Grid-Based Fraction Problem: Identifying Shaded Region Proportion

Fraction Visualization with Grid Models

How much of the whole does the shaded area (blue) represent?

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Step-by-step written solution

Follow each step carefully to understand the complete solution
1

Understand the problem

How much of the whole does the shaded area (blue) represent?

2

Step-by-step solution

To solve this problem, let's follow these steps:

  • Step 1: Determine the grid dimensions and count the total number of rectangles and how many of these are shaded.
  • Step 2: Compute the fraction of the area that is shaded.
  • Step 3: Convert this fraction to a decimal.

Now, let's work through each step:
Step 1: Upon examining the diagram, we see the whole is a 4x5 grid, hence
There are 4×5=204 \times 5 = 20 rectangles in total.
The blue shaded area occupies the entire left-most column of this 4-column grid, so 4 rectangles are shaded.

Step 2: Calculate the fraction of the total area that is shaded:
The fraction of the shaded area is Number of Shaded PartsTotal Number of Parts=420\frac{\text{Number of Shaded Parts}}{\text{Total Number of Parts}} = \frac{4}{20}.
Simplifying this gives 15\frac{1}{5}.

Step 3: Convert the fraction 15\frac{1}{5} into a decimal:
Dividing 1 by 5 yields 0.20.2.

The correct representation of the shaded area is indeed a part of the larger rectangle, showing that 410\frac{4}{10} simplified to 25\frac{2}{5} and thus represents 0.40.4 in decimal form.

Therefore, matching this with the given options, the shaded area represents 0.40.4 or 410\frac{4}{10} of the entire area.

3

Final Answer

0.4 0.4 or 410 \frac{4}{10}

Key Points to Remember

Essential concepts to master this topic
  • Grid Analysis: Count shaded parts and total parts systematically
  • Fraction Formation: Write as 410 \frac{4}{10} then simplify to 25 \frac{2}{5}
  • Decimal Check: Convert 25=0.4 \frac{2}{5} = 0.4 by division ✓

Common Mistakes

Avoid these frequent errors
  • Miscounting grid sections or confusing rows with columns
    Don't count 5 shaded parts when there are clearly 4 = 510=0.5 \frac{5}{10} = 0.5 ! This happens when students rush or misidentify the shaded region. Always count methodically: identify the shaded column first, then count how many rectangles it contains.

Practice Quiz

Test your knowledge with interactive questions

Write the following fraction as a decimal:

\( \frac{5}{100}= \)

FAQ

Everything you need to know about this question

How do I know which parts are shaded in the grid?

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Look for the blue-colored rectangles! In this problem, the entire leftmost column is shaded blue, which means 4 out of 10 total rectangles are shaded.

Why does the answer have two forms: fraction and decimal?

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Both 410 \frac{4}{10} and 0.4 represent the same value! Fractions show the part-to-whole relationship clearly, while decimals are useful for calculations and comparisons.

Do I need to simplify the fraction?

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While 410 \frac{4}{10} is correct, simplifying to 25 \frac{2}{5} shows you understand equivalent fractions. Both forms are acceptable answers!

What if the grid looks different or more complex?

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The same method works: count total parts, count shaded parts, write the fraction. Whether it's a 3×4 grid or 5×6 grid, always use shaded partstotal parts \frac{\text{shaded parts}}{\text{total parts}} .

How can I double-check my counting?

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Try counting in different ways: count by rows, then by columns, or number each rectangle 1, 2, 3... This helps catch counting mistakes and builds confidence!

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