Problem 24 - Algebra Form and Function

Question

To calculate the balance after investing \(P\) dollars for two years at \(5 \%\) interest, Sharif adds \(5 \%\) of \(P\) to \(P\), and then adds \(5 \%\) of the result of this calculation to itself. Donald multiplies \(P\) by \(1.05,\) and then multiplies the result of this by 1.05 again. (a) Write expressions for each calculation. (b) Do the expressions in (a) define the same function?

Step-by-Step Solution

Answer

Answer: Yes, both Sharif's and Donald's methods define the same function, which is \((1.05P)(1.05)\) for calculating the balance after investing P dollars for two years at 5% interest.

Step 1: (Expression for Sharif's calculation)

To find the expression for Sharif's calculation, we first add \(5\%\) of P to P, and then add \(5\%\) of the result to itself. In other words: 1. Add \(5\%\) of P to P: \(P + 0.05P = P(1+0.05) = P(1.05)\) 2. Add \(5\%\) of the result to itself: \((1.05P) + 0.05(1.05P) = (1.05P)(1+0.05) = (1.05P)(1.05)\) So the expression for Sharif's calculation is \((1.05P)(1.05)\).

Step 2: (Expression for Donald's calculation)

To find the expression for Donald's calculation, we multiply P by 1.05, and then multiply the result by 1.05 again. In other words: 1. Multiply P by 1.05: \(P\cdot 1.05 = 1.05P\) 2. Multiply the result by 1.05: \((1.05P) \cdot 1.05 = (1.05P)(1.05)\) So the expression for Donald's calculation is \((1.05P)(1.05)\).

Step 3: (Determine if expressions define the same function)

We can see that both Sharif's and Donald's expressions are equal, i.e., they define the same function: Sharif's expression: \((1.05P)(1.05)\) Donald's expression: \((1.05P)(1.05)\) Thus, the expressions in (a) define the same function.

Key Concepts

Algebraic Expressions

An algebraic expression is a mathematical phrase that can include numbers, variables, and operation symbols. It's a way to represent quantities and their relationships using mathematical symbols.
In the context of the given exercise, Sharif's and Donald's methods are each expressed as algebraic expressions to calculate compound interest for two years.Sharif's method of adding 5% of the principal and then adding 5% of the new amount translates into:

• Add 5% of P to P: \( P + 0.05P = 1.05P \)
• Add 5% of the new result: \( (1.05P) + 0.05(1.05P) = 1.05P \times 1.05 \)

This expression simplifies to \((1.05P)(1.05)\), an example of how algebra helps in crafting formulas.
Donald directly multiplies by \(1.05\) twice, leading to the same expression without transitional steps: \((1.05P)(1.05)\).
Both algebraic expressions ultimately represent the same process mathematically.

Equivalent Functions

Equivalent functions are fundamentally different expressions that yield the same output when given the same input.
In this exercise, both Sharif and Donald use methods that suggest different strategies. However, they lead to equivalent functions because they produce identical outputs from identical inputs.Sharif breaks down the interest calculation into two additive steps: adding interest to the principal and reapplying the percentage. This step-by-step approach helps in explicitly understanding the increment process.
Donald opts for a more straightforward multiplication strategy, aiming for efficiency. By multiplying \(P\) by \(1.05\) twice directly, his method achieves the same overall multiplicative effect.Though visually and methodologically distinct, the functions from Donald's and Sharif's processes simplify to \((1.05P)(1.05)\) each time the calculation is done. This makes them equivalent, mathematically indicating they are essentially two expressions of the same underlying rule.

Percentage Increase

A percentage increase represents how much a quantity grows in relation to its original size. This is a crucial concept in understanding compound interest.
In terms of Sharif's method, the initial increase of 5% of the principal amount means you are adding 0.05 times the original \(P\) to itself, equating to a multiplier of \(1.05\).
After this first step, another percentage (again 5%) is applied to the new total, not just the original \(P\).Donald, on the other hand, immediately uses the cumulative effect of repeated percentage increases by multiplying everything by \(1.05\) twice in sequence. This can be simply considered as finding the future value of \(P\) after a 5% increase, not once, but twice consecutively.Using repeated multipliers to achieve a compounded total sum is a hallmark of percentage increases applied in financial calculations like interest, allowing the initial amount \(P\) to grow exponentially over time due to continual application of the percentage increase.