Today our topic of discussion is Proportional Division.
Proportional Division
Proportional Division
Division of a quantity into fixed ratio is called proportional division. If S is to be divided in a given ratio a: b:c: d, dividing S by a + b + c + d ,the parts a, b, c and d need to be taken.
Therefore, 1st part s of =a/(a + b + c + d) = (Sa)/(a + b + c + d)
2nd parts of b ᏚᏂ = a+b+c+d a+b+c+d
3rd part = S of C Sc = a+b+c+d a+b+c+d
4th part = S of d Sd a+b+c+d a+b+c+d
In this way, any quantity may be divided into any fixed ratio.
Example 15. The area of a rectangular land is 12 hectors and length of its diagonal is 500 metres. The ratio of length and breadth of this land to that of
other land are 3: 4 and 2:3, respectively.
1) What is the area of the given land in square metre?
2) What is the area of the other land?
3) What is the breadth of the given land?
Solution:
1) We know, 1 hector = 10,000 square metres.
∴12 hectors 12 x 10,000 120000 square metres
2) Given that, the ratio of length and breadth of the given land to that of other land are 3: 4 and 2: 3, respectively.
Let length of the given land be 3z metres and breadth be 2y metres.
Therefore, the length of the other land is 4 metres and breadth is 3y metres
∴ the area of the given land s = 3x * 2y = 6xy square metres and the area of the other land is = 4x * 3y = 12xy square metres.
According to the given question, 6xy = 120000or , xy = 20000 the area of the other land is = 12xy = 12 * 20000 = 240000 square metres.
3) Let the length of the given land be 3r metres and the breadth be 2y metres Therefore, the length of its one diagonal is √((3x)² + (2y)² ) metres.
We get from the previous (b) problem, xy = 20000
According to the question, √((3x) ² + (2y)²) = 500 or, 9x² + 4y² = 250000
or, (3x + 2y)² – 2 * 3x * 2y = 250000
or, (3x + 2y)² – 12xy = 250000
or, (3x + 2y)² – 12 * 20000 = 250000
or, (3x + 2y)² = 250000 + 240000
or, (3x + 2y)² = 490000
or, 3x + 2y =700***(1)
Again, (3x – 2y)² = (3x + 2y)² – 4 * 3x * 2y
or, (3x – 2y)² = (3x + 2y)² – 24xy
or (3x – 2y)² = (700)² – 24 * 20000
or, (3x – 2y) ² = 490000 – 480000
or, (3x – 2y)² = 10000
or, 3x – 2y = 100²
By subtracting (2) from (1) we get,
4y = 600
or, y = 150
∴ the breadth of the given land is 2y = 2 * 150 = 300 metres.
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