Question

Of the two square fields, the area of one is 1 hectare while the other one is broader by 1%. What is the difference in their areas?

Answer 1

Hint: To find the difference in the area of the fields we first find the increased percentage value of the area as given in the question and then add to the area or the original area of the field as given in the question. The formula for the difference in area is given as:
\[1%\] of Original Area of the field \[+\] Original Area of the field \[-\] Original Area of the field.

Complete step by step answer:
According to the question given, the length of the side of the first field is taken as “x” and as the field is square both the sides are calculated as \[x\]. Now, the area of the field as given in the question is 1 hectare. Hence, using the square field area formula, the value of sides of the square field is given as:
\[x\times x=1\] hectare or the unit hectare can be changed into m sq. as \[1\] hectare is equal to \[10000\]m.sq. Therefore, placing the value of the area of the field in terms of m.sq, we have the area of the field as:
\[x\times x=10000\] m.sq
Now we find the value of the side of the first square field as:
\[\Rightarrow {{x}^{2}}=10000\] m.sq
\[\Rightarrow x=\sqrt{10000}\] m
\[\Rightarrow x=100\] m
Now that we know the length of the sides of the square field, we increase the length of the side of the square field by \[1%\].
\[\Rightarrow 100+\dfrac{1}{100}\times 100\]
\[\Rightarrow 101\] m
Hence, after increasing the sides by \[1%\], we get the length of the sides of a square field as \[101\] m and using this increase length value we find the area of the square field with one side of the square field as \[{{x}_{1}}\] m, we get the area as:
\[\Rightarrow {{x}_{1}}\times {{x}_{1}}\] m.sq
\[\Rightarrow 101\times 101=10201\] m.sq
\[\Rightarrow {{x}^{2}}=10201\] m.sq
With the area measurement of both the fields we get the difference in the area of the fields as:
Area of the second field \[-\] Area of the first field \[=\] Remaining Area/Difference Area.
Placing the values in the formula we get the difference in the area as:
\[\Rightarrow 10201-10000=201\] m.sq
Therefore, the difference area of both the fields from the second to the first one is given as:
\[\Rightarrow 10201-10000=201\] m.sq
\[\Rightarrow 201\] m.sq

Note: Another formula to find the difference is by using the formula as:
\[\Rightarrow \dfrac{\text{increase}%}{100}\times \text{Original Area}+\text{Original Area}-\text{Original Area}\] m.sq
\[\dfrac{1}{100}\times 10000+10000-10000=201\] m.sq