Answer
The amount that Diana's sister would buy the remaining 720 square inches piece of cloth = $5
Explanation
We are told that Diana bought a piece of cloth that was 48 inches wide and 1 yard long for $12.
Then, she had different parts of the cloth taken to do different stuff until the remaining piece of the cloth was to be bought by her sister at the same rate that she bought the original full piece of cloth.
Before we proceed, we need to establish and find the initial rate of the cloth, that is, how much is 1 unit of area.
Area of a rectangular figure (the cloth) = Length × Breadth
Length of the cloth = 1 yard = 36 inches
Breadth of the cloth = 48 inches
Area of the cloth = 36 × 48 = 1728 square inches.
1728 square inches of the cloth = $12
1 square inch of the cloth = (12/1728) = (1/144) dollars
So, now, we can now find how much of the cloth is removed.
- She cut off one-fourth of the cloth to make a tablecloth.
One-fourth of the 1728 square inches piec of cloth = (1/4) × 1728 = 432 square inches
- She used a piece that was 12 inches wide and 12 inches long to make a scarf.
Area of the piece she used to make a scarf = 12 × 12 = 144 square inches
- She used a piece that was 1.5 feet by 2 feet to make a cover for a pillow.
To find the area of this piece, we need to convert the dimensions into inches.
1 feet = 12 inches
So,
1.5 feet = 18 inches
2 feet = 24 inches
Area of this piece of cloth = 18 × 24 = 432 square inches
So, her sister now wants to buy the remaining piece of cloth from Diana.
Originally, the piece of cloth was 1728 square inches.
The total area removed from the original piece of cloth sum up to
432 + 144 + 432 = 1008 square inches.
The remaining piece of cloth will now have an area calculated by subtracting the area of pieces removed from the original area of the piece of cloth.
The area of the remaining piece of cloth = 1728 - 1008 = 720 square inches
Recall that the Diana's sister wants to but the remaining piece of cloth at the rate that Diana bought the original piece of cloth.
We have calculated this rate at which Diana bough the original piece of clothing to be (1/144) dollars for 1 square inch. So, we can then use this rate to calculate the amount Diana's sister will pay for the remaining piece of cloth
Area of the remaining piece of cloth = 720 square inch
1 square inch costs (1/144) dollars.
Let the cost of 720 square inches be x.
1 square inch = (1/144) dollars
720 square inches = x dollars
We can from a mathematical relationship simply by cross multiplying
1 × x = 720 × (1/144)
x = 5 dollars.
Hence, the amount that Diana's sister would buy the remaining 720 square inches piece of cloth = $5
Hope this Helps!!!
For the guess and check method, we would still need to first find the cost of 1 square inch of material which we obtained to be (1/144) dollars per square inch of cloth.
We would also still need to calculate the total area removed from the cloth, which we calculated to be 1008 square inches.
But now, we will now use the cost of 1 square inch, to calculate the cost of the pieces that have been removed from the cloth.
1008 × (1/144) = 7 dollars.
We can now guess how many dollars will top up the 7 dollars so that it gives the original 12 dollars that the cloth was bought.
After a series of guess, we will end up to get that it is (7 + 5) that is equal to 12 dollars.
So, guess and check would give us 5 dollars too as the answer.