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.

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\(\frac{1}{2}\)%

To divide terms with exponents, the base of both exponents must be the same. In this case they are so divide the coefficients and subtract the exponents:

\( \frac{y^5}{7y^2} \)
\( \frac{1}{7} \) y^{(5 - 2)}
\(\frac{1}{7}\)y³

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 8 are [8, 16, 24, 32, 40, 48, 56, 64, 72, 80]. The first few multiples they share are [8, 16, 24, 32, 40] making 8 the smallest multiple 4 and 8 share.

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

\( \frac{2 x 2}{4 x 2} \) + \( \frac{8 x 1}{8 x 1} \)

\( \frac{4}{8} \) + \( \frac{8}{8} \)

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

\( \frac{4 + 8}{8} \) = \( \frac{12}{8} \) = 1\(\frac{1}{2}\)

Christine scored 88% on the test meaning she earned 88% of the possible points on the test. There were 180 possible points on the test so she earned 180 x 0.88 = 159 points. Each question is worth 3 points so she got \( \frac{159}{3} \) = 53 questions right.