If a:b = 2:3 and b:c = 4:5 and c:d = 3:6 .then a:b:c:d is

We are given the following ratios:
- \( a : b = 2 : 3 \)
- \( b : c = 4 : 5 \)
- \( c : d = 3 : 6 \)

We want to express \( a : b : c : d \) in a simplified form. To do this, we need to adjust these ratios so that the common terms (like \( b \) and \( c \)) match across the different ratios.

### Step 1: Combine the ratios for \( a : b : c \)

From \( a : b = 2 : 3 \) and \( b : c = 4 : 5 \), we need to make the value of \( b \) consistent. To do this, find the least common multiple (LCM) of the values for \( b \) in both ratios. The values for \( b \) are 3 and 4, and their LCM is 12.

- Multiply \( a : b = 2 : 3 \) by 4 to get \( a : b = 8 : 12 \).
- Multiply \( b : c = 4 : 5 \) by 3 to get \( b : c = 12 : 15 \).

Now, we have:
\[ a : b : c = 8 : 12 : 15 \]

### Step 2: Incorporate the ratio \( c : d \)

From \( c : d = 3 : 6 \), simplify this to \( c : d = 1 : 2 \). We need to adjust this ratio so that \( c \) matches the value from \( a : b : c = 8 : 12 : 15 \), where \( c = 15 \).

- Multiply \( c : d = 1 : 2 \) by 15 to get \( c : d = 15 : 30 \).

### Step 3: Final ratio for \( a : b : c : d \)

Now, combine all the ratios:
\[ a : b : c : d = 8 : 12 : 15 : 30 \]

Thus, the ratio \( a : b : c : d \) is \( 8 : 12 : 15 : 30 \).

Would you like more details on any part of the solution?

Here are 5 related questions:
1. How do you find the LCM of two numbers in general?
2. Can ratios always be simplified in the same way?
3. How can this method be applied to three or more terms in a ratio?
4. What is the relationship between ratios and fractions?
5. How do we know when to simplify ratios?

**Tip:** Always ensure that the common terms between ratios match before combining them. This helps in getting the correct proportional relationship.

Discover how to solve the problem of finding the ratio a:b:c:d when given a:b = 2:3, b:c = 4:5, and c:d = 3:6. This solution includes key concepts like proportionality, LCM, and ratio simplification. Suitable for students in Grades 6-8.

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