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Divide a Number into Three Parts in a Given Ratio

To divide a number into three parts in a given ratio

Let the number be p. It is to be divided into three parts in the ratio a : b : c.

Let the parts be x, y and z. Then, x + y + z = p .................... (i)

and        x = ak, y =bk, z = ck.................... (ii)

Substituting in (i), ak + bk + ck = p

  ⟹ k(a + b + c) = p

Therefore, k = pa+b+c

Therefore, x = ak = apa+b+c, y = bk = bpa+b+c, z = ck = cpa+b+c.

The three parts of p in the ratio a : b : c are

apa+b+c, bpa+b+c, cpa+b+c.

Solved Examples on Dividing a Number into Three Parts in a Given Ratio:

1. Divide 297 into three parts that are in the ratio 5 : 13 : 15

Solution:

The three parts are 55+13+15 ∙ 297, 135+13+15 ∙ 297 and 155+13+15 ∙ 297

 i.e., 533 ∙ 297, 1333 ∙ 297 and 1533 ∙ 297 i.e., 45, 117 and 135.

 

2. Divide 432 into three parts that are in the ratio 1 : 2 : 3

Solution:

The three parts are 11+2+3 ∙ 432, 21+2+3 ∙ 432 and 31+2+3 ∙ 432

i.e., 16 ∙ 432, 26 ∙ 432 and 36 ∙ 432

i.e., 72, 144 and 216.

 

3. Divide 80 into three parts that are in the ratio 1 : 3 : 4.

Solution:

The three parts are 11+3+4 ∙ 80, 31+3+4 ∙ 80 and 41+3+4 ∙ 80

i.e., 18 ∙ 80, 38 ∙ 80 and 48 ∙ 80

i.e., 10, 30 and 40.


4. If the perimeter of a triangle is 45 cm and its sides are in the ratio 2: 3: 4, find the sides of the triangle.

Solution:

Perimeter of the triangle = 45 cm

Ratio of the sides of the triangle = 2 : 3 : 4

Sum of ratio terms = (2 + 3 + 4) = 9

The sides of the triangle 29 × 45 cm, 39 × 45 cm and 49 × 45 cm,

i.е., 10 cm, 15 cm and 20 cm. 

Hence, the sides of the triangle are 10 cm, 15 cm and 20 cm.

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● Ratio and proportion




10th Grade Math

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