Reduce the following ratio to its lowest terms and to an equivalent ratio having 1 as the smallest term.
69 : 207 : 161

To reduce the ratio \( 69 : 207 : 161 \) to its lowest terms and to an equivalent ratio having 1 as the smallest term, follow these steps:

### Step 1: Find the GCD (Greatest Common Divisor)
We first find the greatest common divisor (GCD) of the numbers 69, 207, and 161.

- The factors of 69 are: \( 1, 3, 23, 69 \)
- The factors of 207 are: \( 1, 3, 69, 207 \)
- The factors of 161 are: \( 1, 7, 23, 161 \)

The common factor between these numbers is \( 23 \).

### Step 2: Divide each number by their GCD
Now divide each number in the ratio by the GCD (23):

\[
\frac{69}{23} : \frac{207}{23} : \frac{161}{23} = 3 : 9 : 7
\]

### Step 3: Find the ratio with 1 as the smallest term
To convert this into an equivalent ratio with 1 as the smallest term, we divide all terms by the smallest value (which is 3 in this case):

\[
\frac{3}{3} : \frac{9}{3} : \frac{7}{3} = 1 : 3 : \frac{7}{3}
\]

Since \( \frac{7}{3} \) is a fraction, this is the simplest ratio where 1 is the smallest term.

### Final Result:
- The ratio in its lowest terms is \( 3 : 9 : 7 \).
- The equivalent ratio with 1 as the smallest term is \( 1 : 3 : \frac{7}{3} \).

Would you like more details or have any questions about this process?

Here are 5 related questions to explore:

1. What if the original ratio involved negative numbers?
2. How do we find the GCD of larger numbers?
3. What are some applications of ratios in real life?
4. How can we simplify complex ratios involving fractions?
5. What if the terms of the ratio were not integers?

**Tip:** When simplifying ratios, always look for the greatest common divisor (GCD) to ensure the lowest terms.

Reduce the following ratio to its lowest terms and to an equivalent ratio having 1 as the smallest term: 69 : 207 : 161

Discover how to reduce the ratio 69 : 207 : 161 to its lowest terms and find an equivalent ratio with 1 as the smallest term. This problem involves finding the greatest common divisor (GCD) and applying ratio simplification methods. Ideal for students in Grades 6-8.

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