⚠️ Don't copy these < / p > < p >
Item 1:
[tex]{1. \: James \: sold \: \( \frac{7}{20} \) \: of \: the \: company \: to \: Paul.}[/tex]
[tex]{- From \: Part \: 4A, \: it's \: mentioned \: that \: James \: sold \: \( \frac{2}{5} \) \: of \: his \: \( \frac{7}{8} \) \: share \: in \: 2022.}[/tex]
- Calculating this sale:
[tex] \[
\text{Share sold} = \left( \frac{2}{5} \right) \times \left( \frac{7}{8} \right) = \frac{2 \times 7}{5 \times 8} = \frac{14}{40} = \frac{7}{20}
\][/tex]
[tex]{2. \: James \: held \: \( \frac{21}{40} \) \: share \: after the \: sale \: to \: Paul.}[/tex]
[tex]{- Initially, James \: had \: \( \frac{7}{8} \) \: of \: the
\: company.}[/tex]
[tex]{- After \: selling \: \( \frac{7}{20} \) \: of \: the \: company \: to \: Paul, \: calculate \: James's \: remaining \: share:}[/tex]
[tex] { \[
\text{James's remaining share} = \left( \frac{7}{8} \right) - \left( \frac{7}{20} \right)
\]}[/tex]
To subtract these fractions, find a common denominator, which is 40:
[tex] {\[
\frac{7}{8} = \frac{35}{40}\quad \text{and} \quad \frac{7}{20} = \frac{14}{40}
\]}[/tex]
[tex]Then:
{ \[
\text{James's remaining share} = \frac{35}{40} - \frac{14}{40} = \frac{21}{40}
\]}[/tex]
Item 2:
[tex]1. If \: Bella's \: age \: is \: taken \: to \: be \: \( x \), \: write \: down \: Paul's \: and \: James's \: age \: in \: terms \: of \: \( x \).[/tex]
[tex]- Bella's age: {\( x \)}[/tex]
[tex]- James's age: {\( 1.55x + 1 \)}[/tex]
[tex]- Paul's age: {\( 0.95x - 3 \)}[/tex]
2. How old are James, Paul, and Bella?
- We are given:
[tex]{\[
\text{Sum of ages} = \text{James's age} + \text{Paul's age} + \text{Bella's age} = 138
\]}[/tex]
[tex]Substituting \: the \: given \: expressions:
{\[
(1.55x + 1) + (0.95x - 3) + x = 138
\]}[/tex]
[tex]Combine \: like \: terms:
{ \[
1. \: 55x + 0.95x + x + 1 - 3 = 138
\]}[/tex]
[tex]Simplify:
{\[
3.5x - 2 = 138
\]}[/tex]
[tex]Solve \: for \: \( x \):
{ \[
3.5x = 140 \implies x = \frac{140}{3.5} = 40
\]}[/tex]
[tex]- So, Bella's age {(\( x \)) \: is \: \( 40 \).}[/tex]
[tex]- James's age:
{ \[
1. \: 55x + 1 = 1.55(40) + 1 = 62 + 1 = 63
\]}[/tex]
[tex]- Paul's age:
{ \[
0.95x - 3 = 0.95(40) - 3 = 38 - 3 = 35
\]}
[/tex]
Therefore, the ages are as follows:
[tex]- Bella: \boxed{\( 40 \) \: years \: old}[/tex]
[tex]- James: \boxed {\( 63 \) \: years \: old}[/tex]
[tex]- Paul: \boxed{\( 35 \) \: years \: old}[/tex]
