A ratio of 4:1 means that there are 4 home fans for every one visiting fan. So, of every 5 fans, 4 are home fans and \( \frac{4}{5} \) of every fan in the stadium is a home fan:

38,000 fans x \( \frac{4}{5} \) = \( \frac{152000}{5} \) = 30,400 fans.

To convert a negative exponent to a positive exponent, calculate the positive exponent then take the reciprocal.

To subtract these fractions, first find the lowest common multiple of their denominators. The first few multiples of 4 are [4, 8, 12, 16, 20, 24, 28, 32, 36, 40] and the first few multiples of 10 are [10, 20, 30, 40, 50, 60, 70, 80, 90]. The first few multiples they share are [20, 40, 60, 80] making 20 the smallest multiple 4 and 10 share.

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

\( \frac{7 x 5}{4 x 5} \) - \( \frac{3 x 2}{10 x 2} \)

\( \frac{35}{20} \) - \( \frac{6}{20} \)

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

\( \frac{35 - 6}{20} \) = \( \frac{29}{20} \) = 1\(\frac{9}{20}\)

The hourly error rate for this machine is the error rate in parts per 100 multiplied by the number of parts produced per hour:

\( \frac{5}{100} \) x 5 = \( \frac{5 \times 5}{100} \) = \( \frac{25}{100} \) = 0.25 errors per hour

So, in an average hour, the machine will produce 5 - 0.25 = 4.75 error free parts.

The machine ran for 24 - 9 = 15 hours yesterday so you would expect that 15 x 4.75 = 71.3 error free parts were produced yesterday.

The factors of 40 are [1, 2, 4, 5, 8, 10, 20, 40] and the factors of 60 are [1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60]. They share 6 factors [1, 2, 4, 5, 10, 20] making 20 the greatest factor 40 and 60 have in common.