Sum of items = Mean x Number of items.

Mean = 17.6

Number of girls in the class = t

Sum of the ages of girls = 17.6 x t = 17.6t

When the girls where dismissed, the sum of the ages reduced to 17.6t - (16 + 19 + 20 + 17) and the number of girls reduced to t - 4

Note: 4 girls were dismissed.

New sum of ages = 17.6t - (16 + 19 + 20 + 17)

New sum of ages = 17.6t - 72

New number of girls = t - 4

New mean = $\frac{\mathrm{17.6t\; -\; 72}}{\mathrm{t\; -\; 4}}$

The new mean age of girls in the class became 0.2 less than the original mean.

Original mean = 17.6

New mean = 17.6 - 0.2 = 17.4

$\frac{\mathrm{17.6t\; -\; 72}}{\mathrm{t\; -\; 4}}$ = 17.4

Get rid of the fraction by multiplying both sides by the denominator t - 4

$\frac{\mathrm{17.6t\; -\; 72}}{\mathrm{t\; -\; 4}}$ x (t - 4) = 17.4 x (t - 4)

17.6t - 72 = 17.4t - 69.6

17.6t - 17.4t = 72 - 69.6

0.2t = 2.4

Divide both sides by 0.2

t = $\frac{2.4}{0.2}$ = 12

Let m = tens digit and n = unit digit

Thus the number being represented as mn which is the same as 10 x m + n which equals 10m + n

For example 21 = 2 x 10 + 1, 52 = 5 x 10 + 2, etc.

The unit digit is 1 less than the tens digit means when you add 1 to the unit digit, you should get the tens digit.

m = n + 1 ----------- eqn 1

The number is increased by 8 means 8 is added to the number to become (10m + n + 8)

Divided by the sum of the digits (m + n) gives 8

$\frac{\mathrm{10m\; +\; n\; +\; 8}}{\mathrm{m\; +\; n}}$ = 8

Get rid of the fraction by multiplying both sides by the denominator (m + n).

10m + n + 8 = 8(m + n)

10m + n + 8 = 8m + 8n

10m - 8m + 8 = 8n - n

2m + 8 = 7n --------- eqn 2

Substitute m = n + 1 into ------- eqn 2

2(n + 1) + 8 = 7n

2n + 2 + 8 = 7n

10 = 7n - 2n

10 = 5n

Divide both sides by 5

n = $\frac{10}{5}$ = 2

Substitute n = 2 into ------ eqn 1

m = 2 + 1 = 3

The number = 10 x m + n

m = 3 and n = 2

The number = 10 x 3 + 2 = 32

∴ the number is 32

Let y = the son's age.

The man is five times as old as his son means when you multiply the son's age by 5, you will get the man's age.

Man's age = 5 x y = 5y

In three years time, the man will be (5y + 3) years and the son will be (y + 3) years.

In three years time, the product of their ages will be 380 means when you multiply their ages in three years time, you will get 380

Son's age in three years time x man's ange in three years time = 380

(y + 3) x (5y + 3) = 380

Expan the bracket and solve for y

y(5y + 3) + 3(5y + 3) = 380

5y² + 3y + 15y + 9 = 380

5y² + 18y + 9 - 380 = 0

5y² + 18y - 371 = 0

Use the almighty formula to calculate for y

a = 5

b = 18

c = -371

y = $\frac{\mathrm{-18\; +}\sqrt{{18}^2-\mathrm{4\; x\; 5\; x\; (-371)}}}{\mathrm{2\; x\; 5}}$ or $\frac{\mathrm{-18\; -}\sqrt{{18}^2-\mathrm{4\; x\; 5\; x\; (-371)}}}{\mathrm{2\; x\; 5}}$

y = $\frac{\mathrm{-18\; +}\sqrt{324+7420}}{10}$ or $\frac{\mathrm{-18\; -}\sqrt{324+7420}}{10}$

y = $\frac{\mathrm{-18\; +}\sqrt{7744}}{10}$ or $\frac{\mathrm{-18\; -}\sqrt{7744}}{10}$

y = $\frac{\mathrm{-18\; +\; 88}}{10}$ or $\frac{\mathrm{-18\; -\; 88}}{10}$

y = $\frac{70}{10}$ or $\frac{-106}{10}$

y = 7 or -$\frac{53}{5}$

But age is positive hence y = 7

The son is 7 years old.

The man's age = 5 x son's age = 5 x 7 years = 35 years.

Let number of plates purchased = y.

Cost per plate = $\frac{150}{\mathrm{y}}$

Selling price = Cost Price + Profit

Cost price of the plates purchased = GH₵ 150

Profit made = GH₵ 6

Amount sold = GH₵ 150 + GH₵ 6

Amount sold = GH₵ 156

Remaining plates = y - 4

Selling price of each remaining plates = $\frac{\mathrm{Amount\; sold}}{\mathrm{Remaining\; plates}}$

Selling price of each remaining plates = $\frac{156}{\mathrm{y\; -\; 4}}$

Selling price of each - Cost price of each = Profit of each

Profit on each remaining plates = GH₵ 1

$\frac{156}{\mathrm{y\; -\; 4}}$ - $\frac{150}{\mathrm{y}}$ = 1

Multiply both sides by the L.C.M of y and (y - 4) which is y x (y - 4)

y x 156 - (y - 4) x 150 = 1 x y(y - 4)

156y - 150y + 600 = y² - 4y

6y + 600 = y² - 4y

y² - 4y - 6y - 600 = 0

y² - 10y - 600 = 0

Split the -10 such that two numbers when you multiply you will get -600 and when you add you will get -10. The numbers are 20 and -30

y² - 30y + 20y - 600 = 0

y(y - 30) + 20(y - 30)

(y - 30)(y + 20) = 0

When y - 30 = 0, y = 30

When y + 20 = 0, y = -20

Since the number of plates is positive, y = 30. Hence number of plates bought is 30