Calculating Marks Using Ratios

This problem involves finding the marks obtained by Rakesh and Ramesh given the ratio of marks obtained by three individuals and the actual marks of one individual, Rajesh.

Understanding the Given Ratio

The ratio of marks for Rajesh : Rakesh : Ramesh is given as 2 : 4 : 9.

This means that for every 2 parts of marks Rajesh scored, Rakesh scored 4 parts, and Ramesh scored 9 parts.

Finding the Value of One Ratio Part

We are given that Rajesh scored 30 marks. According to the ratio, Rajesh's marks correspond to 2 parts.

So, we can set up the relationship:

$$2 \text{ parts} = 30 \text{ marks}$$

To find the value of one part, we divide Rajesh's marks by his corresponding ratio part:

$$1 \text{ part} = \frac{30 \text{ marks}}{2} = 15 \text{ marks}$$

So, each part of the ratio represents 15 marks.

Calculating Rakesh's Marks

Rakesh's share in the ratio is 4 parts.

To find Rakesh's marks, we multiply his ratio parts by the value of one part:

$$\text{Rakesh's marks} = 4 \text{ parts} \times 15 \text{ marks/part}$$

$$\text{Rakesh's marks} = 60 \text{ marks}$$

Calculating Ramesh's Marks

Ramesh's share in the ratio is 9 parts.

To find Ramesh's marks, we multiply his ratio parts by the value of one part:

$$\text{Ramesh's marks} = 9 \text{ parts} \times 15 \text{ marks/part}$$

$$\text{Ramesh's marks} = 135 \text{ marks}$$

Summary of Marks Obtained

Based on the ratio and Rajesh's score, the marks obtained are:

• Rajesh: 30 marks (given)
• Rakesh: 60 marks (calculated)
• Ramesh: 135 marks (calculated)

Let's verify if the ratio of these marks is indeed 2 : 4 : 9:

$$30 : 60 : 135$$

Divide all numbers by the greatest common divisor, which is 15:

$$\frac{30}{15} : \frac{60}{15} : \frac{135}{15}$$

$$2 : 4 : 9$$

The ratio matches the given ratio, confirming our calculations are correct.

Therefore, Rakesh scored 60 marks and Ramesh scored 135 marks.

Revision Table: Key Concepts

Additional Information: Ratios and Proportion

Ratios are fundamental in mathematics for comparing quantities. When two ratios are equal, they form a proportion.

• Direct Proportion: If two quantities increase or decrease together at the same rate, they are in direct proportion. In this problem, the marks obtained are directly proportional to their respective parts in the ratio. If one person's marks increase, everyone else's marks (maintaining the same ratio) would also increase.
• Using Proportion to Solve: We could also solve this using proportion. Let Rakesh's marks be $M_{\text{Rakesh}}$ and Ramesh's marks be $M_{\text{Ramesh}}$. The ratio of Rajesh's marks to Rakesh's marks is 2:4. We know Rajesh scored 30 marks. So, we can write:

$$\frac{2}{4} = \frac{30}{M_{\text{Rakesh}}}$$

Cross-multiplying gives:

$$2 \times M_{\text{Rakesh}} = 4 \times 30$$

$$M_{\text{Rakesh}} = \frac{4 \times 30}{2} = \frac{120}{2} = 60 \text{ marks}$$

• Similarly, for Ramesh, the ratio of Rajesh's marks to Ramesh's marks is 2:9.

$$\frac{2}{9} = \frac{30}{M_{\text{Ramesh}}}$$

Cross-multiplying gives:

$$2 \times M_{\text{Ramesh}} = 9 \times 30$$

$$M_{\text{Ramesh}} = \frac{9 \times 30}{2} = \frac{270}{2} = 135 \text{ marks}$$

Both methods (finding the unit value or using proportion) yield the same results and are valid ways to solve ratio problems involving known values.