Let us first practise the skill of converting worded problems to mathematical ones with appropriate variables:

\begin{examplebox}
    \subsection{Examples:}
    \begin{questions}
        \Question[2] Three children earn weekly pocket money. Andrew earns \(\$ 2\) more than Gina, and Katya earns twice the amount Gina earns. The total of the weekly pocket money is \(\$ 22\).
        \begin{parts}
            \part How much money does Gina earn?
            \begin{solutionordottedlines}[1in]
                Let \(\$ m\) be the amount of pocket money Gina earns in a week.

                Then Andrew earns \(\$(m+2)\) and Katya earns \(\$(2 m)\),

                so \(\quad m+(m+2)+2 m=22\) (The total weekly pocket money is \(\$ 22\).)

                \[
                4 m+2=22
                \]

                \(4 m=20\)

                \[
                m=5
                \]

                So Gina earns \(\$ 5\) per week.
            \end{solutionordottedlines}
            \part How much money do Andrew and Katya earn?
            \begin{solutionordottedlines}[1in]
                Andrew earns \(\$ 7\) and Katya earns \(\$ 10\) per week.
            \end{solutionordottedlines}
        \end{parts}
        \Question[2] Ali and Jasmine each have a number of swap cards. Jasmine has 25 more cards than Ali, and in total the two children have 149 cards.
        \begin{parts}
            \part How many cards does Ali have?
            \begin{solutionordottedlines}[1in]
                Let \(x\) be the number of cards Ali has.

                Jasmine has \((x+25)\) cards.

                Total number of cards is 149 ,

                \[
                \begin{aligned}
                & \text { so } x+(x+25)=149 \\
                & 2 x+25=149 \\
                & 2 x=124 \\
                & x=62
                \end{aligned}
                \]

                So Ali has 62 cards.
            \end{solutionordottedlines}
            \part How many cards does Jasmine have?
            \begin{solutionordottedlines}[1in]
                Jasmine has \(62+25=87\) cards.
            \end{solutionordottedlines}
        \end{parts}
        \question[] Speed is one of the most familiar rates. In problems involving speed, we use the relationship:

        \[\text{average speed} =\frac{\text { distance travelled }}{\text { time taken }}\]
        which we can remember conveniently with the following triangle:
        \begin{center}
            \begin{tikzpicture}
              % Define the points of a triangle
              \coordinate (A) at (0,0);
              \coordinate (B) at (4,0);
              \coordinate (C) at (2,3.46);

              % Draw the triangle
              \draw (A) -- (B) -- (C) -- cycle;

              % Label the vertices
              \draw (2,0) -- (2,1.73);
              \draw (1,1.73) -- (2,1.73);
              \draw (3,1.73) -- (2,1.73);
              \node at (2,0.75*3.46) {$d$};
              \node at (1.5,1) {$s$};
              \node at (2.5,1) {$t$};
            \end{tikzpicture}
        \end{center}
        \Question[2] For a training run, a triathlete covers \(50 \mathrm{~km}\) in \(4 \frac{1}{4}\) hours. She runs part of the way at a speed of \(10 \mathrm{~km} / \mathrm{h}\), cycles part of the way at a speed of \(40 \mathrm{~km} / \mathrm{h}\) and swims the remaining distance at a speed of \(2 \frac{1}{2} \mathrm{~km} / \mathrm{h}\). The athlete runs for twice the time it takes to complete the cycle leg. How long did she take to complete the cycle leg?
            \begin{solutionordottedlines}[1in]
                Let \(t\) hours be the time for the cycle leg.

                Then \(2 t\) hours is the time for the running leg and \(\left(4 \frac{1}{4}-t-2 t\right)\) hours is the time for the swim leg.

                Now, distance of run + distance of cycle + distance of swim \(=\) total distance,

                so \(10 \times 2 t+40 \times t+2 \frac{1}{2} \times\left(4 \frac{1}{4}-t-2 t\right)=50\)

                \[
                \begin{aligned}
                20 t+40 t+\frac{5}{2}\left(\frac{17}{4}-3 t\right) & =50 \\
                60 t+\frac{85}{8}-\frac{15 t}{2} & =50 \\
                480 t+85-60 t & =400 \\
                420 t & =315 \\
                t & =\frac{315}{420} \\
                t & =\frac{3}{4}
                \end{aligned}
                \]

                The athlete takes \(\frac{3}{4}\) hour, or 45 minutes, to complete the cycle leg.
            \end{solutionordottedlines}
    \end{questions}
\end{examplebox}


\begin{exercisebox}
    \subsection{Exercises:}
    \begin{questions}
        \Question[1] Jacques thinks of a number \(x\). When he adds 17 to his number, the result is 32 . What is the value of \(x\) ?
            \begin{solutionordottedlines}[2cm]
            \end{solutionordottedlines}
        \Question[1] When 16 is added to twice Simone's age, the answer is 44 . How old is Simone?
            \begin{solutionordottedlines}[2cm]
            \end{solutionordottedlines}
        \Question[1] When 14 is added to half of Suzette's weight in kilograms, the result is 42 . How much does Suzette weigh?
            \begin{solutionordottedlines}[2cm]
            \end{solutionordottedlines}
        \Question[4] Derek is presently 20 years older than his daughter, Alana.
        \begin{parts}
            \part If \(x\) represents Alana's present age, express each of the following in terms of \(x\).
            \begin{subparts}
                \subpart Derek's present age 
            \begin{solutionordottedlines}[2cm]
            \end{solutionordottedlines}
                \subpart Alana's age in 12 years' time
            \begin{solutionordottedlines}[2cm]
            \end{solutionordottedlines}
                \subpart Derek's age in 12 years' time
            \begin{solutionordottedlines}[2cm]
            \end{solutionordottedlines}
            \end{subparts}
            \part If Derek's age 12 years from now is twice Alana's age 12 years from now, find their present ages.
            \begin{solutionordottedlines}[2cm]
            \end{solutionordottedlines}
        \end{parts}
        \Question[4] Alan, Brendan and Calum each have a number of plastic toys from a fast food store. Brendan has 5 more toys than Alan, and Calum has twice as many toys as Alan.
        \begin{parts}
            \part If \(x\) represents the number of toys Alan has, express each of the following in terms of \(x\) :
            \begin{subparts}
                \subpart the number of toys Brendan has
            \begin{solutionordottedlines}[2cm]
            \end{solutionordottedlines}
                \subpart the number of toys Calum has
            \begin{solutionordottedlines}[2cm]
            \end{solutionordottedlines}
                \subpart the total number of toys the three boys have
            \begin{solutionordottedlines}[2cm]
            \end{solutionordottedlines}
            \end{subparts}
            \part If the boys have 37 toys in total, determine how many toys each boy has.
            \begin{solutionordottedlines}[2cm]
            \end{solutionordottedlines}
        \end{parts}
    \end{questions}
\end{exercisebox}
