Penny and rolls

For one hundred crowns I need to buy exactly one hundred pieces of pastry at the following prices: rolls at 0.50 crowns each, bread at 10 crowns each, and buns at 3 crowns each. How many of each should I buy?

(The Slovak koruna was the currency in Slovakia before the euro; the Czech koruna is still used in the Czech Republic.)

Final Answer:

a₁ =  80
b₁ =  0
c₁ =  20
a₂ =  94
b₂ =  5
c₂ =  1

Step-by-step explanation:

$$\begin{gathered} a+b+c=100 \\ 0.50a+10b+3c=100 \\ a+b+c=100 \\ a+20b+6c=200 \\ a+20b+6(100-a-b)=200 \\ a+20b+600-6a-6b=200 \\ 14b+400-5a=0 \\ 5a-14b=400 \\ {a}_1=80 \end{gathered}$$

$b_1=0$

$c_1=100-b_1-a_1=100-0-80=20$

$a_2=94$

$b_2=5$

$c_2=100-a_2-b_2=100-94-5=1$

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