To find the ratio $a : d$, we need to combine the given ratios step-by-step. The ratios are as follows:

1. $a : b = 3 : 4$
2. $b : c = 5 : 6$
3. $c : d = 12 : 17$

Step 1: Find $a : c$

To find the ratio $a : c$, we can multiply $a : b$ and $b : c$:

$a : b = 3 : 4 \quad \text{and} \quad b : c = 5 : 6$

To make the ratios compatible, we adjust them so that the values of $b$ are the same in both ratios.

Multiply the terms in $a : b$ by 5 and the terms in $b : c$ by 4:

$a : b = 3 \times 5 : 4 \times 5 = 15 : 20$ $b : c = 5 \times 4 : 6 \times 4 = 20 : 24$

Now, we have $a : b : c = 15 : 20 : 24$. Therefore, $a : c = 15 : 24$.

Step 2: Simplify $a : c$

The ratio $a : c = 15 : 24$ can be simplified by dividing both terms by 3:

$a : c = 5 : 8$

Step 3: Find $a : d$

Now that we have $a : c = 5 : 8$ and $c : d = 12 : 17$, we can find $a : d$ by making the values of $c$ in both ratios the same.

Multiply the terms in $a : c$ by 12 and the terms in $c : d$ by 8:

$a : c = 5 \times 12 : 8 \times 12 = 60 : 96$ $c : d = 12 \times 8 : 17 \times 8 = 96 : 136$

Now, we have $a : c : d = 60 : 96 : 136$. Therefore, $a : d = 60 : 136$.

Step 4: Simplify $a : d$

The ratio $a : d = 60 : 136$ can be simplified by dividing both terms by 4:

$a : d = 15 : 34$

Final Answer

The ratio $a : d$ is $\boxed{15 : 34}$.

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Would you like additional details, or do you have any questions? Here are a few related questions you may explore:

1. If $d = 34$, what is the value of $a$?
2. What is the value of $c$ if $a = 15$?
3. How would the solution change if $b : c$ were $7 : 9$ instead of $5 : 6$?
4. Can you express $a$, $b$, $c$, and $d$ in terms of a single variable?
5. How do you find a combined ratio for more than three ratios in general?

Tip: When finding a combined ratio, aligning common terms across ratios makes it easier to solve accurately.