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If the ratio of some numbers is [tex]\frac{3}{9} : \frac{15}{21} : \frac{18}{30}[/tex], and the difference between the highest and the lowest number is 80, find the respective numbers.

A. 80, 160, 210
B. 70, 126, 150
C. 140, 126, 135
D. 150, 126, 75

Answer :

To solve this problem and find the actual numbers corresponding to the given ratios, let's break it down step by step.

The given ratio is:
[tex](\frac{3}{9}) : (\frac{15}{21}) : (\frac{18}{30})[/tex]

First, simplify each fraction to find the simplest ratio:

  1. [tex]\frac{3}{9} = \frac{1}{3}[/tex]

  2. [tex]\frac{15}{21} = \frac{5}{7}[/tex]

  3. [tex]\frac{18}{30} = \frac{3}{5}[/tex]

The simplified ratio becomes:
[tex]\frac{1}{3} : \frac{5}{7} : \frac{3}{5}[/tex]

To compare these ratios, find a common denominator:

  • The least common multiple of 3, 7, and 5 is 105.

Convert each fraction to have a denominator of 105:

  1. [tex]\frac{1}{3} = \frac{35}{105}[/tex]
  2. [tex]\frac{5}{7} = \frac{75}{105}[/tex]
  3. [tex]\frac{3}{5} = \frac{63}{105}[/tex]

The new ratios are:
[tex]35 : 75 : 63[/tex]

Let these be represented as [tex]35x, 75x,[/tex] and [tex]63x[/tex]. The problem states that the difference between the highest and the lowest number is 80:

[tex]75x - 35x = 80[/tex]

[tex]40x = 80[/tex]

Solve for [tex]x[/tex]:

[tex]x = 2[/tex]

Now, calculate the actual numbers:

  1. [tex]35x = 35 \times 2 = 70[/tex]
  2. [tex]75x = 75 \times 2 = 150[/tex]
  3. [tex]63x = 63 \times 2 = 126[/tex]

Therefore, the respective numbers are 70, 150, and 126. The correct answer choice is B. 70, 126, 150.

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