Find the unit price of each item which was paid as whole

Question

I am trying to help my daughter in her math and there is this question I can't quite get my head around, The sum is:

Three friends go into a book shop. Salma buys a cook book and a novel, she pays \$ 20.75. Isla buys the same novel and a dictionary. Her bill comes to \$ 26.65. Josh buys a cook book and a Dictionary and pays \$ 30.90.

What is the price of each book?

Answer 1

Hint: Setting $C=$ price of cook book; $N=$price of novel and $D=$price of dictionary, we get the equations $$ \left\{ \begin{array}{crccc} C& + N & & = &20.75\\ & N & + D & = & 26.65 \\ C &&+D &=& 30.90 \end{array} \right. $$ This is fairly easy to solve. Adding the first and last equations together, we obtain $$ \left\{ \begin{array}{crccc} 2C& + N & +D & = &51.65\\ & N & + D & = & 26.65 \\ \end{array} \right. $$ From which we can easily see (by subtracting the equations from each other) that $$ 2C = 25.00 \qquad \Rightarrow \qquad C = 12.50 $$ now we can plug in this value to the original last equation, and we obtain $D = 30.90 - 12.50 = 18.40$. Similarly, plugging this result in the second equation of the original set, we get $N=26.65 - 18.40 = 8.25$

The final answers are therefore $$ C = 12.50 \qquad N = 8.25 \qquad D = 18.40 $$

Answer 2

I'll try to put it in a way, so you may explain it to your daughter.

Let's denote $C$ price for cook book, $N$ novel, $D$ dictionary.

So you have $C + N = 20.75$, $N + D = 20.65$ and $C + D = 30.90$. Now the cost of buying a cook book and selling a dictionary $ C - D = C + N - (N + D) = 20.75 -20.65 = 0.10$. And finally buying two cookbooks: $2C = 2C - D + D = C + D + (C - D) = 30.90 + 0.10 = 31.00$. So $C = 15.50$. The rest should by easy.