Solving the Two-Number Problem: Sum and Difference Relationship

The question asks us to find the larger of two numbers, given that their sum is five times their difference and the smaller number is 24.

Let's define the two numbers:

• Let the larger number be \(L\).
• Let the smaller number be \(S\).

We are given the value of the smaller number:

• \(S = 24\)

We are also given a relationship between the sum and the difference of the two numbers:

The sum of the two numbers is \(L + S\).

The difference between the two numbers (assuming \(L > S\)) is \(L - S\).

According to the problem statement, the sum is 5 times their difference. We can write this relationship as an equation:

\(L + S = 5 \times (L - S)\)

Now, we can substitute the given value of the smaller number, \(S = 24\), into this equation:

\(L + 24 = 5 \times (L - 24)\)

Next, we need to solve this equation for \(L\).

First, distribute the 5 on the right side of the equation:

\(L + 24 = 5L - 5 \times 24\)

\(L + 24 = 5L - 120\)

Now, we want to isolate the variable \(L\) on one side of the equation. Let's move the \(L\) term from the left side to the right side by subtracting \(L\) from both sides:

\(24 = 5L - L - 120\)

\(24 = 4L - 120\)

Next, move the constant term (-120) from the right side to the left side by adding 120 to both sides:

\(24 + 120 = 4L\)

\(144 = 4L\)

Finally, solve for \(L\) by dividing both sides by 4:

\(\frac{144}{4} = L\)

\(36 = L\)

So, the larger number is 36.

Let's verify this solution with the original condition: \(L + S = 5 \times (L - S)\)

If \(L = 36\) and \(S = 24\):

• Sum: \(L + S = 36 + 24 = 60\)
• Difference: \(L - S = 36 - 24 = 12\)
• 5 times the difference: \(5 \times 12 = 60\)

Since the sum (60) is indeed 5 times the difference (12), our value for \(L = 36\) is correct.

Comparing our answer to the given options, the larger number is 36, which corresponds to one of the choices.

Revision Table: Key Information

Additional Information: Setting up Equations from Word Problems

Word problems like this one require translating the given information into mathematical equations. Here are some tips:

• Read Carefully: Understand what is given and what needs to be found.
• Define Variables: Use letters (like \(L\) and \(S\)) to represent the unknown quantities. Clearly state what each variable represents.
• Identify Relationships: Look for keywords that indicate mathematical operations (e.g., "sum" means addition, "difference" means subtraction, "times" or "product" means multiplication, "is" often means equals).
• Formulate Equations: Write down the relationships you identified as equations using the variables.
• Solve the Equation(s): Use algebraic techniques to solve for the unknown variable(s).
• Check Your Answer: Plug your solution back into the original word problem or equation to make sure it makes sense and satisfies all conditions.

In this specific problem, the key was recognizing the relationship "sum of two numbers is 5 times their difference" and translating it directly into the equation \(L + S = 5(L - S)\). The rest was solving a linear equation.