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An algebraic expression is a mathematical phrase that includes numbers, variables, and operation symbols. In the context of percentage problems, algebraic expressions help us represent relationships between quantities concisely.
For example, in the problem where a number \( X \) is increased by 10%, we express this increase using an algebraic expression: \( Y = X + 0.1X = 1.1X \).
This expression means that the original number \( X \) is multiplied by 1.1 to find the new increased amount, \( Y \).

Similarly, for a number \( Z \) decreased by 10%, the expression becomes \( Y = Z - 0.1Z = 0.9Z \). This expresses that the original number \( Z \) is multiplied by 0.9 to find the new decreased amount.

• "Coefficient" refers to the number that multiplies the variable (e.g., 1.1 or 0.9).
• Using algebraic expressions helps simplify the depiction of changes without needing lengthy descriptions.
• Understanding these expressions is essential for solving problems that involve percentage changes.

"Percentage change" deals with how much a number has increased or decreased in relative terms. It is expressed as a percentage based on the original number. Understanding percentage change is key to solving many real-world problems, such as understanding sales growth or the effect of discounts.

In the exercise, we explore both a 10% increase and a 10% decrease.
To find the result of a 10% increase on a number \( X \), we calculate:

• Increase as a decimal: 0.1
• New value \( Y = X + 0.1X = 1.1X \)

This means the new value is 110% of the original.

For a 10% decrease on a number \( Z \):

• Decrease as a decimal: 0.1
• New value \( Y = Z - 0.1Z = 0.9Z \)

Here, the new value is 90% of the original.

When calculating how much \( X \) must increase to become equal to \( Z \), we use:

• \( \text{Percentage Increase} = \left(\frac{Z - X}{X}\right) \times 100\% \)
• Substituting \( Z = \frac{1.1X}{0.9} \) gives the final percentage \( \approx 22.22\% \).

This helps to see how these changes relate just by evaluating percentages.

Mastering problem-solving requires understanding concepts, identifying strategies, and applying solutions methodically. For percentage problems, a few key strategies can simplify the process.
First, convert percentage changes to algebraic expressions. Express both increases and decreases using multipliers (like 1.1 for a 10% increase or 0.9 for a 10% decrease).
Then, set equations to compare the expressions directly and solve for unknowns.

In this exercise:

• We started with understanding the problem by defining \( X+0.1X \) and \( Z-0.1Z \) as expressions for \( Y \).
• The next technique was equating these to find a relationship between \( X \) and \( Z \) using: \( 1.1X = 0.9Z \). This equation allows us to solve for \( Z \) in terms of \( X \).
• Next, substitute \( Z \) in the \( \text{Percentage Increase} \) formula to find how much \( X \) should increase.
• Finally, be systematic: Calculate and simplify step-by-step to find the percentage required, approximately 22.22%.

Approaching problems with this structure helps in maneuvering complex tasks, making each step manageable.