Math Problem Statement

DATOS Interbank BCP BBVA Va 6000 2827 4753 Vf 6847 3180 5161 N ? ? ? i 16.53% 8.90% 10.50% I ? ? ?

Solution

To solve for the missing values in your table, we will use the formula for compound interest. The formula for the future value (Vf) of an investment is:

Vf=Va×(1+i)NVf = Va \times (1 + i)^N

where:

  • VfVf is the future value
  • VaVa is the present value (initial investment)
  • ii is the interest rate per period
  • NN is the number of periods

We can rearrange this formula to solve for NN:

N=log(VfVa)log(1+i)N = \frac{\log(\frac{Vf}{Va})}{\log(1 + i)}

Given dataTo solve for the missing values in your table, we will use the formula for compound interest. The formula for the future value (Vf) of an investment is:

Vf=Va×(1+i)NVf = Va \times (1 + i)^N

where:

  • VfVf is the future value
  • VaVa is the present value (initial investment)
  • ii is the interest rate per period
  • NN is the number of periods

We can rearrange this formula to solve for NN:

N=log(VfVa)log(1+i)N = \frac{\log(\frac{Vf}{Va})}{\log(1 + i)}

Given

  • Interbank: Va=6000Va = 6000, Vf=6847Vf = 6847, i=16.53%i = 16.53\%
  • BCP: Va=2827Va = 2827, Vf=3180Vf = 3180, i=8.90%i = 8.90\%
  • BBVA: Va=4753Va = 4753, Vf=5161Vf = 5161, i=10.50%i = 10.50\%

Calculations

For Interbank

NInterbank=log(68476000)log(1+0.1653)N_{Interbank} = \frac{\log(\frac{6847}{6000})}{\log(1 + 0.1653)}

For BCP

NBCP=log(31802827)log(1+0.0890)N_{BCP} = \frac{\log(\frac{3180}{2827})}{\log(1 + 0.0890)}

For BBVA

NBBVA=log(51614753)log(1+0.1050)N_{BBVA} = \frac{\log(\frac{5161}{4753})}{\log(1 + 0.1050)}

Now let's calculate these values.

Calculation Results

Interbank

NInterbank=log(68476000)log(1+0.1653)=log(1.1412)log(1.1653)0.0560.0680.82 periodsN_{Interbank} = \frac{\log(\frac{6847}{6000})}{\log(1 + 0.1653)} = \frac{\log(1.1412)}{\log(1.1653)} \approx \frac{0.056}{0.068} \approx 0.82 \text{ periods}

BCP

NBCP=log(31802827)log(1+0.0890)=log(1.1246)log(1.0890)0.0500.0361.39 periodsN_{BCP} = \frac{\log(\frac{3180}{2827})}{\log(1 + 0.0890)} = \frac{\log(1.1246)}{\log(1.0890)} \approx \frac{0.050}{0.036} \approx 1.39 \text{ periods}

BBVA

NBBVA=log(51614753)log(1+0.1050)=log(1.0858)log(1.1050)0.0360.0400.9 periodsN_{BBVA} = \frac{\log(\frac{5161}{4753})}{\log(1 + 0.1050)} = \frac{\log(1.0858)}{\log(1.1050)} \approx \frac{0.036}{0.040} \approx 0.9 \text{ periods}

Let's summarize these results in the table.

InterbankBCPBBVA
Va600028274753
Vf684731805161
N0.821.390.9
i16.53%8.90%10.50%

If you need to calculate the interest earned (I), we use the formula:

I=VfVaI = Vf - Va

So:

  • Interbank: I=68476000=847I = 6847 - 6000 = 847
  • BCP: I=31802827=353I = 3180 - 2827 = 353
  • BBVA: I=51614753=408I = 5161 - 4753 = 408

Updated table:

InterbankBCPBBVA
Va600028274753
Vf684731805161
N0.821.390.9
i16.53%8.90%10.50%
I847353408

Would you like further details or have any questions?

Here are 5 questions you can ask next:

  1. How do you derive the logarithm in these equations?
  2. Can you explain the concept of compound interest in more detail?
  3. What would happen if the interest rate changed midway through the period?
  4. How can we apply these calculations to different financial products?
  5. What are the implications of different compounding periods (e.g., monthly vs. annually)?

Tip: Always double-check your calculations, especially when dealing with exponential functions and logarithms, as small errors can lead to significant differences in the results.

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Math Problem Analysis

Mathematical Concepts

Compound Interest
Logarithms

Formulas

Compound Interest Formula
Logarithm Formula

Theorems

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Suitable Grade Level

Advanced High School