Six pears and three apples cost $3.90. Two pears and five apples cost$3.30. How much does one pear cost?

Sep 28, 2015

Let's first translate that in "the language".

Explanation:

$6 \cdot p + 3 \cdot a = 3.90 \mathmr{and} 2 \cdot p + 5 \cdot a = 3.30$
We multiply the second equation (all of it) by 3, because that will also give us the price of 6 pears plus apples (as in the first), and the price difference will only be caused by the different number of apples:

$6 \cdot p + 15 \cdot a = 9.90$ now subtract the other equation:
$- \left(6 \cdot p + 3 \cdot a = 3.90\right)$
So: $12 \cdot a = 6.00 \to a = 0.50$ (=price of an apple)

Put this into one of the equations:
$2 \cdot p + 5 \cdot 0.50 = 3.30 \to 2 \cdot p = 3.30 - 2.50 = 0.80 \to$
$p = 0.40$ (price of a pear)

Check!
$6 \cdot 0.40 + 3 \cdot 0.50 = 3.90 \mathmr{and} 2 \cdot 0.40 + 5 \cdot 0.50 = 3.30$