Let’s work through this problem step by step.

Given Information

• $A$ and $B$ are given pens in the ratio $6 : 7$.
• $B$ and $C$ are given pens in the ratio $14 : 15$.

Part (a): Find the ratio of pens given to $A$ and $C$.

To find the ratio of $A : B : C$, we need to make the ratios consistent across all three participants.

1. Matching $B$'s values in both ratios:

   • In the ratio $A : B = 6 : 7$, $B$ is represented by $7$.
   • In the ratio $B : C = 14 : 15$, $B$ is represented by $14$.

2. Scaling $A : B$ to match $B : C$:

   • To make $B$ consistent in both ratios, we scale the first ratio $A : B = 6 : 7$ by $2$, giving: $A : B = 12 : 14$

3. Combining to find $A : B : C$:

   • Now we have $A : B : C = 12 : 14 : 15$.

4. Answer for Part (a):

   • The ratio of pens given to $A$ and $C$ is $12 : 15$.
   • Simplifying this, $12 : 15 = 4 : 5$.

Part (b): If the total number of pens given to $A$ and $C$ is 36, find the ratio of the number of pens $A$, $B$, and $C$ get.

1. Finding the value of one unit in the ratio:

   • From the ratio $A : B : C = 12 : 14 : 15$, the total units for $A$ and $C$ is $12 + 15 = 27$.
   • We are given that the sum of $A$ and $C$'s pens is $36$.
   • So, each unit is worth: $\frac{36}{27} = 1.33$

2. Calculating the actual number of pens for each:

   • $A = 12 \times 1.33 = 16$ pens
   • $B = 14 \times 1.33 = 18.67 \approx 19$ pens (if rounded to whole pens)
   • $C = 15 \times 1.33 = 20$ pens

3. Answer for Part (b):

   • The ratio of the number of pens $A$, $B$, and $C$ get is approximately $16 : 19 : 20$.

Would you like further details on any part of the solution, or do you have any questions?

Here are five related questions you might find useful:

1. What if the total number of pens given to $B$ and $C$ was a different value?
2. How would the ratio change if the initial ratios between $A$, $B$, and $C$ were different?
3. How would you solve if each person was given a specific total instead of a ratio?
4. What is the impact of rounding on the final distribution in real-life applications?
5. How can this type of ratio problem be used in budget allocation?

Tip: When dealing with multiple ratios, try to equalize the common terms first to simplify the problem.