If a : b = 5 : 6 and b : c = 12 : 25 , find a : c

Answer

To solve the problem of finding the ratio \( a : c \) given \( a : b = 5 : 6 \) and \( b : c = 12 : 25 \), we can follow these steps:

### Step 1: Write the Ratios as Fractions
From the given ratios, we can express them as fractions:
- \( \frac{a}{b} = \frac{5}{6} \) (Equation 1)
- \( \frac{b}{c} = \frac{12}{25} \) (Equation 2)

### Step 2: Multiply the Two Ratios
To find \( \frac{a}{c} \), we can multiply the two fractions:
\[ \frac{a}{b} \times \frac{b}{c} = \frac{5}{6} \times \frac{12}{25} \]

### Step 3: Cancel Out \( b \)
When we multiply these fractions, the \( b \) in the numerator and denominator cancels out:
\[ \frac{a}{c} = \frac{5 \times 12}{6 \times 25} \]

### Step 4: Simplify the Right Side
Now, we simplify the right side:
- Calculate the numerator: \( 5 \times 12 = 60 \)
- Calculate the denominator: \( 6 \times 25 = 150 \)

Thus, we have:
\[ \frac{a}{c} = \frac{60}{150} \]

### Step 5: Reduce the Fraction
Now we can simplify \( \frac{60}{150} \):
- Both 60 and 150 can be divided by 30:
\[ \frac{60 \div 30}{150 \div 30} = \frac{2}{5} \]

### Step 6: Write the Final Ratio
Thus, the ratio \( a : c \) is:
\[ a : c = 2 : 5 \]

### Final Answer
The final answer is \( a : c = 2 : 5 \).

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