In a fruit shop, Jaime noticed that an orange costs the same as half an apple plus half a real, and he also noticed that a third of an apple costs the same as a quarter of an orange plus half a real. With the value of 5 oranges plus 5 reals, how many apples can Jaime buy?

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Solution

If an orange costs the same as half an apple plus half a real, then, 2 oranges cost the same as 1 apple plus 1 real; we also have that if a third of an apple costs the same as a quarter of an orange plus half a real, then, multiplying each part by 12, we have that 4 apples cost the same as 3 oranges plus 6 reals. Combining the two pieces of information, we conclude that 5 oranges plus 5 reals cost the same as 5 apples. We can solve this problem using the system of equations:

$$
\left\{\begin{array} { c } 
{ L = \frac { M } { 2 } + \frac { 1 } { 2 } } \\
{ \frac { M } { 3 } = \frac { L } { 4 } + \frac { 1 } { 2 } }
\end{array} \rightarrow \left\{\begin{array} { c } 
{ 2 L = M + 1 } \\
{ 4 M = 3 L + 6 }
\end{array} \rightarrow \left\{\begin{array}{llc}
2 L-1= & M \\
3 L+6= & 4 M
\end{array}\right.\right.\right.
$$

Adding the equations of the last system, we arrive at $5 L+5=5 M$, that is, 5 oranges plus 5 reals cost the same as 5 apples.

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