A bag of rubies and diamonds contains 10 jewels. Ruby's are worth $300 and diamonds are worth 2/3 as much as rubies. How many rubies are in the bag if the whole bag has a value of $2300?

Let's solve this step by step.

### Step 1: Define Variables
Let:
- $ r $ be the number of rubies in the bag.
- $ d $ be the number of diamonds in the bag.

### Step 2: Set Up Equations
We know that:
1. The total number of jewels is 10:
   \[
   r + d = 10
   \]

2. The value of each ruby is $300, and the value of each diamond is $ \frac{2}{3} $ of $300, which is $200.

3. The total value of the jewels is $2300, so:
   \[
   300r + 200d = 2300
   \]

### Step 3: Solve the System of Equations

From the first equation, solve for $ d $:
\[
d = 10 - r
\]

Substitute $ d = 10 - r $ into the second equation:
\[
300r + 200(10 - r) = 2300
\]

Distribute and simplify:
\[
300r + 2000 - 200r = 2300
\]
\[
100r + 2000 = 2300
\]

Subtract 2000 from both sides:
\[
100r = 300
\]

Divide by 100:
\[
r = 3
\]

### Step 4: Conclusion
So, there are 3 rubies in the bag.

Would you like more details or have any other questions?

Here are some related questions:

1. How many diamonds are in the bag if there are 3 rubies?
2. What is the total value of the diamonds in the bag?
3. If the number of rubies were doubled, what would be the new total value of the bag?
4. How much would the bag be worth if all the jewels were rubies?
5. What if the value of each ruby was $350, how many rubies would be in the bag for a total value of $2300?
6. If the bag had 5 rubies, how many diamonds would it have, and what would be the total value?
7. How would the total value change if there was one less diamond?
8. If diamonds were worth $250 each, how many rubies would be needed for the bag to be worth $2300?

**Tip:** When dealing with systems of equations, always check that your solution satisfies both original equations.

Learn how to solve the problem involving a bag of rubies and diamonds worth $2300. Understand the concepts of systems of equations and value calculation. Suitable for students in Grades 9-12.

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