Solving for the Fraction xy / yz Given x/y 2 and y/z 3

The problem at hand is to find the value of the fraction xy / yz given that x/y 2 and y/z 3. We'll approach this step-by-step to ensure a clear and comprehensive understanding.

Understanding the Given Information

We are provided with two pieces of information:

x/y 2 y/z 3

Our goal is to find the value of the fraction xy / yz.

Solving Step-by-Step

Step 1 Expressing x in Terms of y

Starting with the first equation:

x/y 2

This implies that x 2y.

Step 2 Expressing y in Terms of z

Now, using the second equation:

y/z 3

This implies that y 3z.

Substitute y 3z into x 2y to express x in terms of z:

x 2(3z) 6z

Evaluating the Fraction xy / yz

Now that we have expressions for x and y in terms of z, we can substitute these into the fraction xy / yz to find its value.

Step 3 Substituting the Values

Recall that:

x 6z y 3z

Substituting these into xy / yz:

xy / yz (6z × 3z) / (3z × z)

Which simplifies to:

xy / yz (18z^2) / (3z^2) 18 / 3 6 / 2 9 / 4

Therefore, the value of xy / yz is 9/4.

Conclusion

In conclusion, the fraction xy / yz, when given that x/y 2 and y/z 3, simplifies to 9/4.

Understanding and solving algebraic expressions and fractions is a fundamental skill in mathematics. By breaking down the problem into smaller, manageable steps, we can ensure clarity and accuracy in our solution.

Key Points:

Express initial variables in terms of a third variable. Substitute these expressions into the given fraction. Simplify the resulting expression to find the value.

Our final answer is:

xy / yz 9/4

This method can be applied to similar algebraic problems, making us better prepared to tackle more complex mathematical challenges.