2-6 Find the Whole

Property

In percent word problems, identify three key components:

• Part: The portion or amount being compared
• Whole: The total amount (often follows the word "of")
• Percent: The rate per 100 (includes % symbol or words like "percent")

Examples

Property

Visual models like the bar (tape) model and the double number line help illustrate percent problems. A bar is divided into equal parts representing friendly percentages (like $10\%$ or $25\%$). A double number line draws a correspondence between the percent scale (from $0\%$ to $100\%$) and the quantity scale (from 0 to the whole amount), allowing for visual calculation.

Examples

• To find $40\%$ of 30, draw a bar model with 10 sections. The whole bar is 30, so each section is $30 \div 10 = 3$. Shading 4 sections for $40\%$ gives $4 \times 3 = 12$.
• A recipe needs 2 cups of sugar, which is $25\%$ of the total ingredients. Using a bar model with 4 sections (each $25\%$), one section is 2 cups. The total is $4 \times 2 = 8$ cups.
• On a double number line, 15 minutes corresponds to $20\%$. To find how long it takes to reach $100\%$, you can see that $100\%$ is $5 \times 20\%$. So the total time is $5 \times 15 = 75$ minutes.

Explanation

Models like bar diagrams and double number lines turn tricky percent problems into pictures. They help you visually connect the part, the whole, and the percent, making it much easier to see the relationship and find the answer.

Property

A ratio table can be used to find the whole by creating a series of equivalent ratios. You can scale the given part and percent up or down by multiplying or dividing both quantities by the same number. The goal is to find the value (the whole) that corresponds to $100\%$.

Examples

• 30 is 60% of what number?

We can scale down from 60% to 10% by dividing by 6, and then scale up to 100% by multiplying by 10.

So, 30 is 60% of 50.

• 45 is 75% of what number?

We can scale down from 75% to 25% by dividing by 3, and then scale up to 100% by multiplying by 4.

So, 45 is 75% of 60.

Explanation

A ratio table helps organize the relationship between the part and the percent. Start by writing the known part and percent in the table. Then, find a convenient "benchmark" percent (like 1%, 5%, 10%, or 25%) by dividing both the part and the percent by the same number. Finally, multiply your new part and percent by a number that scales the percent to 100% to find the whole.