If `A :B = 7 : 5` and `B : C = 9 : 14` find `A : C`

Answer

To find the ratio \( A : C \) given that \( A : B = 7 : 5 \) and \( B : C = 9 : 14 \), we can follow these steps:

### Step 1: Write the given ratios in fractional form
From the problem, we have:
- \( A : B = 7 : 5 \) can be written as \( \frac{A}{B} = \frac{7}{5} \)
- \( B : C = 9 : 14 \) can be written as \( \frac{B}{C} = \frac{9}{14} \)

### Step 2: Express \( A \) in terms of \( B \)
From the ratio \( \frac{A}{B} = \frac{7}{5} \), we can express \( A \) as:
\[ A = \frac{7}{5} B \]

### Step 3: Express \( C \) in terms of \( B \)
From the ratio \( \frac{B}{C} = \frac{9}{14} \), we can express \( C \) as:
\[ C = \frac{14}{9} B \]

### Step 4: Substitute \( B \) in the expression for \( A \) and \( C \)
Now, we can substitute the expressions for \( A \) and \( C \) in terms of \( B \):
- We have \( A = \frac{7}{5} B \)
- We have \( C = \frac{14}{9} B \)

### Step 5: Find the ratio \( A : C \)
Now, we can find the ratio \( A : C \):
\[ A : C = \frac{A}{C} = \frac{\frac{7}{5} B}{\frac{14}{9} B} \]

### Step 6: Simplify the ratio
The \( B \) in the numerator and denominator cancels out:
\[ \frac{A}{C} = \frac{7/5}{14/9} = \frac{7}{5} \times \frac{9}{14} \]

### Step 7: Perform the multiplication
Now, multiply the fractions:
\[ \frac{7 \times 9}{5 \times 14} = \frac{63}{70} \]

### Step 8: Simplify the fraction
Now, we can simplify \( \frac{63}{70} \):
Both 63 and 70 can be divided by 7:
\[ \frac{63 \div 7}{70 \div 7} = \frac{9}{10} \]

### Final Result
Thus, the ratio \( A : C \) is:
\[ A : C = 9 : 10 \]

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