5. Part-Whole Ratio Problems

In part-whole ratio problems, the ratio parts add to the total number of equal parts.

Example: Blue and green tiles are in the ratio \(3:2\). There are 40 tiles total. How many are blue?

The total parts are:
\[ 3+2=5. \]
Each part represents:
\[ 40/5=8\text{ tiles}. \]
Blue uses 3 parts:
\[ 3(8)=24\text{ blue tiles}. \]

This is why a tape diagram is powerful: it shows how many equal parts make the whole.

If the problem asks for one part, multiply the part value by the number of parts requested. If it asks for the difference between parts, subtract the part counts first. In a \(7:4\) ratio with each part equal to 6, the difference is \((7-4)(6)=18\).

Common misconception: using \(3/2\) of the total when the ratio is \(3:2\). The blue share is \(3/5\), not \(3/2\), because the whole has 5 parts.

Academic habit: always add ratio parts before using a total. The sum of ratio parts represents the whole.