Question

Given the equations \(x30y=4\) and \(1797+3y=15x\), compare Quantity A and Quantity B.
Quantity A: y
Quantity B: 1

• A. The relationship cannot be determined from the information given.
• B. The quantities are equal.
• C. Quantity A is greater.
• D. Quantity B is greater.
• E.

Hint:

In quantitative comparison questions, you don't always need to find the exact numerical value. Analyzing the constraints and relationships between variables can often lead to the answer more quickly. Here, checking the signs of x and y was the key.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
We are given a system of two equations with two variables, x and y. We need to find the value of y, or at least determine its relationship to 1, to compare Quantity A (y) and Quantity B (1).
Step 2: Key Formula or Approach:
We will use substitution to solve the system of equations. We can express one variable in terms of the other from one equation and substitute it into the second equation.
Step 3: Detailed Explanation:
The given equations are: \[ 30xy = 4 \quad \Rightarrow \quad xy = \frac{4}{30} = \frac{2}{15} \quad \text{(Equation 1)} \] \[ 1797 + 3y = 15x \quad \Rightarrow \quad 15x - 3y = 1797 \] Dividing the second equation by 3, we get: \[ 5x - y = 599 \quad \Rightarrow \quad y = 5x - 599 \quad \text{(Equation 2)} \] Now, we substitute the expression for y from Equation 2 into Equation 1: \[ x(5x - 599) = \frac{2}{15} \] This gives a quadratic equation, which might be complicated to solve due to the large numbers. Let's try to analyze the possible values of y.
From Equation 1, \(xy = 2/15>0\). This implies that x and y must have the same sign (both positive or both negative).
Case 1: x and y are both positive.
If y>0, then from Equation 2, we must have \(5x - 599>0\).
This means \(5x>599\), or \(x>119.8\).
Now, let's look at Equation 1 again: \(y = \frac{2}{15x}\).
Since \(x>119.8\), the value of y will be: \[ y<\frac{2}{15 \times 119.8} = \frac{2}{1797} \] In this case, y is a very small positive number, which is clearly less than 1.
Case 2: x and y are both negative.
If y is negative, it is automatically less than 1. For completeness, let's check if this case is possible. Let \(x<0\). From Equation 2, \(y = 5x - 599\). Since 5x is negative, \(5x - 599\) will be a large negative number. This is consistent with y being negative.
In both possible cases, the value of y is less than 1.
Step 4: Final Answer:
Since y is always less than 1, Quantity B (which is 1) is greater than Quantity A (y).