Problem 51 - Algebra 1

Question

You earn \(10 \%\) more money at your summer job than your sister earns at her summer job. Does this mean that your sister earns \(10 \%\) less money than you? Explain your answer.

Step-by-Step Solution

Answer

No, the sister does not earn 10% less. A 10% increase of a certain number does not equate to a 10% decrease of the increased number.

Step 1: Understanding the problem

The student earns 10% more than the sister. Let's say the sister's earnings are $100 (a rounded number for ease of calculation). This means the student's earnings would be $110 (10% more).

Step 2: Apply the concept of percentage decrease

A 10% decrease from the student's earnings of $110 would leave them with $99 (not $100). This shows that a 10% increase in one direction does not equate to a 10% decrease in the other direction.

Step 3: Formulation of the conclusion

So, it can be concluded that if you earn 10% more than someone else, it does not mean that they earn 10% less than you. The percentages of increase and decrease between two numbers are not symmetrical.

Key Concepts

Percentage Calculation

Percentage calculation is a fundamental concept in understanding the relationship between two different values. When you calculate a percentage, you are essentially determining how one number relates to another, expressed as a part of 100. This can help you compare different quantities more easily.

In the exercise, the student's earnings are said to be 10% more than the sister's. To find this new amount, you take the original earnings (let’s say, \(100 for the sister) and add 10% of this value to it. Mathematically, this is done as follows:

1. Find 10% of \)100, which is \(0.10 \times 100 = 10\).
2. Add this amount to the original, resulting in \(100 + 10 = 110\).

This shows that the student earns $110.

Understanding percentages allows you to perform such calculations quickly and helps in various scenarios involving financial transactions, discounts, tips, and much more.

Mathematical Reasoning

Mathematical reasoning involves a logical thought process to understand concepts and solve problems. It is crucial in explaining why a 10% increase and a 10% decrease don't equate between two values.

Let's examine the problem using logical steps. When an amount increases by 10%, you accrue a higher value based on the initial figure. For example, increasing from \(100 to \)110 adds \(10. However, when you decrease \)110 by 10%, you are taking 10% off the new higher total.

Calculate 10% of \(110, which is \(0.10 \times 110 = 11\).
Subtract this \)11 from \(110 to yield \)99.

This reasoning demonstrates that a 10% decrease cannot bring you back to the original $100, thus proving the non-symmetrical nature of percentage changes.

Problem Solving in Algebra

Algebra provides tools to generalize these percentages to any number. Let's say the sister earns \(x\) dollars. The student's earnings are 10% more than that, calculated by:

\( \text{Student's earnings} = x + 0.10x = 1.1x \)

To determine if the sister earns 10% less than this new amount, we would express it in algebraic form.

1. Compute 10% less of the student's earnings:
\(1.1x - 0.10 \times 1.1x = 1.1x - 0.11x = 0.99x\)
2. The resulting \(0.99x\) is not equal to \(x\). Thus, a 10% increase does not reverse to a 10% decrease smoothly.

Using algebra illustrates how percentage changes can be applied to any variable and emphasizes why inverse percentage calculations do not return to the original figure.