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Let's break this problem down step by step to find how many intermediates are in the sample of 86 members.

### Step 1: Find the total ratio of Beginners (B), Intermediates (I), and Experts (E)
We are given:
- Ratio of Beginners to Intermediates is 9:2, so \( B : I = 9:2 \).
- Ratio of Intermediates to Experts is 3:5, so \( I : E = 3:5 \).

To combine these ratios into a single ratio for B, I, and E, we need to adjust them so that the number of Intermediates (I) is the same in both ratios.

In the first ratio, \( B : I = 9:2 \), so Intermediates are represented as 2 parts.
In the second ratio, \( I : E = 3:5 \), so Intermediates are represented as 3 parts.

We need to make the number of Intermediates in both ratios the same. To do this, we can find the least common multiple (LCM) of 2 and 3, which is 6. Now, adjust both ratios:

- Multiply \( B : I = 9:2 \) by 3 to get \( B : I = 27:6 \).
- Multiply \( I : E = 3:5 \) by 2 to get \( I : E = 6:10 \).

Now we can combine the two ratios:
\[ B : I : E = 27 : 6 : 10. \]

### Step 2: Calculate the total number of parts in the ratio
The total number of parts is:
\[
27 + 6 + 10 = 43.
\]

### Step 3: Proportion in the sample of 86 members
The ratio 27:6:10 means that out of every 43 members, 6 are Intermediates. In a sample of 86 members, the proportion remains the same. To find how many intermediates are in the sample, we set up the following proportion:
\[
\frac{6}{43} \times 86 = \frac{6 \times 86}{43} = 12.
\]

So, there are **12 Intermediates** in the sample of 86 members.

### Final Answer:
There are **12 Intermediates** in the sample.

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Do you want more details or have any questions?

Here are 5 related questions to expand on this:
1. How many beginners are there in the sample of 86 members?
2. How many experts are there in the sample?
3. If the total club had 430 members, how many intermediates would there be?
4. What is the ratio of beginners to experts in the club?
5. If the ratio of beginners to intermediates changed, how would that affect the total distribution?

**Tip:** When working with ratios, it’s often useful to scale them up to a common base before solving problems related to total or partial quantities.

The members of a martial arts club are all ranked as beginner, intermediate or expert. The ratio of beginners to intermediates is 9 : 2 and the ratio of intermediates to experts is 3 : 5. The head of the club takes a sample of 86 members of the club so that the proportions of beginners, intermediates and experts in the sample are the same as the proportions in the whole club. How many intermediates are in their sample?

Learn how to solve a ratio problem involving a martial arts club where the members are categorized into beginners, intermediates, and experts. The solution explains how to maintain proportional distribution in a sample of 86 members using ratios. Suitable for students in Grades 8-10.

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