Problem 84 - Elementary Algebra Within Reach

Question

Reasoning Would you rather receive a \(3 \%\) raise followed by a \(9 \%\) raise or a \(9 \%\) raise followed by a \(3 \%\) raise? Explain.

Step-by-Step Solution

Answer

It doesn't matter if you receive a 3% raise followed by a 9% raise, or a 9% raise followed by a 3% raise. Both result in the same final salary.

Step 1: Define Baseline Salary

Take an initial salary as a baseline for comparison, say \(100 \) units. This will not affect the relative percentages but provides a concrete example for calculations.

Step 2: Calculate First Scenario Increase

For the first scenario, give a \(3 \%\) raise first. That would mean \(100 *(1 + 0.03) = 103\) units.

Step 3: Apply Second Raise First Scenario

Then apply a \(9 \%\) raise on the increased salary. That would mean \(103 *(1 + 0.09) = 112.27\) units. This is the final salary in the first scenario.

Step 4: Calculate Second Scenario Increase

For the second scenario, give a \(9 \%\) raise first. That means \(100 *(1 + 0.09) = 109\) units.

Step 5: Apply Second Raise Second Scenario

Then apply a \(3 \%\) raise on the increased salary. That would mean \(109 * (1 + 0.03) = 112.27\) units. This is the final salary in the second scenario.

Step 6: Compare Results

Compare the final salaries from both scenarios. If they're the same, it doesn't matter in which order the raises are applied, because percentage raises are compounding.

Key Concepts

Compounding Percentages

When dealing with compounding percentages, it's crucial to understand the concept of interest on interest. This means that each percentage increase is calculated based on the new total, not just the original amount.

In our exercise example, we consider two salary raises. Given a baseline salary, say of \(100\) units, a \(3\%\) raise is not just an additional \(3\) units every time but rather 3% of whatever the current salary happens to be after any previous raises.

So, when deciding between sequential percentage raises, the order of these raises matters because the second percentage is calculated on the increased amount from the first raise. This is the essence of compounding—the increase 'builds upon' itself. However, in our specific exercise, because multiplication is commutative (changing the order doesn't affect the result), a \(3\%\) raise followed by a \(9\%\) raise results in the same final salary as a \(9\%\) raise followed by a \(3\%\) raise.

Sequential Percentage Increases

Sequential percentage increases occur when you have multiple percentage adjustments applied one after the other. These situations are common in finance, such as when applying annual interest rates, or in retail, during marked discount periods.

To grasp this concept, consider a tagged price of an item that undergoes a series of discounts. A \(10\%\) discount followed by an additional \(20\%\) discount doesn't equal a \(30\%\) discount overall. Instead, the second discount is computed on the already reduced price, resulting in a compounding effect.

This principle also applies to salary raises, as seen in the textbook exercise. After receiving a raise, any subsequent raises are applied to the already increased salary, not the original amount.

Algebraic Reasoning

At its heart, algebraic reasoning involves using mathematical concepts and techniques to solve problems. In the context of percentage raises, algebra helps us to understand and quantify sequential increases.

For example, let's take an algebraic approach to analyze our raise problem. We define the baseline salary as \(100\) units to simplify the calculation. Using algebraic expressions, we can then apply the percentage raises. In the form of \(100 * (1 + \(raise1\)) * (1 + \(raise2\))\), it becomes clear that the order of multiplicative factors doesn't influence the final result, highlighting the commutative property of multiplication.

Mastering this type of algebraic reasoning allows students to tackle a range of real-world problems where sequential percentage changes occur. Such knowledge is also beneficial for making informed financial decisions.