The first few multiples of 3 are [3, 6, 9, 12, 15, 18, 21, 24, 27, 30] and the first few multiples of 7 are [7, 14, 21, 28, 35, 42, 49, 56, 63, 70]. The first few multiples they share are [21, 42, 63, 84] making 21 the smallest multiple 3 and 7 have in common.

Average speed in miles per hour is the number of miles traveled divided by the number of hours:

Solving for distance:

distance = \( \text{speed} \times \text{time} \)
distance = \( 35mph \times 3h \)
105 miles

To simplify this fraction, first find the greatest common factor between them. The factors of 24 are [1, 2, 3, 4, 6, 8, 12, 24] and the factors of 64 are [1, 2, 4, 8, 16, 32, 64]. They share 4 factors [1, 2, 4, 8] making 8 their greatest common factor (GCF).

Next, divide both numerator and denominator by the GCF:

\( \frac{24}{64} \) = \( \frac{\frac{24}{8}}{\frac{64}{8}} \) = \( \frac{3}{8} \)

To add 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 6 are [6, 12, 18, 24, 30, 36, 42, 48, 54, 60]. The first few multiples they share are [12, 24, 36, 48, 60] making 12 the smallest multiple 4 and 6 share.

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

\( \frac{8 x 3}{4 x 3} \) + \( \frac{9 x 2}{6 x 2} \)

\( \frac{24}{12} \) + \( \frac{18}{12} \)

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

\( \frac{24 + 18}{12} \) = \( \frac{42}{12} \) = 3\(\frac{1}{2}\)

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

\( \frac{9}{100} \) x 9 = \( \frac{9 \times 9}{100} \) = \( \frac{81}{100} \) = 0.81 errors per hour

So, in an average hour, the machine will produce 9 - 0.81 = 8.19 error free parts.

The machine ran for 24 - 9 = 15 hours yesterday so you would expect that 15 x 8.19 = 122.9 error free parts were produced yesterday.