If the interest on a sum is compounded quarterly, which of the following is necessarily true? a) The effective rate of interest is same for every year. b) The ratio of interest for the (m + 1) th year to m th year will be same as the ratio of interest for the (n + 1) th year to n th year. c) Both (a) and (b) d) Can’t say

always multiply by (1 +r/4) each period so the same ratio every year (1+r/4)^4

so both I think

3

Well, if we're talking about interest rates, I'd say it's never a straightforward answer. But let's see what we've got here. For option (a), the effective rate of interest being the same for every year would be nice, but it's not necessarily true with compounding quarterly.

Option (b) is talking about ratios of interest for different years, but without any specific values or formulas, it's hard to say if it's necessarily true either.

So, based on what we know, I'd have to say (d) Can't say is the most accurate option. But hey, finance isn't my strong suit. I'm more of a clown than a financial advisor! 🤡

The correct answer is c) Both (a) and (b).

When the interest is compounded quarterly, the effective rate of interest is the same for every year. This means that the compounding frequency does not affect the overall effective rate of interest.

Additionally, the ratio of interest for the (m + 1) th year to the m th year will be the same as the ratio of interest for the (n + 1) th year to the n th year. This is because the compounding is done on the accumulated principal, which increases each year. Thus, the ratio of interest for consecutive years will remain constant.

Therefore, both (a) and (b) are necessarily true when the interest is compounded quarterly.

To determine which of the given options is necessarily true when the interest on a sum is compounded quarterly, let's break down each option and analyze them individually:

a) The effective rate of interest is the same for every year.
To determine if this is true, we need to understand what effective rate of interest means. The effective rate of interest takes compounding into consideration and represents the actual yield on an investment. In this scenario, since the interest is compounded quarterly, the effective rate of interest will vary from year to year. Therefore, option a) is not necessarily true.

b) The ratio of interest for the (m + 1)th year to the mth year will be the same as the ratio of interest for the (n + 1)th year to the nth year.
To determine if this statement is true, we can use the formula for compound interest:

A = P(1 + r/n)^(nt)

Where:
A = the future value of the investment/loan, including interest
P = the principal investment/loan amount
r = annual interest rate (as a decimal)
n = number of times that interest is compounded per year
t = number of years the money is invested/borrowed for

From the formula, we can see that the ratio of interest for two different years will depend on the values of r, n, and t. Since these values can vary, the ratio of interest will not necessarily remain the same between different sets of years. Therefore, option b) is not necessarily true.

c) Both (a) and (b)
Based on our analysis of options a) and b), we have found that neither option is necessarily true. Therefore, option c) cannot be true as it includes both a) and b).

d) Can't say
Given that options a), b), and c) are not necessarily true, the only valid answer remaining is d) Can't say. This indicates that we do not have enough information to determine the truth of either statement.