To add these fractions, first find the lowest common multiple of their denominators. The first few multiples of 5 are [5, 10, 15, 20, 25, 30, 35, 40, 45, 50] 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 [35, 70] making 35 the smallest multiple 5 and 7 share.

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

\( \frac{2 x 7}{5 x 7} \) + \( \frac{4 x 5}{7 x 5} \)

\( \frac{14}{35} \) + \( \frac{20}{35} \)

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

\( \frac{14 + 20}{35} \) = \( \frac{34}{35} \) = \(\frac{34}{35}\)

The factors of 52 are [1, 2, 4, 13, 26, 52] and the factors of 40 are [1, 2, 4, 5, 8, 10, 20, 40]. They share 3 factors [1, 2, 4] making 4 the greatest factor 52 and 40 have in common.

The number of students taking German or Spanish is 13 + 11 = 24. Of that group of 24, 5 are taking both languages so they've been counted twice (once in the German group and once in the Spanish group). Subtracting them out leaves 24 - 5 = 19 who are taking at least one language. 29 - 19 = 10 students who are not taking either language.

A factor is a positive integer that divides evenly into a given number. For example, the factors of 8 are 1, 2, 4, and 8.

By buying two shirts, Charlie will save $12 x \( \frac{15}{100} \) = \( \frac{$12 x 15}{100} \) = \( \frac{$180}{100} \) = $1.80 on the second shirt.

So, his total cost will be
$12.00 + ($12.00 - $1.80)
$12.00 + $10.20
$22.20