Reduce the ratio in simplest form
` 1/6 : (1)/(10) : (1)/(15)`

Answer

To reduce the ratio \( \frac{1}{6} : \frac{1}{10} : \frac{1}{15} \) to its simplest form, follow these steps:

### Step 1: Identify the Denominators
The denominators in the given ratios are 6, 10, and 15.

### Step 2: Find the LCM of the Denominators
To simplify the ratios, we need to find the Least Common Multiple (LCM) of the denominators 6, 10, and 15.

- The prime factorization of the numbers is:
- \( 6 = 2 \times 3 \)
- \( 10 = 2 \times 5 \)
- \( 15 = 3 \times 5 \)

- The LCM is found by taking the highest power of each prime factor:
- From 6: \( 2^1 \) and \( 3^1 \)
- From 10: \( 5^1 \)
- From 15: \( 3^1 \) and \( 5^1 \)

Thus, the LCM is \( 2^1 \times 3^1 \times 5^1 = 30 \).

### Step 3: Convert Each Fraction to Have the LCM as the Denominator
Now we will convert each fraction to have a denominator of 30.

- For \( \frac{1}{6} \):
\[ \frac{1}{6} = \frac{1 \times 5}{6 \times 5} = \frac{5}{30} \]

- For \( \frac{1}{10} \):
\[ \frac{1}{10} = \frac{1 \times 3}{10 \times 3} = \frac{3}{30} \]

- For \( \frac{1}{15} \):
\[ \frac{1}{15} = \frac{1 \times 2}{15 \times 2} = \frac{2}{30} \]

### Step 4: Write the New Ratio
Now that all fractions have the same denominator, we can write the new ratio:
\[ \frac{5}{30} : \frac{3}{30} : \frac{2}{30} \]
This simplifies to:
\[ 5 : 3 : 2 \]

### Final Answer
The simplest form of the ratio \( \frac{1}{6} : \frac{1}{10} : \frac{1}{15} \) is:
\[ 5 : 3 : 2 \]

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