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

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

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

i.e., \(\frac{1}{8}\) ∙ 80, \(\frac{3}{8}\) ∙ 80 and \(\frac{4}{8}\) ∙ 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 \(\frac{2}{9}\) × 45 cm, \(\frac{3}{9}\) × 45 cm and \(\frac{4}{9}\) × 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.

● Ratio and proportion

• Basic Concept of Ratios
• Important Properties of Ratios
• Ratio in Lowest Term
  
• Types of Ratios
• Comparing Ratios
• Arranging Ratios
  
• Dividing into a Given Ratio
  
• Divide a Number into Three Parts in a Given Ratio
• Dividing a Quantity into Three Parts in a Given Ratio
  
• Problems on Ratio
  
• Worksheet on Ratio in Lowest Term
  
• Worksheet on Types of Ratios
  
• Worksheet on Comparison on Ratios
• Worksheet on Ratio of Two or More Quantities
  
• Worksheet on Dividing a Quantity in a Given Ratio
• Word Problems on Ratio
  
• Proportion
  
• Definition of Continued Proportion
  
• Mean and Third Proportional
  
• Word Problems on Proportion
  
• Worksheet on Proportion and Continued Proportion
  
• Worksheet on Mean Proportional
  
• Properties of Ratio and Proportion

10th Grade Math

From Divide a Number into Three Parts in a Given Ratio to HOME PAGE