The problem consists of two parts. Part (a): Simplify the expression $\frac{3\frac{1}{12} + \frac{7}{8}}{2\frac{1}{4} - \frac{1}{6}}$. Part (b): (i) Make $n$ the subject of the formula $p = \frac{m}{2} - \frac{n^2}{5m}$. (ii) Find the value of $n$, correct to three significant figures, when $p = 14$ and $m = -8$.

1. Problem Description

The problem consists of two parts.

Part (a): Simplify the expression $\frac{3\frac{1}{12} + \frac{7}{8}}{2\frac{1}{4} - \frac{1}{6}}$.

Part (b):

(i) Make $n$ the subject of the formula $p = \frac{m}{2} - \frac{n^2}{5m}$.

(ii) Find the value of $n$, correct to three significant figures, when $p = 14$ and $m = -8$.

2. Solution Steps

(a) Simplify the expression $\frac{3\frac{1}{12} + \frac{7}{8}}{2\frac{1}{4} - \frac{1}{6}}$.

First, we convert the mixed numbers to improper fractions:

$3\frac{1}{12} = \frac{3 \times 12 + 1}{12} = \frac{36 + 1}{12} = \frac{37}{12}$

$2\frac{1}{4} = \frac{2 \times 4 + 1}{4} = \frac{8 + 1}{4} = \frac{9}{4}$

So the expression becomes:

$\frac{\frac{37}{12} + \frac{7}{8}}{\frac{9}{4} - \frac{1}{6}}$

We find a common denominator for the fractions in the numerator. The least common multiple of 12 and 8 is

2

4. $\frac{37}{12} = \frac{37 \times 2}{12 \times 2} = \frac{74}{24}$

$\frac{7}{8} = \frac{7 \times 3}{8 \times 3} = \frac{21}{24}$

So, $\frac{37}{12} + \frac{7}{8} = \frac{74}{24} + \frac{21}{24} = \frac{74 + 21}{24} = \frac{95}{24}$

We find a common denominator for the fractions in the denominator. The least common multiple of 4 and 6 is

1

2. $\frac{9}{4} = \frac{9 \times 3}{4 \times 3} = \frac{27}{12}$

$\frac{1}{6} = \frac{1 \times 2}{6 \times 2} = \frac{2}{12}$

So, $\frac{9}{4} - \frac{1}{6} = \frac{27}{12} - \frac{2}{12} = \frac{27 - 2}{12} = \frac{25}{12}$

The expression is now:

$\frac{\frac{95}{24}}{\frac{25}{12}} = \frac{95}{24} \div \frac{25}{12} = \frac{95}{24} \times \frac{12}{25} = \frac{95 \times 12}{24 \times 25} = \frac{95}{2 \times 25} = \frac{95}{50} = \frac{19}{10} = 1.9$

(b) (i) Make $n$ the subject of the formula $p = \frac{m}{2} - \frac{n^2}{5m}$.

$p = \frac{m}{2} - \frac{n^2}{5m}$

$\frac{n^2}{5m} = \frac{m}{2} - p$

$n^2 = 5m (\frac{m}{2} - p)$

$n^2 = \frac{5m^2}{2} - 5mp$

$n = \pm \sqrt{\frac{5m^2}{2} - 5mp}$

(ii) Find the value of $n$, correct to three significant figures, when $p = 14$ and $m = -8$.

$n = \pm \sqrt{\frac{5(-8)^2}{2} - 5(-8)(14)}$

$n = \pm \sqrt{\frac{5(64)}{2} + 40(14)}$

$n = \pm \sqrt{\frac{320}{2} + 560}$

$n = \pm \sqrt{160 + 560}$

$n = \pm \sqrt{720}$

$n = \pm 26.83281573$

Correct to three significant figures, $n = \pm 26.8$.

3. Final Answer

(a) $1.9$

(b) (i) $n = \pm \sqrt{\frac{5m^2}{2} - 5mp}$

(ii) $n = \pm 26.8$