Using Grid Models to Solve Percentage Problems

How to using grid models to solve percentage problems, step by step: the rule in plain language, solved examples worked one line at a time, and practice questions so you can try it yourself before moving on.

A \(\color{blue}{10 \times 10}\) grid is a powerful visual tool for understanding percentage problems. Because the grid has exactly 100 squares, each square represents 1%. You can use it to find the part, the percent, or the whole, the three quantities in every percentage problem.

What Is a Grid Model for Percentages?

A grid model is a square divided into 100 equal cells. Shading a certain number of cells represents that percentage of the whole. The relationship can be written as:

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\(\color{blue}{\text{ Part } = \text{ Percent } \times \text{ Whole }}\)

Every percentage problem asks you to find one of these three values when the other two are known.

How to Use Grid Models to Solve Percentage Problems

Finding the Part

Shade (percent) cells out of 100 to represent the percent. The shaded cells correspond to the part of the whole.

Example: 30% of 50. The grid shows 30 shaded cells out of 100. Scale: \(\color{blue}{0.30 \times 50 = 15}\).

Finding the Percent

Count the shaded cells (or equivalent cells out of 100). That count is the percent.

Example: 24 out of 80. Scale down: \(\color{blue}{\frac{24}{80} = \frac{30}{100} = 30\%}\).

Finding the Whole

The percent \(\color{blue}{\text{ shaded } = \text{ part }}\) ÷ \(\color{blue}{\text{ whole } \times 100}\). Rearrange: \(\color{blue}{\text{ whole } = \text{ part }}\) ÷ (\(\color{blue}{\frac{\text{ percent }}{100}}\)).

Example: 60% of what number is 18? \(\color{blue}{18 &\text{ div }; 0.60 = 30}\).

Step-by-Step Summary

1. Identify which value is missing: part, percent, or whole.
2. Write the formula: \(\color{blue}{\text{ Part } = \text{ Percent } \times \text{ Whole }}\).
3. Substitute the two known values and solve for the third.
4. On a grid: count or shade cells to visualize and verify.

Watch: Finding a Percent of a Number (Video Lesson)

Math Antics explains the step-by-step approach to finding a percent of a number using visual methods:

Worked Examples

Example 1: What is 30% of 50?

Formula: \(\color{blue}{\text{ Part } = 0.30 \times 50 = 15}\).
Answer: 15

Example 2: 24 is what percent of 80?

Formula: \(\color{blue}{\text{ Percent } = \frac{24}{80} \times 100 = 30\%}\).
Answer: 30%

Example 3: 60% of a number is 18. What is the number?

Formula: \(\color{blue}{\text{ Whole } = 18 &\text{ div }; 0.60 = 30}\).
Answer: 30

Example 4: A grid has 45 cells shaded. The full grid represents 200 items. How many items does the shaded portion represent?

Percent shaded: \(\color{blue}{45\%}\). Part: \(\color{blue}{0.45 \times 200 = 90}\).
Answer: 90 items

More Practice: Percent Word Problems

Math with Mr. J applies these exact three-part percentage strategies to word problems:

Exercises

1. What is 25% of 80?
2. 36 is what percent of 90?
3. 75% of a number is 60. What is the number?
4. A grid has 55 shaded squares. The whole represents 300 people. How many people does the shaded part represent?
5. What is 40% of 125?
6. 18 is 45% of what number?

Answers

1. \(\color{blue}{20}\)
2. \(\color{blue}{40\%}\)
3. \(\color{blue}{80}\)
4. \(\color{blue}{165 \text{ people }}\)
5. \(\color{blue}{50}\)
6. \(\color{blue}{40}\)

Frequently Asked Questions

What is a grid model?

A grid model for percents is a 10 × 10 square split into 100 equal cells. Since there are 100 cells, each one stands for 1% of the whole, and shading cells shows a percent. For example, shading 3 full rows and 5 more cells shows 35%. Grid models help students see that a percent is just a part out of 100.

What percent is modeled by the grid?

Count the shaded cells on the 100-cell grid; that number is the percent. Counting full rows of 10 speeds things up: 6 full rows plus 4 cells is 64 shaded cells, so the grid models 64%. If a grid does not have 100 cells, write shaded over total and change it to a fraction out of 100. For example, 12 out of 25 cells equals 48/100, or 48%.

How do you use a 10 by 10 grid to model the percent?

Shade one cell for each percent point. For 72%, shade 7 full rows of 10 and then 2 more cells. To show a percent of a number, let the whole grid stand for that number. If the grid represents 50 students, each cell is 0.5 students, so 72% of 50 is 72 × 0.5 = 36 students.

How do you find a percent of a number with a grid?

Let the full grid equal the whole, find the value of one cell, then multiply by the number of shaded cells. To find 30% of 80, each cell is 80 ÷ 100 = 0.8, and 30 cells make 30 × 0.8 = 24. This matches the shortcut part = percent × whole: 0.30 × 80 = 24. The grid simply shows why the shortcut works.

Related Topics

• Representing Percentage
• Using Decimals, Grid Models and Fractions to Represent Percent
• Word Problems Involving Percentage of a Number
• Solving Percentage Word Problems
• Fractional and Decimal Percentages