frac{72\text{ pages}}{3\text{ minutes}} = 24\text{ pages per minute}$.
   2. Multiply the unit rate by the new time: $24 \times 8 = 192\text{ pages}$.

   Method 2: Using a Ratio Table

   Set up the relationship between time and pages:

   • Minutes: 3 → divide by 3 → 1 → multiply by 8 → 8
   • Pages: 72 → divide by 3 → 24 → multiply by 8 → 192

   By finding the value for 1 minute first, you can scale directly to any target amount.

   Comparing Rates

   To determine the "better buy" between two packages, calculate the unit price for each option:

   • Brand A: 12 ounces for 3.60 → \(\frac{3.60}{12} = \0.30\) per ounce
   • Brand B: 16 ounces for 4.00 → \(\frac{4.00}{16} = \0.25\) per ounce

   Because $0.25 is less than$0.30, Brand B offers the lower price per ounce.