3. Four friends rent a cottage for a total of $£ 300$ for the weekend. The first friend pays half of the sum of the amounts paid by the other three friends. The second friend pays one third of the sum of the amounts paid by the other three friends. The third friend pays one quarter of the sum of the amounts paid by the other three friends.
How much money does the fourth friend pay?
Solution
Let $f_{a}$ represent the amount of money the $a^{\text {th }}$ friend pays, in pounds. Therefore the sum of the amounts paid by the other three friends is $300-f_{a}$ pounds.
The first friend pays half of the amount paid by the other three friends so $f_{1}=\frac{1}{2}\left(300-f_{1}\right)$. Multiplying both sides by 2 gives $2 f_{1}=300-f_{1}$. Adding $f_{1}$ to both sides gives $3 f_{1}=300$ and so $f_{1}=100$.
Similarly, the second friend pays one third of the amount paid by the other three friends so $f_{2}=\frac{1}{3}\left(300-f_{2}\right)$. Multiplying both sides by 3 gives $3 f_{2}=300-f_{2}$. Adding $f_{2}$ to both sides gives $4 f_{2}=300$ and so $f_{2}=75$.
Finally, the third friend pays a quarter of the amount paid by the other three friends so $f_{3}=\frac{1}{4}\left(300-f_{3}\right)$. Multiplying both sides by 4 gives $4 f_{3}=300-f_{3}$. Adding $f_{3}$ to both sides gives $5 f_{3}=300$ and so $f_{3}=60$.
This means that the fourth friend must pay $300-100-75-60$ pounds.
So the fourth friend pays $£ 65$.