a. M = {n: 2n - 3 ≤ 37} Where n is a counting number. i). write down all the elements in M.
ii. If a number is selected at random from M what is the probability that it is a:
(α) multiple of 3;
(β) factor of 10.
b. A shop owner gave an end-of-year bonus to two of his attendees, Kontor and Gapson in the ratio of their ages. Gapson's age is one and a half times that of Kontor who is 20 years old. if Kontor received Le 200,000.00, find: i). Find the total amount shared.
ii. Find Gapson's share.
ai. To find all the elements in the set M, we have to solve the inequality.
2n - 3 ≤ 37
2n = 37 + 3 = 40.
2n ≤ 40
n ≤ \(\frac{40}{2}\)
n ≤ 20
so, the set M contains all positive numbers n such that n is less than or equal to 20
Therefore, set M elements are {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20}
(α)
Pr(multiple of 3) = \(\frac{multiples of 3}{ Total number of element in M}\)
But the multiples of 3 in set M are = {3, 6, 9, 12, 15, 18} = 6
ஃ Pr(multiples of 3) = \(\frac{6}{20} = \frac{3}{10}\)
(β)
Pr( factors of 10) = \(\frac{№ of factors of 10}{ Total № of set M elements}\)
factors of 10 in set M = { 1, 2, 5, and 10} = 4
ஃ Pr( factors of 10) = \(\frac{4}{20} = \frac{1}{5}\)
bi. Let the total amount shared = y
Gapson's age = \(\frac{3}{2}\times Kontor's age\). Since kontor is 20yrs old then Gapson's age = \(\frac{3}{2}\times20\) = 30years.
sharing ratio = 20 : 30 = 2 : 3
total ratio = 2 + 3 = 5
Since Kontor received Le200,000
\(\frac{2}{5} \times y\) = 200,000
\(\frac{2y}{5} = 200,000\)
cross multiplying, we have
2y = 5 x 200,000 = 1000,000
y = \(\frac{1000,000}{2}\) = 500,000.
ஃ Total amount shared = Le500,000.
ii. Gapson's share = Total - kontor's share
= 500,000 - 200,000 = Le300,000.
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