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 75% the radius (and, consequently, the total area) increases by \( \frac{75\text{%}}{2} \) = 37\(\frac{1}{2}\)%

Christine scored 94% on the test meaning she earned 94% of the possible points on the test. There were 140 possible points on the test so she earned 140 x 0.94 = 132 points. Each question is worth 2 points so she got \( \frac{132}{2} \) = 66 questions right.

The distributive property for multiplication helps in solving expressions like a(b + c). It specifies that the result of multiplying one number by the sum or difference of two numbers can be obtained by multiplying each number individually and then totaling the results: a(b + c) = ab + ac. For example, 4(10-5) = (4 x 10) - (4 x 5) = 40 - 20 = 20.

A rational number (or fraction) is represented as a ratio between two integers, a and b, and has the form \({a \over b}\) where a is the numerator and b is the denominator. An improper fraction (\({5 \over 3} \)) has a numerator with a greater absolute value than the denominator and can be converted into a mixed number (\(1 {2 \over 3} \)) which has a whole number part and a fractional part.

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

8\( \sqrt{12} \) + 3\( \sqrt{3} \)
8\( \sqrt{4 \times 3} \) + 3\( \sqrt{3} \)
8\( \sqrt{2^2 \times 3} \) + 3\( \sqrt{3} \)
(8)(2)\( \sqrt{3} \) + 3\( \sqrt{3} \)
16\( \sqrt{3} \) + 3\( \sqrt{3} \)

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