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Each big square below represents one whole. An array with $10$ columns and $10$ rows that represents $1$ whole. $10$ rows of $10$ are shaded. An array with $10$ columns and $10$ rows that represents $1$ whole. $10$ rows of $10$ are shaded. An array with $10$ columns and $10$ rows that represents $1$ whole. $1$ column of $10$ is shaded. $1$ column of $10$ $6$ is shaded. An array with $10$ columns and $10$ rows that represents $1$ whole. $1$ column of $10$ is shaded. $1$ column of $10$ $6$ is shaded. What percent is represented by the shaded area?

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Math Problems

Algebra 2

Solve a system of equations using substitution: word problems

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Q. Each big square below represents one whole. An array with

10

columns and

10

rows that represents

1

whole.

10

rows of

10

are shaded. An array with

10

columns and

10

rows that represents

1

whole.

10

rows of

10

are shaded. An array with

10

columns and

10

rows that represents

1

whole.

1

column of

10

is shaded.

1

column of

10

6

is shaded. An array with

10

columns and

10

rows that represents

1

whole.

1

column of

10

is shaded.

1

column of

10

6

is shaded. What percent is represented by the shaded area?

Understand the Problem: Let's first understand the problem. We have four arrays, each with $10$ columns and $10$ rows, representing $1$ whole each. The shaded areas in these arrays represent a certain fraction of the whole, which we need to calculate and then convert to a percentage.
First Array Calculation: For the first array, all $10$ rows of $10$ columns are shaded. This means the entire array is shaded, which is $100\%$ of that array.
Second Array Calculation: For the second array, it's the same as the first one: all $10$ rows of $10$ columns are shaded. So, this array is also $100\%$ shaded.
Third Array Calculation: In the third array, $1$ column of $10$ is shaded, and $1$ column of $9$ is shaded. This means we have $10 + 9 = 19$ squares shaded in this array.
Fourth Array Calculation: In the fourth array, the shading is identical to the third array, with $1$ column of $10$ shaded and $1$ column of $9$ shaded, giving us another $19$ squares shaded.
Total Shaded Squares Calculation: Now, let's add up all the shaded squares. We have two full arrays ( $100\%$ each) and two arrays with $19$ squares shaded each. Since each array has $10$ rows of $10$ columns, that's $100$ squares per array. So, the total shaded squares from the third and fourth arrays are $19 + 19 = 38$ squares.
Total Shaded Percentage Calculation: To find the total shaded percentage, we add the percentages of the fully shaded arrays ( $100\% + 100\%$ ) and the percentage of the partially shaded arrays. The partially shaded arrays have $38$ shaded squares out of $200$ total squares ( $100$ per array), which is $\frac{38}{200} = 0.19$ or $19\%$ .
Total Possible Percentage Calculation: The total shaded percentage is therefore $100\% + 100\% + 19\% + 19\% = 238\%$ .
Shaded Percentage Calculation: However, since we have four arrays, and each array represents $100\%$ , the total possible percentage is $400\%$ . The shaded percentage we calculated ( $238\%$ ) is out of this $400\%$ total.
Relative Shaded Area Percentage Calculation: To find the percentage of the shaded area relative to the total area of all four arrays, we divide the shaded percentage by the total possible percentage: $\frac{238\%}{400\%} = 0.595$ or $59.5\%$ .