714:34

To simplify the ratio \( 714:34 \), divide both numbers by their greatest common divisor (GCD).

First, find the GCD of 714 and 34:

• The prime factorization of 34 is \( 2 \times 17 \).
• To find the GCD, we can also check for divisibility.

714 can be divided by 2 (as it is even), which gives us:

\[ 714 \div 2 = 357 \]

Next, check the divisibility of 34:

• \( 34 \div 2 = 17 \)

Next, check if \( 357 \) (which is \( 714 \div 2 \)) has common factors with \( 17 \). Since \( 357 \) is divisible by \( 3 \) (as the sum of its digits \( 3 + 5 + 7 = 15 \) is divisible by \( 3 \)):

\[ 357 \div 3 = 119 \]

Now, \( 119 \) is \( 7 \times 17 \).

Thus, the GCD of \( 714 \) and \( 34 \) is \( 17 \).

Now divide both numbers by their GCD:

\[ 714 \div 17 = 42 \] \[ 34 \div 17 = 2 \]

The simplified ratio is:

\[ 42:2 \]

This can be further simplified by dividing both numbers by \( 2 \):

\[ 21:1 \]

So, the ratio \( 714:34 \) simplifies to \( 21:1 \).