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 = \(\frac{p}{a + b + c}\)

Therefore, x = ak = \(\frac{ap}{a+ b + c}\), y = bk = \(\frac{bp}{a+ b + c}\), z = ck = \(\frac{cp}{a+ b + c}\).

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

\(\frac{ap}{a+ b + c}\), \(\frac{bp}{a+ b + c}\), \(\frac{cp}{a+ b + c}\).

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 \(\frac{5}{5 + 13 + 15}\) βˆ™ 297, \(\frac{13}{5 + 13 + 15}\) βˆ™ 297 and \(\frac{15}{5 + 13 + 15}\) βˆ™ 297

 i.e., \(\frac{5}{33}\) βˆ™ 297, \(\frac{13}{33}\) βˆ™ 297 and \(\frac{15}{33}\) βˆ™ 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 \(\frac{1}{1 + 2 + 3}\) βˆ™ 432, \(\frac{2}{1 + 2 + 3}\) βˆ™ 432 and \(\frac{3}{1 + 2 + 3}\) βˆ™ 432

i.e., \(\frac{1}{6}\) βˆ™ 432, \(\frac{2}{6}\) βˆ™ 432 and \(\frac{3}{6}\) βˆ™ 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 \(\frac{1}{1 + 3 + 4}\) βˆ™ 80, \(\frac{3}{1 + 3 + 4}\) βˆ™ 80 and \(\frac{4}{1 + 3 + 4}\) βˆ™ 80

i.e., \(\frac{1}{8}\) βˆ™ 80, \(\frac{3}{8}\) βˆ™ 80 and \(\frac{4}{8}\) βˆ™ 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 \(\frac{2}{9}\) Γ— 45 cm, \(\frac{3}{9}\) Γ— 45 cm and \(\frac{4}{9}\) Γ— 45 cm,

i.Π΅., 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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