To add these fractions, first find the lowest common multiple of their denominators. The first few multiples of 2 are [2, 4, 6, 8, 10, 12, 14, 16, 18, 20] 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 [6, 12, 18, 24, 30] making 6 the smallest multiple 2 and 6 share.

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

\( \frac{4 x 3}{2 x 3} \) + \( \frac{3 x 1}{6 x 1} \)

\( \frac{12}{6} \) + \( \frac{3}{6} \)

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

\( \frac{12 + 3}{6} \) = \( \frac{15}{6} \) = 2\(\frac{1}{2}\)

The area of a circle is given by the formula A = πr² where r is the radius of the circle. The radius of a circle is its diameter divided by two so A = π(\( \frac{d}{2} \))². If the diameter of the logo increases by 30% the radius (and, consequently, the total area) increases by \( \frac{30\text{%}}{2} \) = 15%

To add these radicals together their radicands must be the same:

2\( \sqrt{50} \) + 3\( \sqrt{2} \)
2\( \sqrt{25 \times 2} \) + 3\( \sqrt{2} \)
2\( \sqrt{5^2 \times 2} \) + 3\( \sqrt{2} \)
(2)(5)\( \sqrt{2} \) + 3\( \sqrt{2} \)
10\( \sqrt{2} \) + 3\( \sqrt{2} \)

Now that the radicands are identical, you can add them together:

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

A factorial is the product of an integer and all the positive integers below it. To solve a fraction featuring factorials, expand the factorials and cancel out like numbers:

\( \frac{5!}{4!} \)
\( \frac{5 \times 4 \times 3 \times 2 \times 1}{4 \times 3 \times 2 \times 1} \)
\( \frac{5}{1} \)
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