Question

If \(7x = 4y = 9z\), what is the value of \(9x - 5y + 4z\)?
Statement 1: \(3x - y = \frac{5}{14}\)
Statement 2: \(5y + 2z = \frac{53}{18}\)

Hint:

When variables are given in a continuous proportion (e.g., \(ax = by = cz\)), immediately set them equal to a constant \(k\). This simplifies the problem by allowing you to express all variables in terms of one unknown, which you then try to solve for using the information in the statements.

Answer 1

Step 1: Understanding the Concept:
This data sufficiency problem gives a proportional relationship between three variables and asks for the value of a linear combination of them. The key is to express all variables in terms of a single constant of proportionality, \(k\).
Step 2: Analyze the Main Question:
We are given \(7x = 4y = 9z\). Let this common value be \(k\). \[ 7x = k \implies x = \frac{k}{7} \] \[ 4y = k \implies y = \frac{k}{4} \] \[ 9z = k \implies z = \frac{k}{9} \] We need to find the value of the expression \(9x - 5y + 4z\). Let's substitute the expressions in terms of \(k\): \[ 9\left(\frac{k}{7}\right) - 5\left(\frac{k}{4}\right) + 4\left(\frac{k}{9}\right) = k\left(\frac{9}{7} - \frac{5}{4} + \frac{4}{9}\right) \] To find the value of this expression, we need to find the value of \(k\).
Step 3: Analyze the Statements:
Statement (1): \(3x - y = \frac{5}{14}\)
Substitute the expressions for \(x\) and \(y\) in terms of \(k\): \[ 3\left(\frac{k}{7}\right) - \left(\frac{k}{4}\right) = \frac{5}{14} \] \[ \frac{3k}{7} - \frac{k}{4} = \frac{5}{14} \] This is a linear equation with one variable, \(k\). We can solve it for \(k\). \[ \frac{12k - 7k}{28} = \frac{5}{14} \implies \frac{5k}{28} = \frac{5}{14} \implies k=2 \] Since we can find a unique value for \(k\), we can find a unique value for the target expression. Thus, Statement (1) is sufficient.
Statement (2): \(5y + 2z = \frac{53}{18}\)
Substitute the expressions for \(y\) and \(z\) in terms of \(k\): \[ 5\left(\frac{k}{4}\right) + 2\left(\frac{k}{9}\right) = \frac{53}{18} \] \[ \frac{5k}{4} + \frac{2k}{9} = \frac{53}{18} \] This is a linear equation with one variable, \(k\). We can solve it for \(k\). \[ \frac{45k + 8k}{36} = \frac{53}{18} \implies \frac{53k}{36} = \frac{53}{18} \implies k=2 \] Since we can find a unique value for \(k\), we can find a unique value for the target expression. Thus, Statement (2) is sufficient.
Step 4: Final Answer:
Each statement alone is sufficient to determine the value of the expression.