Exercise 2 · Systems of equations — Fruit boxes
A system of two equations in three unknowns: parametric solution and an application.
Maximum score · 2.5 pointsA farmer grows different kinds of fruit: strawberries, apricots and cherries, sold in boxes at different prices. An order of 5 boxes of strawberries, 3 of apricots and 4 of cherries costs 120 € in total. On the other hand, an order of 2 boxes of strawberries, 1 of apricots and 3 of cherries costs 56 € in total.
- Find the price of a box of strawberries and the price of a box of apricots in terms of the price of a box of cherries. With this information, work out the price of an order of 6 boxes of strawberries, 4 of apricots and 2 of cherries. 1.5 p
- If a box of apricots is known to cost 12.50 €, find the price of the other two fruit boxes. 1 p
Step-by-step solution
Key idea: let $m$, $a$ and $c$ be the prices (in €) of the boxes of strawberries, apricots and cherries. We have two equations and three unknowns: the system is underdetermined and we express $m$ and $a$ in terms of $c$.
a) Prices in terms of cherries and the order's price
The two given conditions:
From the second equation, $a = 56 - 2m - 3c$; substituting into the first:
Price of the order of 6 boxes of strawberries, 4 of apricots and 2 of cherries:
The $c$'s cancel: the order's price does not depend on the price of the cherry box.
b) Specific prices
Impose $a = 12.50$:
Check: $5(10.5) + 3(12.5) + 4(7.5) = 52.5 + 37.5 + 30 = 120$ ✓ | $2(10.5) + 12.5 + 3(7.5) = 21 + 12.5 + 22.5 = 56$ ✓