Express the ratios in the simplest form: 85: 561.

To simplify the ratio 85:561, we need to find the greatest common divisor (GCD) of the two numbers and then divide both by that GCD. This process will help us express the ratio in its simplest form.

Finding the GCD

The first step is to determine the GCD of 85 and 561. One effective method to find the GCD is to use the prime factorization of each number.

Prime Factorization

• 85: The prime factors of 85 are 5 and 17, since 85 = 5 × 17.
• 561: To factor 561, we can start by checking for divisibility by smaller prime numbers. It turns out that 561 is divisible by 3 (since 5 + 6 + 1 = 12, which is divisible by 3). Dividing gives us 561 ÷ 3 = 187. Next, we factor 187, which is 11 × 17. Therefore, the prime factorization of 561 is 3 × 11 × 17.

Identifying Common Factors

Now that we have the prime factorizations:

• 85 = 5 × 17
• 561 = 3 × 11 × 17

The common factor here is 17. Thus, the GCD of 85 and 561 is 17.

Simplifying the Ratio

Next, we divide both parts of the ratio by the GCD:

• 85 ÷ 17 = 5
• 561 ÷ 17 = 33

So, the ratio 85:561 simplifies to 5:33.

Final Result

In conclusion, the simplest form of the ratio 85:561 is 5:33. This means that for every 5 parts of one quantity, there are 33 parts of the other quantity, reflecting their proportional relationship in its most reduced form.