1. If one wishes to accumulate a Php 30,000-fund in 5 years, how much should he deposit now at 18% compounded quarterly?
2. If a bank offers a rate of 4.5% compounded semiannually, how much should you deposit to accumulate Php 50,000 in 15 years
3. Suppose that a particular radioactive substance has a half-life of 25 years. Starting with 100 grams of the substance, how many grams of the substance would remain after 15 years?
1
p(1.045)^20 = 30000
p = 30000/1.045^20 = 12439.29
2. same method as #1
3. amount = 100(.5)^(15/25)
= 65.98 g
To answer these questions, we can use formulas related to compound interest and exponential decay.
1. To calculate the amount needed to deposit now to accumulate a Php 30,000-fund in 5 years at 18% compounded quarterly, we can use the formula for compound interest:
A = P * (1 + r/n)^(nt)
Where:
A = the future value of the investment (Php 30,000)
P = the principal amount (to be calculated)
r = annual interest rate (18% or 0.18)
n = number of times the interest is compounded per year (quarterly means 4 times)
t = the number of years (5)
Rearranging the formula to solve for P, we have:
P = A / (1 + r/n)^(nt)
Substituting the values into the formula:
P = 30,000 / (1 + 0.18/4)^(4*5)
= 30,000 / (1.045)^20
≈ 30,000 / 1.932924795
≈ Php 15,520.52
Hence, he should deposit approximately Php 15,520.52 now to accumulate a Php 30,000-fund in 5 years at 18% compounded quarterly.
2. To calculate how much should be deposited to accumulate Php 50,000 in 15 years at a rate of 4.5% compounded semiannually, we can again use the compound interest formula:
A = P * (1 + r/n)^(nt)
Rearranging the formula to solve for P:
P = A / (1 + r/n)^(nt)
Substituting the given values:
P = 50,000 / (1 + 0.045/2)^(2*15)
= 50,000 / (1.0225)^30
≈ 50,000 / 1.85631330
≈ Php 26,917.20
Therefore, approximately Php 26,917.20 should be deposited to accumulate Php 50,000 in 15 years at a rate of 4.5% compounded semiannually.
3. To calculate how many grams of the radioactive substance remain after 15 years with a half-life of 25 years and starting with 100 grams, we can use the exponential decay formula:
N = N₀ * (1/2)^(t / T)
Where:
N = the remaining quantity of the substance (to be calculated)
N₀ = the initial quantity of the substance (100 grams)
t = the number of years (15)
T = the half-life of the substance (25 years)
Substituting the given values into the formula:
N = 100 * (1/2)^(15 / 25)
= 100 * (1/2)^(3/5)
≈ 100 * 0.7355812
≈ 73.55812 grams
Therefore, after 15 years, approximately 73.56 grams of the substance would remain.