In horticulture Rose, one store ordered a total of 120 roses in red and yellow, a second store a total of 105 roses in red and white, and a third store a total of 45 roses in yellow and white. The horticulture fulfilled the order in such a way that it delivered the same number of roses of the same color to each store.

How many red, how many white, and how many yellow roses did the horticulture deliver to these three stores in total?

(M. Volfová)

Hint. Imagine the individual orders within the total number of delivered roses.
If we sum the quantities of roses delivered to all three stores, we get 270 pieces. In this sum, the quantities of roses of each color are included twice.

In the first store, there were red and yellow roses totaling 120 pieces. Double this quantity is 240, so the number of white roses delivered to the remaining two stores was $270-240=30$.

In the second store, there were 105 red and white roses. Double this quantity is 210, so the number of yellow roses delivered to the remaining two stores was $270-210=60$.

In the third store, there were 45 yellow and white roses. Double this quantity is 90, so the number of red roses delivered to the remaining two stores was $270-90=180$.

Note. In each of the stores, the quantities of roses of each color were half, i.e., 15 white, 30 yellow, and 90 red.

![](https://cdn.mathpix.com/cropped/2024_04_17_848c3145926e37c65b12g-4.jpg?height=49&width=1553&top_left_y=1232&top_left_x=297)
The problem can be written as

$$
\check{c}+\check{z}=120, \quad \check{c}+b=105, \quad \check{z}+b=45 .
$$

The reasoning in the above solution corresponds to the following transformations:

$$
2 \check{c}+2 \check{z}+2 b=270, \quad 2 \check{c}+2 \check{z}=240, \quad \text { thus } \quad 2 b=30
$$

and similarly in the other two cases.