(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...

Answer Details

a) (i) To find Ms. Maureen's monthly income:

To find Ms. Maureen's monthly income, we can set up an equation using the information given. Let's call her monthly income "M".

Ms. Maureen spent 1/4 of her monthly income at the shopping mall, so: M * 1/4 = M/4

She spent 1/3 of her monthly income at the open market, so: M * 1/3 = M/3

The remaining amount after spending at the shopping mall and open market is M - M/4 - M/3 = M * (3/4) - M/3.

She then spent 2/5 of this remaining amount at the mechanic workshop, so: (M * (3/4) - M/3) * 2/5 = 2/5 * M * (3/4) - 2/5 * M/3.

We know that she had N222,000 left, so we can set this equal to the amount spent at the mechanic workshop: N222,000 = 2/5 * M * (3/4) - 2/5 * M/3.

Solving for M, we can find her monthly income:

M = N222,000 * 5/2 * 4/3 / (3/4 - 2/5) = N222,000 * 5/2 * 4/3 / (5/4 - 6/5) = N222,000 * 20/3 / (25/20 - 24/20) = N222,000 * 20/3 / (5/20) = N222,000 * 20/3 * 20/5 = N222,000 * 4 = N888,000.

So Ms. Maureen's monthly income is N888,000.

(ii) To find the amount spent at the open market:

To find the amount spent at the open market, we use the equation M * 1/3 = M/3 and substitute M = N888,000:

N888,000 * 1/3 = N888,000/3

N888,000/3 = N296,000

So the amount spent at the open market is N296,000.

b) The third term of an arithmetic progression (A.P.) is 4m - 2n, and the ninth term is 2m - 8n. To find the common difference:

To find the common difference, we can subtract the third term from the ninth term and divide by 6, since there are 6 terms between the third and ninth terms:

(2m - 8n) - (4m - 2n) = -2m + 6n

The common difference is -2m + 6n / 6 = -m + n.

So the common difference in terms of m and n is -m + n.