6. Ivana decided to buy some products in an online store. In the cart, she already has 4 products whose total price is $30 €$. If she adds a jacket to these products, the average price of all selected products will be $20 €$. If instead of the jacket, she chooses boots and pants, the average price would also be $20 €$. It is known that the price of the boots is $54 €$ higher than the price of the pants. Determine the price of the jacket, the price of the boots, and the price of the pants.
6. Ivana decided to buy some products in an online store. In the cart, she already has 4 products whose total price is $30 €$. If she adds a jacket to these products, the average price of all selected products will be $20 €$. If instead of the jacket, she chooses shoes and pants, the average price would also be $20 €$. It is known that the price of the shoes is $54 €$ higher than the price of the pants. Determine the price of the jacket, the price of the shoes, and the price of the pants.

## Solution.

Let $j$ be the price of the jacket.

If the average price of 5 products is $20 €$, we can write this as:

$a+b+c+d=30$

$(a+b+c+d+j): 5=20$

$(30+j): 5=20$

1 POINT

$30+j=20 \cdot 5$

$30+j=100$

1 POINT

$j=100-30$

$j=70$

1 POINT

The price of the jacket is $70 €$.

If instead of the jacket, she chooses shoes and pants, the average price remains $20 €$.

Let $\check{c}$ be the price of the shoes, and $h$ be the price of the pants.

$a+b+c+d=30$

$(a+b+c+d+\check{c}+h): 6=20$

$(30+\check{c}+h): 6=20$

1 POINT

$30+\check{c}+h=20 \cdot 6$

$30+\check{c}+h=120$

1 POINT

$\check{c}+h=120-30$

$\check{c}+h=90$

1 POINT

We know that the price of the shoes is $54 €$ higher than the price of the pants, so we can write:

$\check{c}=h+54$

$\check{c}+h=90$

$h+54+h=90$

1 POINT

$2 h=90-54$

$2 h=36$

1 POINT

$h=18$

1 POINT

The price of the pants is $18 €$.

$\check{c}=h+54$

$\check{c}=72$

1 POINT

The price of the shoes is $72 €$.