Let's break down this ratio and sum problem step-by-step to find the value of z.

We are given the ratio of three sums involving x, y, and z:

$(x +y) : (y + z) : (z + x) = 3 : 5 : 7$

This means we can write these sums in terms of a common constant, let's call it \(k\).

• \(x + y = 3k\)
• \(y + z = 5k\)
• \(z + x = 7k\)

We are also given the total sum of x, y, and z:

\(x + y + z = 45\)

Finding the Value of the Constant \(k\)

We can find the value of the constant \(k\) by adding the three equations we got from the ratio:

\((x + y) + (y + z) + (z + x) = 3k + 5k + 7k\)

Combining the terms on the left side:

\(2x + 2y + 2z = 15k\)

Factor out 2 from the left side:

\(2(x + y + z) = 15k\)

We know that \((x + y + z) = 45\). Substitute this value into the equation:

\(2(45) = 15k\)

\(90 = 15k\)

Now, solve for \(k\):

\(k = \frac{90}{15}\)

\(k = 6\)

Calculating the Individual Sums

Now that we have the value of \(k\), we can find the actual values of \((x+y)\), \((y+z)\), and \((z+x)\):

• \(x + y = 3k = 3 \times 6 = 18\)
• \(y + z = 5k = 5 \times 6 = 30\)
• \(z + x = 7k = 7 \times 6 = 42\)

Finding the Value of z

We have the sum of all three variables, \((x + y + z) = 45\). We also have the sum of two variables, \((x + y) = 18\). We can find the value of z by subtracting the sum \((x + y)\) from the total sum \((x + y + z)\):

\((x + y + z) - (x + y) = 45 - 18\)

\(x + y + z - x - y = 27\)

\(z = 27\)

Verification

We can find the values of x and y as well to verify. Since \((x+y+z) = 45\) and \((y+z)=30\), we get \(x = 45 - 30 = 15\). Since \((x+y+z) = 45\) and \((z+x)=42\), we get \(y = 45 - 42 = 3\). So, \(x=15\), \(y=3\), \(z=27\).

Check the sums:

• \(x+y = 15+3 = 18\)
• \(y+z = 3+27 = 30\)
• \(z+x = 27+15 = 42\)

The sums are 18, 30, and 42. Let's check their ratio:

\(18 : 30 : 42\)

Divide all terms by their greatest common divisor, which is 6:

\(\frac{18}{6} : \frac{30}{6} : \frac{42}{6} = 3 : 5 : 7\)

This matches the given ratio. Also, the sum \(x+y+z = 15+3+27 = 45\), which matches the given sum. So the values are correct.

The value of z is 27.

Revision Table: Key Concepts

Additional Information: Solving Systems of Equations

This problem involves a system of linear equations. Once we found the values of \((x+y)\), \((y+z)\), and \((z+x)\), we effectively had the following system:

1. \(x + y = 18\)
2. \(y + z = 30\)
3. \(z + x = 42\)

And we also know \(x+y+z=45\).

There are several ways to solve such a system:

• Method 1 (Using the total sum): As shown in the solution, use the given total sum. If you want to find \(z\), subtract the equation that doesn't contain \(z\) (which is \(x+y=18\)) from the total sum equation \((x+y+z=45)\). \( (x+y+z) - (x+y) = 45 - 18 \implies z = 27 \). Similarly for x and y.
• Method 2 (Elimination/Substitution):
  • Add all three equations: \((x+y)+(y+z)+(z+x) = 18+30+42 \implies 2x+2y+2z = 90 \implies x+y+z = 45\). This confirms the given total sum.
  • Now you have \(x+y+z = 45\). Subtract \(x+y=18\) from this to find \(z\), subtract \(y+z=30\) to find \(x\), and subtract \(z+x=42\) to find \(y\).

Both methods lead to the same result and confirm the consistency of the given information.