Dinesh Ishwar Ganesh and Harish had a total of 120 Marbles the Marbles with Dinesh was half the total number of Marbles with the other the Marbles with Ishwar  has 1/3rd of total number of Marbles with other the number of Marbles with Ganesh was 1/4th total number of Marbles with the other find the number of Marbles with Harish

We are given that Dinesh, Ishwar, Ganesh, and Harish together have a total of 120 marbles, and each person's number of marbles is defined in relation to the others. Let's define the variables and set up the problem step by step.

### Variables:
- Let \( D \) represent the number of marbles Dinesh has.
- Let \( I \) represent the number of marbles Ishwar has.
- Let \( G \) represent the number of marbles Ganesh has.
- Let \( H \) represent the number of marbles Harish has.

### Equations:
From the problem, we have the following relationships:

1. **Total marbles:**  
   \( D + I + G + H = 120 \)

2. **Dinesh's marbles:**  
   Dinesh has half the total number of marbles the others have. So:
   \[
   D = \frac{1}{2}(I + G + H)
   \]

3. **Ishwar's marbles:**  
   Ishwar has one-third of the total number of marbles the others have. So:
   \[
   I = \frac{1}{3}(D + G + H)
   \]

4. **Ganesh's marbles:**  
   Ganesh has one-fourth of the total number of marbles the others have. So:
   \[
   G = \frac{1}{4}(D + I + H)
   \]

### Solving the system of equations:
We will solve these equations step by step to find the number of marbles each person has, especially Harish.

Let me compute this.The solution to the problem is as follows:

- Dinesh has 40 marbles.
- Ishwar has 30 marbles.
- Ganesh has 24 marbles.
- **Harish has 26 marbles.**

Would you like more details on the steps used to solve this system, or do you have any other questions?

Here are some follow-up questions:

1. How can we verify the consistency of these relationships?
2. How would the solution change if Dinesh had one-third the marbles of the others?
3. Can you set up a similar problem with five people and different marble ratios?
4. What if the total number of marbles were increased to 150 but the ratios stayed the same?
5. How does changing one person's ratio affect the others?

**Tip:** When solving problems involving ratios, translating them into equations helps break down complex relationships.

Dinesh Ishwar Ganesh and Harish had a total of 120 Marbles the Marbles with Dinesh was half the total number of Marbles with the other the Marbles with Ishwar has 1/3rd of total number of Marbles with other the number of Marbles with Ganesh was 1/4th total number of Marbles with the other find the number of Marbles with Harish

Learn how to solve a system of linear equations to find the number of marbles each person has when given ratios among them. This problem covers key algebraic concepts, such as setting up equations based on given relationships and solving them step-by-step. Suitable for students in Grades 8-10.

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