# Solving the equations $x_1= 4 x_2$ and $x_3= 5 x_2$, with the sum of all three being $150$

Here is the problem.

A set X is partitioned into subsets x1, x2, and x3. The number of elements in x1 is 4 times the number in x2. And the number in x3 is 5 times the number in x2. If n(x)=150, find n(x1), n(x2), and n(x3).

So I have been trouble finding anything online for this since I don't know what this process is called. I tried dividing 150 by many numbers with no success in getting the correct answer. Thanks in advance for your help!

• Do you know how to solve systems of equations using substitution? Like when you have two equations with two unknowns or three equations with three unknowns? Jan 21 '15 at 5:18
• Apart from the partition terminology, we have $3$ dogs A, B, and C. The weight of A is $4$ times the weight of B, and the weight of C is $5$ times the weight of B. Together they weigh $150$ pounds. How much does each dog weigh? Jan 21 '15 at 5:30

Number of elements in $X_i$ is $n(X_i)$.
We know that $$n(X) = n(x_1) + n(x_2) + n(x_3) = 150$$ $$n(x_1) = 4 n(x_2)$$ and $$n(x_3) = 5 n(x_2).$$
This system of equations can be solved by substituting $n(x_1)$ and $n(x_3)$ to the first equation.