Solving the Sum of Two Numbers and Percentage Change Problem

This problem involves finding two numbers based on their sum and how they change after applying percentages. Let's break it down step by step.

Defining the Variables

Let the two numbers be $B$ and $S$. According to the problem statement, one number is bigger and the other is smaller. We will assume $B$ is the bigger number and $S$ is the smaller number, so $B > S$.

Setting up the Equations

Based on the information given, we can form two equations:

1. The sum of the two numbers is 680:

   \(B + S = 680 \quad (1)\)

2. The bigger number is decreased by 15%, and the smaller number is increased by 15%. The resultant numbers are equal.
   • Decreasing $B$ by 15% means it becomes \(B - 0.15B = (1 - 0.15)B = 0.85B\).
   • Increasing $S$ by 15% means it becomes \(S + 0.15S = (1 + 0.15)S = 1.15S\).
   Since these resultant numbers are equal:

   \(0.85B = 1.15S \quad (2)\)

Solving the Equations to Find the Smaller Number

We now have a system of two linear equations with two variables:

\(B + S = 680\)

\(0.85B = 1.15S\)

We want to find the smaller number, $S$. We can use the substitution method. From equation (1), we can express $B$ in terms of $S$:

\(B = 680 - S\)

Now, substitute this expression for $B$ into equation (2):

\(0.85(680 - S) = 1.15S\)

Distribute 0.85 on the left side:

\(0.85 \times 680 - 0.85S = 1.15S\)

\(578 - 0.85S = 1.15S\)

Now, isolate the term with $S$ by adding $0.85S$ to both sides of the equation:

\(578 = 1.15S + 0.85S\)

Combine the terms on the right side:

\(578 = (1.15 + 0.85)S\)

\(578 = 2.00S\)

\(578 = 2S\)

To find $S$, divide both sides by 2:

\(S = \frac{578}{2}\)

\(S = 289\)

So, the smaller number is 289.

Verifying the Solution

Let's check if this value of $S$ satisfies the original conditions.

• The smaller number $S = 289$.
• From $B + S = 680$, the bigger number is $B = 680 - S = 680 - 289 = 391$.

Check if $B > S$: $391 > 289$. This is consistent with our assumption that $B$ is the bigger number.

• Decrease the bigger number (391) by 15%:

  \(391 \times (1 - 0.15) = 391 \times 0.85 = 332.35\)

• Increase the smaller number (289) by 15%:

  \(289 \times (1 + 0.15) = 289 \times 1.15 = 332.35\)

The resultant numbers, 332.35 and 332.35, are indeed equal. This confirms that our calculated value for the smaller number is correct.

The smaller number is 289.

Revision Table: Key Concepts

Additional Information: Solving Number Problems

Number problems often involve setting up equations based on the relationships described between numbers. Here are some key ideas:

• Identify the unknowns: Assign variables (like x, y, A, B, etc.) to the numbers you need to find.
• Translate words into math: Look for keywords like "sum", "difference", "product", "quotient", "is", "equals", "more than", "less than", "increased by", "decreased by", "percentage of".
  • "Sum" means addition (+).
  • "Difference" means subtraction (-).
  • "Product" means multiplication (\(\times\) or juxtaposition).
  • "Quotient" means division (\(\div\) or fraction).
  • "Is" or "equals" means equality (=).
  • "Increased by x%" means multiply by \((1 + x/100)\).
  • "Decreased by x%" means multiply by \((1 - x/100)\).
• Formulate equations: Write down the mathematical relationships as equations based on the problem statement.
• Solve the system of equations: Use methods like substitution or elimination to find the values of the variables.
• Check your answer: Substitute the values you found back into the original word problem or equations to make sure they work.

Practice with different types of word problems helps in mastering the translation from language to mathematical expressions and solving techniques.