(a) Ms. Maureen spent \(\frac{1}{4}\) of her monthly income at a shopping mall, \(\frac{1}{3}\) at an open market and \(\frac{2}{5}\) of the remaining amoun...
Question 1 Report
(a) Ms. Maureen spent \(\frac{1}{4}\) of her monthly income at a shopping mall, \(\frac{1}{3}\) at an open market and \(\frac{2}{5}\) of the remaining amount at a Mechanic workshop. If she had N222,000.00 left, find:
(i) her monthly income.
(ii) the amount spent at the open market.
(b) The third term of an Arithmetic Progression (A. P.) is 4m - 2n. If the ninth term of the progression is 2m - 8n. find the common difference in terms of m and n.
Answer Details
(a)(i) Monthly income. At the mall and market she spent \(\tfrac{1}{4} + \tfrac{1}{3} = \tfrac{7}{12}\) of her income, leaving \(\tfrac{5}{12}\). Of this remainder she spent \(\tfrac{2}{5}\) at the mechanic, so she kept \(\tfrac{3}{5}\) of the remainder:
\[\tfrac{3}{5}\times\tfrac{5}{12} = \tfrac{1}{4}\text{ of the income} = \text{N}222{,}000.\]
\[\text{Income} = 222{,}000\times4 = \text{N}888{,}000.\]
(a)(ii) Amount at the open market.
\[\tfrac{1}{3}\times888{,}000 = \text{N}296{,}000.\]
(b) Common difference. Let the first term be a and common difference d.
\[T_3 = a + 2d = 4m - 2n,\qquad T_9 = a + 8d = 2m - 8n.\]
Subtracting the first from the second:
\[6d = (2m - 8n) - (4m - 2n) = -2m - 6n \Rightarrow d = \frac{-2m - 6n}{6} = -\frac{m + 3n}{3}.\]
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