Question

select all the expressions that can be used to 18 ft 9 ft 6 ft 20 ft (9 × 20) − 6 (9 × 18) − (6 × 2) (9 × 18) + (6 × 2) (20 × 6) + (18 × 3) (20 × 9) − (3 × 2)

Explanation:

To find the area of the composite figure, we can use two methods:

Method 1: Subtraction of a small rectangle from a large rectangle

The large rectangle has dimensions \(20\) ft (length) and \(9\) ft (width). The small rectangle that is missing has dimensions \((20 - 18)=2\) ft (length) and \((9 - 6) = 3\) ft (width)? Wait, no, let's re - examine. Wait, actually, if we consider the figure as a large rectangle of \(20\times9\) and then subtract the area of the small rectangle that is "cut out". The cut - out rectangle has length \(20 - 18=2\) ft and width \(9 - 6 = 3\) ft? Wait, no, maybe another way. Wait, the correct way:

Alternatively, we can split the figure into two rectangles. One rectangle with dimensions \(18\times9\) and another rectangle with dimensions \(6\times2\) (since \(20 - 18 = 2\) and the height of the second rectangle is \(6\) ft? Wait, no, the height of the second rectangle: the total height is \(9\) ft, and the first part has height \(9\) ft, the second part has height \(6\) ft? Wait, maybe I made a mistake. Let's re - analyze the figure.

The figure can be thought of as a large rectangle of length \(20\) ft and width \(9\) ft, and then a small rectangle of length \((20 - 18)=2\) ft and width \((9 - 6)=3\) ft is subtracted? No, wait, the vertical side on the right is \(6\) ft, and the left is \(9\) ft, so the difference in height is \(9 - 6 = 3\) ft. The horizontal difference is \(20 - 18=2\) ft. So the area of the cut - out is \(2\times3\). So the area of the figure is \(20\times9-2\times3=(20\times9)-(3\times2)\).

Another way: split the figure into two rectangles. The first rectangle has length \(18\) ft and width \(9\) ft, and the second rectangle has length \(2\) ft (\(20 - 18\)) and width \(6\) ft. Then the area is \(18\times9+2\times6=(9\times18)+(6\times2)\).

Now let's check each option:

1. Option \((9\times20)-6\): The area of the large rectangle is \(9\times20\), and subtracting \(6\) (a length, not an area) is incorrect. So this option is wrong.

2. Option \((9\times18)-(6\times2)\): \(9\times18\) is the area of the left - hand rectangle, and subtracting \(6\times2\) (which would be a subtraction of an area, but in reality, we should be adding the area of the right - hand rectangle, not subtracting). So this option is wrong.

3. Option \((9\times18)+(6\times2)\): The left - hand rectangle has area \(9\times18\), and the right - hand rectangle has length \(20 - 18 = 2\) and width \(6\), so its area is \(6\times2\). Adding these two areas gives the total area of the figure. So this option is correct.

4. Option \((20\times6)+(18\times3)\): The first part \(20\times6\) is the area of a rectangle with length \(20\) and width \(6\), and the second part \(18\times3\) (since \(9 - 6=3\)) is the area of the upper rectangle. Let's calculate: \(20\times6 = 120\), \(18\times3=54\), \(120 + 54=174\). Now, let's calculate the area using the correct method. The area of the figure: if we use the split method \(18\times9+6\times2=162 + 12 = 174\). If we use the subtraction method \(20\times9-3\times2=180 - 6 = 174\). And \((20\times6)+(18\times3)=120 + 54 = 174\). So this option is correct.

5. Option \((20\times9)-(3\times2)\): The area of the large rectangle is \(20\times9 = 180\), the area of the cut - out is \(3\times2=6\), so \(180 - 6 = 174\), which is the correct area. So this option is correct.

Wait, let's re - check the third option \((9\times18)+(6\times2)\): \(9\times18 = 162\), \(6\times2 = 12\), \(162+12 = 174\).

Fourth option \((20\times6)+(18\times3)\): \(20\times6=120\), \(18\times3 = 54\), \(120 + 54=174\).

Fifth option \((20\times9)-(3\times2)\): \(20\times9 = 180\), \(3\times2 = 6\), \(180-6 = 174\).

Now let's re - check the second option \((9\times18)-(6\times2)\): \(9\times18=162\), \(6\times2 = 12\), \(162-12 = 150 eq174\), so it's wrong.

First option \((9\times20)-6\): \(9\times20 = 180\), \(180-6 = 174\)? Wait, no, \(6\) is a length, not an area. The area of the cut - out should be a product of length and width, not just a length. So this is wrong.

So the correct options are \((9\times18)+(6\times2)\), \((20\times6)+(18\times3)\), and \((20\times9)-(3\times2)\). But let's check the original options again. Wait, the options are:

1. \((9\times20)-6\)

2. \((9\times18)-(6\times2)\)

3. \((9\times18)+(6\times2)\)

4. \((20\times6)+(18\times3)\)

5. \((20\times9)-(3\times2)\)

So let's recalculate the area:

Method 1: Split into two rectangles.

Rectangle 1: length \(18\) ft, width \(9\) ft. Area \(A_1=18\times9 = 162\) square feet.

Rectangle 2: length \(20 - 18=2\) ft, width \(6\) ft. Area \(A_2 = 2\times6=12\) square feet.

Total area \(A=A_1 + A_2=162 + 12=(9\times18)+(6\times2)\). So this option is correct.

Method 2: Another split.

Rectangle 1: length \(20\) ft, width \(6\) ft. Area \(A_1=20\times6 = 120\) square feet.

Rectangle 2: length \(18\) ft, width \(9 - 6=3\) ft. Area \(A_2=18\times3 = 54\) square feet.

Total area \(A=A_1+A_2=(20\times6)+(18\times3)\). So this option is correct.

Method 3: Subtraction.

Large rectangle: length \(20\) ft, width \(9\) ft. Area \(A_{large}=20\times9 = 180\) square feet.

Small rectangle to subtract: length \(20 - 18 = 2\) ft, width \(9 - 6=3\) ft. Area \(A_{small}=2\times3=6\) square feet.

Total area \(A = A_{large}-A_{small}=(20\times9)-(3\times2)\). So this option is correct.

The other two options:

For \((9\times20)-6\): \(9\times20\) is area, \(6\) is length, so it's incorrect.

For \((9\times18)-(6\times2)\): \(9\times18-(6\times2)=162 - 12 = 150 eq174\), so incorrect.

Answer:

The correct expressions are:

• \((9\times18)+(6\times2)\)
• \((20\times6)+(18\times3)\)
• \((20\times9)-(3\times2)\)