Ratio Notation and Relationships

You already write ratios like 2:5 and turn one amount into a fraction of another. A ratio compares two quantities measured in the same unit, and you can write it three ways: 2 to 5, 2:5, or the fraction 2/5. All three mean the same thing. Like fractions, ratios can be simplified, so 4:10 is the same comparison as 2:5.

Now connect the two ideas. A ratio does not just compare two amounts once. It tells you the fixed multiplier between them every time. If the red count is always 2/5 of the blue count, then doubling the blue doubles the red. A relationship with a fixed multiplier like this is called direct variation, and mathematicians write it as y = kx, where k is the multiplier.

The same relationship draws a picture. Plot y against x for y = 3x and every point sits on one straight line through the origin (0, 0). That always happens with direct variation: zero x gives zero y, so the line must pass through (0, 0), and the multiplier k is exactly the line's steepness. A ratio, a fraction, and a line through zero are three views of a single idea.