Ratios/Proportions of numbers: problem solving In a bag, there are red and blue cubes in the ratio $4 : 7$. $\quad$ (red : blue) $\;\;4 : 7$.
I add $10$ more red cubes to the bag.
Now there are red and blue cubes in the ratio of $\quad$ (red : blue) $\;\;6 : 7$.

How many blue cubes are in the bag? 

I'm not sure, how would you work it out if you haven't got the amount of cubes altogether?
 A: The following is the most mechanical approach I can think of. 
The current ratio is $4:7$. So there are $4k$ red and $7k$ blue for some unknown number $k$.
When we add $10$ red, we end up with $4k+10$ red. The blues remain unchanged at $7k$.
So the new proportion is $(4k+10): 7k$. We are told that the proportion $(4k+10): 7k$ is $6:7$. So 
$$\frac{4k+10}{7k}=\dfrac{6}{7}.$$ 
If we multiply through by $7k$, we get $4k+10=6k$, and therefore $k=5$. It follows that there are $35$ blues in the bag. 
A: Let $x$ be the number of red cubes and $y$ be the number of blue cubes. 
To start, the ratio of red cubes to blue cubes is 4:7, or for every 4 red cubes, there are 7 blue cubes. Hence, we have: 
$7x = 4y$. 
When 10 more red cubes are added to the bag, the ratio of red cubes to blue cubes shifts to 6:7, or:
$7(x+10) = 6y$.
Expanding, we get a system:
$7x = 4y$
$7x + 70 = 6y$
Can you solve the system of equations from here?
A: Let $4x$ be the number of red cubes. Since the ratio is 4:7, this means we have $7x$ blue cubes. If you add 10 more red cubes to the bag, then we have $4x + 10$ red cubes, and still $7x$ blue cubes.
Now the new ratio is 6:7, so let the number of red cubes be $6y$. Then the number of blue cubes is $7y$. This tells us that
$$
4x + 10 = 6y
$$
$$
7x = 7y.
$$
 Hence we have a system of equations we can solve for. We see that $x = y$ and so
$$
4y + 10 = 6y \rightarrow 10 = 2y \rightarrow y = 5,
$$
hence $x = 5$ as well. So the number of blue cubes in the bag was $\boxed{7 \cdot 5 = 35}$.
A: Let $\color{red}{r}$ be the number of red cubes (at the start), and $\color{blue}{b}$ be the number of blue cubes.  From the first statement, we know that:
$$\frac{\color{red}{r}}{\color{blue}{b}} = \frac 4 7$$
From the second statement, we know that 
$$\frac{\color{red}{r + 10}}{\color{blue}{b}} = \frac 6 7$$
Cross multiplying both of these, we obtain:
$$7\color{red}r = 4\color{blue}b\tag{1}$$
$$7(\color{red}{r+10}) = 6\color{blue}b\tag{2}$$
Distribute the $7$ through in equation $(2)$:
$$7\color{red}{r} + 70 = 6\color{blue}b\tag{3}$$
Now plug the $7\color{red}r$ from equation $(1)$ into equation $(3)$:
$$4\color{blue}b + 70 = 6\color{blue}b\tag{4}$$
Collect like terms:
$$70 = 2\color{blue}b$$
Dividing the two over:
$$\color{blue}b = 35$$
