Problem 159 - Elementary Algebra

Question

After returning from vacation, Alex said he should have packed \(50 \%\) fewer shorts and \(200 \%\) more shirts. Explain what Alex meant.

Step-by-Step Solution

Answer

Alex should have packed half as many shorts and three times as many shirts.

Step 1 - Understand the Problem

Analyze the statement that Alex made about packing fewer shorts and more shirts. Specifically, clarify what it means to pack '50% fewer' and '200% more'.

Step 2 - Define Variables

Let the number of shorts Alex originally packed be denoted by S, and the number of shirts be denoted by H.

Step 3 - Calculate 50% Fewer Shorts

To find out how many fewer shorts Alex should have packed, calculate 50% of S: \[0.5 \times S = 0.5S\] Therefore, he should have packed \[S - 0.5S = 0.5S\] shorts.

Step 4 - Calculate 200% More Shirts

To determine how many more shirts Alex should have packed, calculate 200% of H: \[2 \times H = 2H\] Therefore, he should have packed a total of \[H + 2H = 3H\] shirts.

Key Concepts

Percentage Increase

When Alex mentioned he should have packed 200% more shirts, he referred to a percentage increase. Understanding percentage increase is crucial as it helps in comparing the old quantity to the new one.

To calculate a percentage increase:

• Identify the original quantity.
• Convert the percentage to a decimal by dividing by 100.
• Multiply the original quantity by this decimal value.

In Alex's case, the original quantity of shirts is represented by \(H\).
He intended to pack 200% more shirts, which is equivalent to multiplying \(H\) by 2:
\[ 2 \times H = 2H \]
Adding this to the original amount gives:
\[ H + 2H = 3H \]
So, Alex should pack three times the initial number of shirts.

Percentage Decrease

When Alex recognized he should have packed 50% fewer shorts, he was talking about a percentage decrease. Percent decrease helps us determine how much the original quantity should be reduced.

To calculate a percentage decrease:

• Identify the original quantity.
• Convert the percentage to a decimal by dividing by 100.
• Multiply the original quantity by this decimal value.

In Alex's context, the original number of shorts is represented by \(S\).
He wanted to pack 50% fewer, which means finding 50% of \(S\):
\[ 0.5 \times S = 0.5S \]
Subtracting this from \(S\) gives:
\[ S - 0.5S = 0.5S \]
Therefore, Alex should pack only half the original number of shorts.

Algebraic Expressions

Algebraic expressions help in representing problems concisely using letters and numbers. In the problem about Alex's packing, we used the variables \(S\) and \(H\) for shorts and shirts, respectively.

To work with algebraic expressions:

• Define the variables clearly. For example, let \(S\) be the number of shorts and \(H\) be the number of shirts.
• Use mathematical symbols to show relationships. For example, 50% fewer shorts can be written as \(S - 0.5S\).
• Simplify the expressions when possible. For example, \(S - 0.5S\) simplifies to \(0.5S\).

In solving Alex's problem:
By defining \(S\) and \(H\), then expressing percentage changes using these variables, we arrive at simplified results like \(0.5S\) for shorts and \(3H\) for shirts. This method makes it easier to solve and understand real-world problems using math.