If a:b = 3:4 , b:c= 5:6 and c:d = 12:17 . What is a:d

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 \( \boxed{15 : 34} \).

---

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.

If a:b = 3:4, b:c= 5:6, and c:d = 12:17, what is a:d?

Explore a detailed solution to find the ratio a:d when given a:b = 3:4, b:c = 5:6, and c:d = 12:17. Learn the process of combining and simplifying ratios, suitable for Grades 7-9 math students.

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