Understanding the Marks Problem

The question asks us to find Rohan's marks in English based on the relationships between his marks in three subjects: English, Science, and Mathematics. We are given three pieces of information:

• Rohan scored twice as many marks in English as in Science.
• His total marks in all three subjects (English, Science, Mathematics) are 126.
• The ratio of his marks in English and Mathematics is 2 : 3.

We need to use these clues to set up equations and solve for the marks in English.

Setting Up Equations from Marks Relationships

Let's represent Rohan's marks in each subject with a variable:

• Let \(E\) be the marks in English.
• Let \(S\) be the marks in Science.
• Let \(M\) be the marks in Mathematics.

Now, we can translate the given information into mathematical equations:

1. "Rohan scored twice as many marks in English as he did in Science":
   \(E = 2S\)
2. "His total marks in English, Science and Mathematics are 126":
   \(E + S + M = 126\)
3. "If the ratio of his marks in English and Mathematics is 2 : 3":
   \(\frac{E}{M} = \frac{2}{3}\)

We now have a system of three equations with three variables (\(E\), \(S\), and \(M\)). Our goal is to find the value of \(E\).

Solving for Rohan's English Marks

We can use substitution to solve this system. Let's express \(S\) and \(M\) in terms of \(E\) using equations (1) and (3), and then substitute these into equation (2).

From equation (1):

\(E = 2S\)

Divide both sides by 2 to find \(S\):

\(S = \frac{E}{2}\)

From equation (3):

\(\frac{E}{M} = \frac{2}{3}\)

Cross-multiply:

\(3E = 2M\)

Divide both sides by 2 to find \(M\):

\(M = \frac{3E}{2}\)

Now, substitute the expressions for \(S\) and \(M\) into equation (2):

\(E + S + M = 126\)

\(E + \left(\frac{E}{2}\right) + \left(\frac{3E}{2}\right) = 126\)

Combine the terms involving \(E\). The terms with a denominator of 2 can be added first:

\(E + \frac{E + 3E}{2} = 126\)

\(E + \frac{4E}{2} = 126\)

Simplify the fraction:

\(E + 2E = 126\)

Combine the \(E\) terms:

\(3E = 126\)

Divide both sides by 3 to find the value of \(E\):

\(E = \frac{126}{3}\)

\(E = 42\)

So, Rohan's marks in English are 42.

Verifying the Marks

Let's check if the calculated marks satisfy all the given conditions:

• English marks \(E = 42\).
• Science marks \(S = \frac{E}{2} = \frac{42}{2} = 21\).
• Mathematics marks \(M = \frac{3E}{2} = \frac{3 \times 42}{2} = \frac{126}{2} = 63\).

Now check the original conditions:

• Is \(E = 2S\)? \(42 = 2 \times 21\). Yes, \(42 = 42\).
• Is \(E + S + M = 126\)? \(42 + 21 + 63 = 63 + 63 = 126\). Yes, the total is 126.
• Is \(E : M = 2 : 3\)? \(42 : 63\). Divide both numbers by their greatest common divisor, which is 21. \(42 \div 21 = 2\), \(63 \div 21 = 3\). So, the ratio is 2 : 3. Yes, the ratio is correct.

All conditions are satisfied, confirming that Rohan's marks in English are 42.

Revision Table: Key Concepts for Marks Problems

Additional Information: Ratio and Proportion in Word Problems

Ratio problems like this are common in mathematics. A ratio is a comparison of two quantities. For example, a ratio of 2:3 means that for every 2 units of the first quantity, there are 3 units of the second quantity.

In this problem, the ratio of English to Mathematics marks is 2:3. This means that if Rohan's English marks were 20, his Mathematics marks would be 30 (because 20 is 10 times 2, so 3 times 10 is 30). If his English marks were 40, his Mathematics marks would be 60 (because 40 is 20 times 2, so 3 times 20 is 60). In our case, his English marks are 42. Since 42 is 21 times 2, his Mathematics marks must be 21 times 3, which is 63. This matches our calculation \(M = \frac{3}{2}E = \frac{3}{2} \times 42 = 3 \times 21 = 63\).

Understanding how to represent ratios as fractions and use them in equations is crucial for solving many word problems.