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.

If the tiger has consumed 35 pounds of food in 7 days that's \( \frac{35}{7} \) = 5 pounds of food per day. The tiger needs to consume 70 - 35 = 35 more pounds of food to reach 70 pounds total. At 5 pounds of food per day that's \( \frac{35}{5} \) = 7 more days.

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.

guard shots made = shots taken x \( \frac{\text{% made}}{100} \) = 30 x \( \frac{40}{100} \) = \( \frac{40 x 30}{100} \) = \( \frac{1200}{100} \) = 12 shots

The center makes 30% of his shots so he'll have to take:

shots made = shots taken x \( \frac{\text{% made}}{100} \)
shots taken = \( \frac{\text{shots taken}}{\frac{\text{% made}}{100}} \)

to make as many shots as the guard. Plugging in values for the center gives us:

center shots taken = \( \frac{12}{\frac{30}{100}} \) = 12 x \( \frac{100}{30} \) = \( \frac{12 x 100}{30} \) = \( \frac{1200}{30} \) = 40 shots

to make the same number of shots as the guard and thus score the same number of points.

The equation for this sequence is:

a_n = a_n-1 + 4(n - 1)

where n is the term's order in the sequence, a_n is the value of the term, and a_n-1 is the value of the term before a_n. This makes the next number:

a₆ = a₅ + 4(6 - 1)
a₆ = 41 + 4(5)
a₆ = 61