Finding Two Numbers with Given Product and Sum-Difference Relationship

The problem asks us to find the greater of two numbers based on two given conditions:

• Their product is 900.
• Their sum exceeds their difference by 30.

Let the two numbers be represented by variables, say \(x\) and \(y\). To make it easier to identify the greater number later, let's assume \(x\) is the greater number and \(y\) is the smaller number, so \(x > y\).

Now, let's translate the given conditions into mathematical equations:

Condition 1: Product of the two numbers is 900.

\(x \times y = 900\) (Equation 1)

Condition 2: Their sum exceeds their difference by 30.

The sum is \(x + y\).

The difference is \(x - y\) (since we assumed \(x > y\)).

The sum exceeding the difference by 30 means:

\((x + y) - (x - y) = 30\) (Equation 2)

Now we have a system of two linear equations with two variables:

1. \(xy = 900\)
2. \((x + y) - (x - y) = 30\)

Let's first simplify Equation 2:

\(x + y - x + y = 30\)

Notice that the \(x\) terms cancel out:

\(y + y = 30\)

\(2y = 30\)

Now, we can solve for \(y\):

\(y = \frac{30}{2}\)

\(y = 15\)

So, one of the numbers is 15.

Now substitute the value of \(y\) (which is 15) into Equation 1:

\(x \times 15 = 900\)

Now, solve for \(x\):

\(x = \frac{900}{15}\)

To calculate \(900 \div 15\), we can simplify the fraction or perform division:

\(x = \frac{90 \times 10}{15}\)

\(x = \frac{(6 \times 15) \times 10}{15}\)

Cancel out the 15:

\(x = 6 \times 10\)

\(x = 60\)

So, the two numbers are 60 and 15.

Let's verify if these numbers satisfy both original conditions:

• Product: \(60 \times 15 = 900\). This matches Condition 1.
• Sum: \(60 + 15 = 75\).
• Difference: \(60 - 15 = 45\).
• Does the sum exceed the difference by 30? \(75 - 45 = 30\). Yes, this matches Condition 2.

The two numbers are indeed 60 and 15. The question asks for the greater of the two numbers. Comparing 60 and 15, the greater number is 60.

Therefore, the greater of the two numbers is 60.

Let's look at the given options:

• 60
• 75
• 90
• 100

Our calculated greater number, 60, matches the first option.

The greater number is 60.

Revision Table: Two Numbers Problem

Here's a quick review of the key information used in this problem:

Understanding how to translate word problems into algebraic equations is crucial for solving such quantitative aptitude questions.

Additional Information: Solving Number Relation Problems

Problems involving relations between numbers often require setting up and solving equations. Here are some related concepts:

• Defining Variables: Always start by assigning variables (like \(x\), \(y\), \(a\), \(b\)) to the unknown quantities you need to find. Clearly state what each variable represents.
• Translating Words to Equations:
  • "Sum" means addition (\(x+y\)).
  • "Difference" means subtraction (\(x-y\) or \(y-x\)).
  • "Product" means multiplication (\(xy\)).
  • "Exceeds by" means one quantity is greater than another by a certain amount, leading to a subtraction or an equation like \(A = B + \text{amount}\) or \(A - B = \text{amount}\). In this problem, "sum exceeds difference by 30" means \(\text{Sum} - \text{Difference} = 30\).
• Solving Systems of Equations: When you have multiple conditions, you will likely end up with a system of equations. Common methods to solve systems include:
  • Substitution Method: Solve one equation for one variable and substitute that expression into the other equation. This is what we primarily did in this problem (we found \(y\) directly from the second equation and substituted it into the first).
  • Elimination Method: Multiply equations by constants so that when you add or subtract them, one variable cancels out.
• Checking Your Answer: Always plug your final numbers back into the original word problem conditions to ensure they satisfy everything stated in the question. This helps catch errors.

This problem specifically yielded a direct value for one variable from the second equation, simplifying the substitution step significantly.