Ratios & Proportions Study Guide & Hub

1. Checking Tables for Proportionality

Calculate the ratio y/x for each pair. If all ratios are equal to the same constant k, the relationship is proportional.

2. Checking Graphs for Proportionality

A graph represents a proportional relationship when both conditions are true:
1. It is a straight line.
2. It passes through the origin (0,0).

💡 How to Decide Which Quantity is x and Which is y

• Table Position Rule: In tables, the 1st column (or top row) is always the independent input ($x$), and the 2nd column (or bottom row) is the dependent output ($y$).
• Real-World Dependency Rule: Ask "Which quantity depends on the other?" The output ($y$) depends on the input ($x$). For example, Total Cost ($c$) depends on Quantity ($n$), so $c = y$ and $n = x$.

1. Table Example

✓ Table: k = y/x

Ratios: $12/1 = 24/2 = 36/3 = 12$. Constant $k = 12$ ($y = 12x$).

2. Graph Example

Line passing through (0,0) and (1, 4)

Point (1, 4) gives $k = 4/1 = 4$ ($y = 4x$).

3. Equation Example

In $y = 5x$, the constant of proportionality is $k = 5$.

4. Double Number Line Diagram

Matching tick marks $x = 1$ and $y = 6$ show $k = 6/1 = 6$.

5. Word Problem Example

A car travels 150 miles in 3 hours at a constant speed. What is k in miles per hour?

Unit rate: $k = 150 \div 3 = 50$ miles per hour.

💡 How to Decide Which Quantity is x and Which is y

• Table Position Rule: In tables, the 1st column (or top row) is always the independent input ($x$), and the 2nd column (or bottom row) is the dependent output ($y$).
• Real-World Dependency Rule: Ask "Which quantity depends on the other?" The output ($y$) depends on the input ($x$). For example, Total Cost ($c$) depends on Quantity ($n$), so $c = y$ and $n = x$.

Swapping y and x with Single-Letter Quantity Variables

In real-world problems, standard variables $y$ (dependent output) and $x$ (independent input) are replaced by quantity variables:

• Determining $x$ and $y$: The input ($x$) is the cause or base quantity (e.g., time, quantity, hours) — in tables, it is the 1st column (or top row). The output ($y$) is the result or total (e.g., distance, cost, pay) — in tables, it is the 2nd column (or bottom row).
• Distance ($d$) and Time ($t$): d = kt (e.g., $d = 60t$)
• Total Cost ($c$) and Quantity ($n$): c = kn (e.g., $c = 3.50n$)
• Total Pay ($p$) and Hours ($h$): p = kh (e.g., $p = 15h$)
• Volume ($v$) and Time ($t$): v = kt (e.g., $v = 2.5t$)
• Total Weight ($w$) and Boxes ($b$): w = kb (e.g., $w = 4b$)

1. Writing an Equation from a Table

Given table with Time $t$ (hours) and Distance $d$ (miles):

• Constant rate: $k = \frac{45}{1} = 45$.
• Equation substituting quantity variables: d = 45t.

2. Writing an Equation from a Graph

Given a line on a graph with x-axis Time $t$ (hours) and y-axis Distance $d$ (miles) passing through (0,0) and (1, 60):

• Unit rate / slope: $k = 60$.
• Equation substituting quantity variables: d = 60t.

3. Writing an Equation from a Word Problem

• Output variable: $p$, Input variable: $h$, Unit rate: $k = 25$.
• Equation substituting quantity variables: p = 25h.

1. What does any point (x, y) represent?

The point $(x, y)$ indicates that x units of the horizontal quantity correspond to y units of the vertical quantity.

2. Special Point: (0, 0) (The Origin)

The point $(0, 0)$ shows that when input $x = 0$, output $y = 0$. For example, 0 hours worked results in $0 earned.

3. Special Point: (1, r) (The Unit Rate)

The point $(1, r)$ explicitly reveals the unit rate r. For example, $(1, 90)$ means traveling for 1 hour covers 90 km (speed is 90 km/h).