If `a : b = 2(1)/(2) : 1(2)/(3)` and `b : c = 1(1)/(4): 3(1)/(2)`, then find a : b : c.

Options

A. `7 : 6 : 18`
B. `5 : 9 : 14`
C. `15 : 11 : 14`
D. `15 : 10 : 28`

Answer

To solve the problem, we need to find the ratio \( a : b : c \) given the ratios \( a : b = 2\frac{1}{2} : 1\frac{2}{3} \) and \( b : c = 1\frac{1}{4} : 3\frac{1}{2} \).

### Step 1: Convert mixed numbers to improper fractions

1. Convert \( 2\frac{1}{2} \) to an improper fraction:
\[ 2\frac{1}{2} = \frac{2 \times 2 + 1}{2} = \frac{5}{2} \]

2. Convert \( 1\frac{2}{3} \) to an improper fraction:
\[ 1\frac{2}{3} = \frac{1 \times 3 + 2}{3} = \frac{5}{3} \]

So, \( a : b = \frac{5}{2} : \frac{5}{3} \).

### Step 2: Simplify the ratio \( a : b \)

To simplify \( \frac{5}{2} : \frac{5}{3} \):
\[ a : b = \frac{5}{2} \div \frac{5}{3} = \frac{5}{2} \times \frac{3}{5} = \frac{3}{2} \]
Thus, \( a : b = 3 : 2 \).

### Step 3: Convert the second ratio \( b : c \)

1. Convert \( 1\frac{1}{4} \) to an improper fraction:
\[ 1\frac{1}{4} = \frac{1 \times 4 + 1}{4} = \frac{5}{4} \]

2. Convert \( 3\frac{1}{2} \) to an improper fraction:
\[ 3\frac{1}{2} = \frac{3 \times 2 + 1}{2} = \frac{7}{2} \]

So, \( b : c = \frac{5}{4} : \frac{7}{2} \).

### Step 4: Simplify the ratio \( b : c \)

To simplify \( \frac{5}{4} : \frac{7}{2} \):
\[ b : c = \frac{5}{4} \div \frac{7}{2} = \frac{5}{4} \times \frac{2}{7} = \frac{10}{28} = \frac{5}{14} \]
Thus, \( b : c = 5 : 14 \).

### Step 5: Combine the ratios \( a : b \) and \( b : c \)

Now we have:
- \( a : b = 3 : 2 \)
- \( b : c = 5 : 14 \)

To combine these ratios, we need to express them in terms of a common value for \( b \).

### Step 6: Find a common value for \( b \)

From \( a : b = 3 : 2 \), we can express \( b \) as \( 2k \) for some \( k \). Then:
\[ a = 3k \]

From \( b : c = 5 : 14 \), we can express \( b \) as \( 5m \) for some \( m \). Then:
\[ c = 14m \]

### Step 7: Set the values of \( b \) equal

Set \( 2k = 5m \):
\[ k = \frac{5m}{2} \]

### Step 8: Substitute \( k \) into the expressions for \( a \) and \( c \)

Substituting \( k \) into \( a \) and \( c \):
\[ a = 3k = 3 \left(\frac{5m}{2}\right) = \frac{15m}{2} \]
\[ c = 14m \]

### Step 9: Write the combined ratio \( a : b : c \)

Now we have:
\[ a : b : c = \frac{15m}{2} : 5m : 14m \]

To eliminate \( m \), we can multiply the entire ratio by 2:
\[ = 15 : 10 : 28 \]

### Final Answer

Thus, the ratio \( a : b : c \) is \( 15 : 10 : 28 \).

---