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}