Question

1. The water/cement ratio of a concrete mix is the ratio of the mass of water in the mix to the mass of cement in the mix. If the ratio of a mix is specified as 0.40, how much water should be added to a mixture with 25 kg of cement.

2. The profit y of selling a product is partly constant and party varies directly as the number of products sold x. When x=1, the profit y is $15. When x=100, y=$213.

(a) Find the relation between the number of items sold x and the profit y.
(b) Find the profit when the number of items sold is 5.

Answer 1

1) Let $m_w$ the mass of water in the mix, $m_c$ the mass of cement in the mix.

The ration of water in the mix is $\cfrac{m_w}{m_c}=0.4=\cfrac{m_w}{25 kg}$ , $m_w=25*0.4=10kg$ .

Answer: 10 kg water should be added to a mixture with 25 kg of cement and the ratio of a mix will be 0.4. 

2) If profit directly depends on sales, then profit can be written through the equation $y=kx+b$ , where $x$ is number of products sold, and $y$ is profit, b is a constant.

If $x=1$ then $y=15$ ;

If $x=100$ then $y=213$ .

These values $x,y$ must satisfy the equation $y=kx+b$ :

$\begin{alignedat}{2} &15 = k+b \\ &213 = 100k+b \end{alignedat}$ .

$\begin{alignedat}{2} &15 = k+b \\ &213 = 99k +15 \end{alignedat}$ , $\begin{alignedat}{2} &15 = k+b \\ &198=99k \end{alignedat}$ , $\begin{alignedat}{2} &b=13 \\ &k=2 \end{alignedat}$ .

So, the relation between $x$ and $y$ is $y=2x+13$ .

If the number of items sold is 5, the profit will be $y = 2*5+13 = 23$ .

Answer: a) the ralation is $y=2x+13$ ; b) 23.