Question

3 (a) solve the following equations, (i) $\frac{3}{5}(f - 1)=\frac{1}{5}f$ 2 (ii) $(2\frac{2}{3}-p)^2 = 1\frac{7}{9}$ 3 (b) factorise completely, (i) $10m^2 - tr - 2mt+5mr$ 2 (ii) $p^3 - 16p$ 2 (c) a carpenters wage depends on the number of hours she works. she is paid at a rate of $3.65 per hour. calculate her wage for working a 38 - hour week. 2

Explanation:

Step1: Solve equation (i)

Multiply both sides by 5 to get rid of denominators.
$3(f - 1)=f$
Expand the left - hand side: $3f-3 = f$.
Subtract $f$ from both sides: $3f - f-3=f - f$, which simplifies to $2f-3 = 0$.
Add 3 to both sides: $2f-3 + 3=0 + 3$, so $2f=3$.
Divide both sides by 2: $f=\frac{3}{2}$.

Step2: Solve equation (ii)

First, rewrite the mixed numbers. $2\frac{2}{3}=\frac{8}{3}$ and $1\frac{7}{9}=\frac{16}{9}$.
The equation $(\frac{8}{3}-p)^2=\frac{16}{9}$.
Take the square root of both sides: $\frac{8}{3}-p=\pm\frac{4}{3}$.
Case 1: $\frac{8}{3}-p=\frac{4}{3}$. Subtract $\frac{8}{3}$ from both sides: $-p=\frac{4}{3}-\frac{8}{3}=-\frac{4}{3}$, so $p = \frac{4}{3}$.
Case 2: $\frac{8}{3}-p=-\frac{4}{3}$. Subtract $\frac{8}{3}$ from both sides: $-p=-\frac{4}{3}-\frac{8}{3}=-4$, so $p = 4$.

Step3: Factorise (i)

Group the terms: $(10m^{2}-2mt)+(5mr - tr)$.
Factor out common factors from each group: $2m(5m - t)+r(5m - t)$.
Factor out $(5m - t)$: $(5m - t)(2m + r)$.

Step4: Factorise (ii)

Factor out the common factor $p$ first: $p(p^{2}-16)$.
Since $p^{2}-16$ is a difference of squares ($a^{2}-b^{2}=(a + b)(a - b)$ with $a = p$ and $b = 4$), the factorised form is $p(p + 4)(p - 4)$.

Step5: Calculate the carpenter's wage

The wage rate is $3.65$ per hour and the number of hours is 38.
The wage $W=3.65\times38=138.7$.

Answer:

(i) $f=\frac{3}{2}$
(ii) $p=\frac{4}{3}$ or $p = 4$
(b)(i) $(5m - t)(2m + r)$
(b)(ii) $p(p + 4)(p - 4)$
(c) $138.7$