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Representing
Ratios

Learn to represent ratio relationships using tables and graphs, and discover what each point on a ratio graph really means.

Pre-Algebra

What Is a Ratio?

A ratio compares two quantities by showing how much of one thing there is relative to another. If a recipe uses 2 cups of flour for every 3 cups of sugar, the ratio of flour to sugar is \( 2 : 3 \).

Building Equivalent Ratios

You create equivalent ratios by multiplying (or dividing) both quantities by the same number. The relationship stays the same: the quantities just scale up or down together.

\( 4 : 6 \)
\( 2 : 3 \)
\( 6 : 9 \)
\( \times 2 \)
\( \times 3 \)
 All equivalent to 2 : 3 2 blue : 3 green

Why Represent Ratios?

Writing a single ratio like \( 2 : 3 \) is useful, but what if you need to scale a recipe for 20 people, or predict future sales? Tables and graphs let you see all the equivalent ratios at once and spot patterns quickly.

Tables
Organize exact values in rows and columns: easy to compute.
Graphs
Plot points on a coordinate plane: easy to visualize trends.
Predictions
Extend the pattern to values you haven't calculated yet.

The Ratio Table

A ratio table lists pairs of values that share the same ratio. Each row is an equivalent ratio created by multiplying both quantities by the same factor.

Flour (cups) Sugar (cups) Factor
23\( \times 1 \)
46\( \times 2 \)
69\( \times 3 \)
812\( \times 4 \)
1015\( \times 5 \)

The Ratio Graph

Plot each pair from the table as a point \( (x, y) \) on a coordinate plane. For equivalent ratios, all points lie on a straight line through the origin.

0 3 6 9 12 15 2 4 6 8 10 Flour (cups) Sugar (cups) (2, 3) (4, 6) (6, 9) (8, 12) (10, 15) (0, 0)

The Straight-Line Rule

Origin
The line starts at \( (0,0) \): zero of one means zero of the other.
Slope
The steepness of the line equals the unit rate \( k = \frac{y}{x} \).

What the Points Mean

Each plotted point represents one specific pair of quantities that keeps the ratio true. For the flour-to-sugar ratio \( 2 : 3 \):

1
The point \( (2, 3) \) means 2 cups flour and 3 cups sugar.
2
The point \( (6, 9) \) means 6 cups flour and 9 cups sugar: same ratio, bigger batch.
3
The point \( (0, 0) \) means no flour, no sugar: the recipe hasn't started.
4
Any point not on the line (like \( (4, 5) \)) breaks the ratio.

Ratio vs. Fraction

Ratios and fractions look similar but describe different things:

Ratio Fraction
Compares Part to part or part to whole Part to whole only
Example \( 2 : 3 \) (flour to sugar) \( \frac{2}{5} \) (flour out of total)
On a graph Straight line through origin Not typically graphed this way
Scaling Multiply both parts Equivalent fractions

Step-by-Step: Table to Graph

1
Identify the ratio. Write it as \( a : b \).
2
Build the table. Multiply both \( a \) and \( b \) by \( 1, 2, 3, \ldots \) to generate rows.
3
Set up axes. Label the \( x \)-axis for the first quantity, the \( y \)-axis for the second.
4
Plot each pair as a point \( (a, b) \). Start with \( (0, 0) \).
5
Draw a straight line through all points. If it passes through the origin, the ratio is correct.

Example 1: Miles per Hour

A cyclist rides at a constant rate of 12 miles every 2 hours. Build a table and graph the ratio.

Hours Miles
00
212
424
636
848

Example 1: Graphed

0 12 24 36 48 2 4 6 8 Hours Miles (2, 12) (4, 24) (6, 36) (8, 48)

Example 2: Lemonade Recipe

A lemonade recipe calls for 3 lemons for every 5 cups of water. Complete the table and identify the point that represents making lemonade for a party.

Lemons Water (cups)
35
610
915
1220
1525

Example 3: Reading a Ratio Graph

The graph below shows the ratio of blue paint to red paint needed for a specific shade of purple. What ratio does it represent?

0 2 4 6 8 1 2 3 4 5 Blue Paint (oz) Red Paint (oz) (1, 2) (2, 4) (3, 6) (4, 8)

Common Mistakes to Avoid

✕ Forgetting the Origin
A ratio graph must pass through \( (0, 0) \). If you start plotting at \( (1, 3) \), you miss this anchor point.
✕ Swapping the Axes
Plotting \( x \) and \( y \) in the wrong order gives the reciprocal ratio. Always match the axis to the correct quantity.
✕ Scaling Only One Side
Multiplying just one quantity (e.g., \( 2 : 3 \to 4 : 3 \)) breaks the ratio. You must multiply both parts.

Summary: Tables vs. Graphs

Ratio Table Ratio Graph
Format Rows of number pairs Points on a coordinate plane
Strength Exact values, easy arithmetic Visual pattern recognition
Key feature Constant multiplier between rows Straight line through origin
Prediction Extend the table with more rows Extend the line and read off values
Best for Precise calculations Seeing the big picture

Question 1 of 5

A table shows the ratio \( 4 : 7 \). Which row belongs in the table?

Question 2 of 5

On a ratio graph, all points representing equivalent ratios form:

Question 3 of 5

The point \( (5, 15) \) is on a ratio graph. What is the unit rate \( \frac{y}{x} \)?

Question 4 of 5

A recipe has a ratio of \( 2 : 5 \) (eggs to cups of milk). You want to use 8 eggs. How many cups of milk do you need?

Question 5 of 5

Which point would not lie on the graph of the ratio \( 3 : 4 \)?