If a single cook can bake 5 large cakes per hour and the kitchen is available for 3 hours, a single cook can bake 5 x 3 = 15 large cakes during that time. 22 large cakes are needed for the party so \( \frac{22}{15} \) = 1\(\frac{7}{15}\) cooks are needed to bake the required number of large cakes.

If a single cook can bake 12 small cakes per hour and the kitchen is available for 3 hours, a single cook can bake 12 x 3 = 36 small cakes during that time. 340 small cakes are needed for the party so \( \frac{340}{36} \) = 9\(\frac{4}{9}\) cooks are needed to bake the required number of small cakes.

Because you can't employ a fractional cook, round the number of cooks needed for each type of cake up to the next whole number resulting in 2 + 10 = 12 cooks.

To subtract these fractions, first find the lowest common multiple of their denominators. The first few multiples of 6 are [6, 12, 18, 24, 30, 36, 42, 48, 54, 60] and the first few multiples of 10 are [10, 20, 30, 40, 50, 60, 70, 80, 90]. The first few multiples they share are [30, 60, 90] making 30 the smallest multiple 6 and 10 share.

Next, convert the fractions so each denominator equals the lowest common multiple:

\( \frac{8 x 5}{6 x 5} \) - \( \frac{2 x 3}{10 x 3} \)

\( \frac{40}{30} \) - \( \frac{6}{30} \)

Now, because the fractions share a common denominator, you can subtract them:

\( \frac{40 - 6}{30} \) = \( \frac{34}{30} \) = 1\(\frac{2}{15}\)

The least common multiple (LCM) is the smallest positive integer that is a multiple of two or more integers.

You have 31 out of the total of 450 raffle tickets sold so you have a (\( \frac{31}{450} \)) x 100 = \( \frac{31 \times 100}{450} \) = \( \frac{3100}{450} \) = 7% chance to win the raffle.

To multiply fractions, multiply the numerators together and then multiply the denominators together:

\( \frac{4}{9} \) x \( \frac{1}{8} \) = \( \frac{4 x 1}{9 x 8} \) = \( \frac{4}{72} \) = \(\frac{1}{18}\)