If (x +y) ∶ (y + z) ∶ (z + x) = 3 ∶ 5 ∶ 7 and (x + y + z) = 45, then what is the value of z?
27
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.
| Given Information | Derived Information |
|---|---|
| \((x+y):(y+z):(z+x) = 3:5:7\) | \(x+y = 18\) |
| \((x+y+z) = 45\) | \(y+z = 30\) |
| \(z+x = 42\) | |
| \(k = 6\) |
Revision Table: Key Concepts
| Concept | Explanation | Application in Problem |
|---|---|---|
| Ratio | A comparison of two or more quantities. \(a:b:c = d:e:f\) means \(\frac{a}{d}=\frac{b}{e}=\frac{c}{f}=k\) (a constant). | \((x+y):(y+z):(z+x) = 3:5:7\) implies \(x+y=3k, y+z=5k, z+x=7k\). |
| Algebraic Sums | Adding equations to eliminate variables or find combined sums. | Adding \((x+y)\), \((y+z)\), and \((z+x)\) gives \(2(x+y+z)\). |
| Substitution | Replacing a variable or expression with its equivalent value in another equation. | Substituting \((x+y+z) = 45\) into \(2(x+y+z) = 15k\). Substituting \((x+y) = 18\) into \((x+y+z) = 45\) to find z. |
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:
- \(x + y = 18\)
- \(y + z = 30\)
- \(z + x = 42\)
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.
Important Questions from Simplification
Simplify the following expression.
\(\left(\frac{7}{16} \div \frac{1}{2}\:of\: \frac{1}{5}\right)\times \frac{4}{5}-\frac{1}{3}\times\frac{5}{8}\div \frac{1}{2}+\frac{3}{4}\)
The value of \(\left( {2\frac{6}{7}of4\frac{1}{5} \div \frac{2}{3}} \right) \times 5\frac{1}{9} \div \left( {\frac{3}{4} \times 2\frac{2}{3}of\frac{1}{2} \div \frac{1}{4}} \right)\) is:
The value of \(\left[ {\frac{4}{7}\rm \;of\;2\frac{4}{5} \times 1\frac{2}{3} - \left( {3\frac{1}{2} - 2\frac{1}{6}} \right)} \right] \div \left( {3\frac{1}{5} \div 4\frac{1}{2}\;\rm of\;\;5\frac{1}{3}} \right)\) is:
The value of \(\frac{{0.0203 \times 2.92}}{{0.7 \times 0.0365 \times 2.9}} \div \frac{{{{\left( {12.12} \right)}^2} - {{\left( {8.12} \right)}^2}}}{{{{\left( {0.25} \right)}^2} + \left( {0.25} \right)\left( {19.99} \right)}}\) is:
The value of 4 ÷ 12 of [3 ÷ 4 of {(4 - 2) × 6 ÷ 2}] - 2 × 6 ÷ 8 + 3 is:
