★ Concept :-

Here the concept of Rule 72 or Accountancy has been used. We see that here we have to present the uses and derivativion of the Rule 72. Here in order to begin derivation, we shall start with the relationship between Future Value and Present Value. There we shall apply the logarithmic function and then attain simple values on the basis of close approximations. And then we can derive the value of the Rule 72.

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★ Solution :-

Rule 72 : It is the rule in accountacy which defines the time period in which the principal value will get doubled of itself. This is given by the ratio of 72 by the rate of interest in percentage. Here the principal can be considered as investment.

Mathematically it is given as,

Let's begin with Derivation now ::

• Let the Future Value of the principal amount be FV.

• Let the Present Value of the principal amount be PV.

• Let the Time Period of attaining the future value be T.

• Let the Rate of Interest of the principal amount for investment be R (not in percentage form).

Now here, according to the relationship of Future Value, we know that

Let us assume that the Present Value (PV) is 1. Then,

Now if the Future Value (FV) is to be doubled, then the condition will be as follows :-

This condition should fulfil then only the cáse will satisfy.

From here we shall take out the value of T. Since, here there are variable values, so the relationship should be in terms of T.

In order to get that, the easiest way is using logarithm. So we shall apply both sides by ln where ln is the natural logarithm.

Now the power as exponent will come in front.

• Here the power of 2 is 1.

• And the power of (1 + R) is T.

Then from this, we will get

On applying the properties of logarithm, we get

Let's begin some important assumptions here.

We know that, if R is much smaller, then clearly ln(1 + R) ≈ R. This means this can be replaced with R. This is clearly defined Taylor's Series' first term.

Now here when T is very small, then the the fractional part will be treated as a function of R that is f(R). This means we will get,

Now applying this in the main equation, we get

It has been proved that when T is very small with positive rate of Interest (R), then f(R) = f(0.8). And from experiments, the value of f(0.08) is approximately equal to 1.0395

Now applying this in the equation, we get

Applying Logarithmic Function here, we get

This will give us,

Now multiplying both numerator and denominator by 100, we get

Now here 100R will be the percentage of rate of Interest. So,

• Here the Rate of Interest is in Percentage Form.

Thus this derives the mathematical formulation of Rule 72.

• Uses of Rule 72 ::

» Determines the amount when it is received at maturity of an investment after certain time period.

» Determines the time period when the investment will get double of itself.

» Gives an estimate to the investors about their profit in the definite time period.

Verified answer

Explanation:

TOPIC:-

ᴀᴄᴄᴏᴜɴᴛᴀɴᴄʏ

UNDERSTANDING THE CONCEPT:-

ᴛʜᴇ ʀᴜʟᴇ ᴏғ 72 ɪs ᴀ ɢʀᴇᴀᴛ ᴍᴇɴᴛᴀʟ ᴍᴀᴛʜ sʜᴏʀᴛᴄᴜᴛ ᴛᴏ ᴇsᴛɪᴍᴀᴛᴇ ᴛʜᴇ ᴇғғᴇᴄᴛ ᴏғ ᴀɴʏ ɢʀᴏᴡᴛʜ ʀᴀᴛᴇ, ғʀᴏᴍ ǫᴜɪᴄᴋ ғɪɴᴀɴᴄɪᴀʟ ᴄᴀʟᴄᴜʟᴀᴛɪᴏɴs ᴛᴏ ᴘᴏᴘᴜʟᴀᴛɪᴏɴ ᴇsᴛɪᴍᴀᴛᴇs.

DERIVATION:-

ʟᴇᴛ’s sᴛᴀʀᴛ ᴡɪᴛʜ $1 sɪɴᴄᴇ ɪᴛ’s ᴇᴀsʏ ᴛᴏ ᴡᴏʀᴋ ᴡɪᴛʜ (ᴛʜᴇ ᴇxᴀᴄᴛ ᴠᴀʟᴜᴇ ᴅᴏᴇsɴ’ᴛ ᴍᴀᴛᴛᴇʀ). sᴏ, sᴜᴘᴘᴏsᴇ ᴡᴇ ʜᴀᴠᴇ $1 ᴀɴᴅ ᴀ ʏᴇᴀʀʟʏ ɪɴᴛᴇʀᴇsᴛ ʀᴀᴛᴇ ʀ. ᴀғᴛᴇʀ ᴏɴᴇ ʏᴇᴀʀ ᴡᴇ ʜᴀᴠᴇ:

1 * (1+ʀ)

ғᴏʀ ᴇxᴀᴍᴘʟᴇ, ᴀᴛ 10% ɪɴᴛᴇʀᴇsᴛ, ᴡᴇ’ᴅ ʜᴀᴠᴇ $1 * (1 + 0.1) = $1.10 ᴀᴛ ᴛʜᴇ ᴇɴᴅ ᴏғ ᴛʜᴇ ʏᴇᴀʀ. ᴀғᴛᴇʀ 2 ʏᴇᴀʀs, ᴡᴇ’ᴅ ʜᴀᴠᴇ

1 * (1+ʀ) * (1+ʀ)

= 1 * (1+ʀ)^2

ᴀɴᴅ ᴀᴛ 10% ɪɴᴛᴇʀᴇsᴛ, ᴡᴇ ʜᴀᴠᴇ $1 * (1.1)2 = $1.21 ᴀᴛ ᴛʜᴇ ᴇɴᴅ ᴏғ ʏᴇᴀʀ 2.

ᴇxᴛᴇɴᴅɪɴɢ ᴛʜɪs ʏᴇᴀʀ ᴀғᴛᴇʀ ʏᴇᴀʀ, ᴀғᴛᴇʀ ɴ ʏᴇᴀʀs ᴡᴇ ʜᴀᴠᴇ

1 * (1+ʀ)^ɴ

ɴᴏᴡ, ᴡᴇ ɴᴇᴇᴅ ᴛᴏ ғɪɴᴅ ʜᴏᴡ ʟᴏɴɢ ɪᴛ ᴛᴀᴋᴇs ᴛᴏ ᴅᴏᴜʙʟᴇ — ᴛʜᴀᴛ ɪs, ɢᴇᴛ ᴛᴏ 2 ᴅᴏʟʟᴀʀs. ᴛʜᴇ ᴇǫᴜᴀᴛɪᴏɴ ʙᴇᴄᴏᴍᴇs:

1 * (1+ʀ)^ɴ = 2

USE:-

• ᴛʜɪs ғᴏʀᴍᴜʟᴀ ɪs ᴜsᴇғᴜʟ ғᴏʀ ғɪɴᴀɴᴄɪᴀʟ ᴇsᴛɪᴍᴀᴛᴇs ᴀɴᴅ ᴜɴᴅᴇʀsᴛᴀɴᴅɪɴɢ ᴛʜᴇ ɴᴀᴛᴜʀᴇ ᴏғ ᴄᴏᴍᴘᴏᴜɴᴅ ɪɴᴛᴇʀᴇsᴛ. ᴇxᴀᴍᴘʟᴇs:

1. ᴀᴛ 6% ɪɴᴛᴇʀᴇsᴛ, ʏᴏᴜʀ ᴍᴏɴᴇʏ ᴛᴀᴋᴇs 72/6 ᴏʀ 12 ʏᴇᴀʀs ᴛᴏ ᴅᴏᴜʙʟᴇ.

2. To ᴅᴏᴜʙʟᴇ ʏᴏᴜʀ ᴍᴏɴᴇʏ ɪɴ 10 ʏᴇᴀʀs, ɢᴇᴛ ᴀɴ ɪɴᴛᴇʀᴇsᴛ ʀᴀᴛᴇ ᴏғ 72/10 ᴏʀ 7.2%.

3. ɪғ ʏᴏᴜʀ ᴄᴏᴜɴᴛʀʏ’s ɢᴅᴘ ɢʀᴏᴡs ᴀᴛ 3% ᴀ ʏᴇᴀʀ, ᴛʜᴇ ᴇᴄᴏɴᴏᴍʏ ᴅᴏᴜʙʟᴇs ɪɴ 72/3 ᴏʀ 24 ʏᴇᴀʀs.

EXTRA:-

ᴛʜɪs ᴅᴇᴄᴇᴘᴛɪᴠᴇʟʏ sᴍᴀʟʟ, ᴄᴜᴍᴜʟᴀᴛɪᴠᴇ ɢʀᴏᴡᴛʜ ᴍᴀᴋᴇs ᴄᴏᴍᴘᴏᴜɴᴅ ɪɴᴛᴇʀᴇsᴛ ᴇxᴛʀᴇᴍᴇʟʏ ᴘᴏᴡᴇʀғᴜʟ – ᴇɪɴsᴛᴇɪɴ ᴄᴀʟʟᴇᴅ ɪᴛ ᴏɴᴇ ᴏғ ᴛʜᴇ ᴍᴏsᴛ ᴘᴏᴡᴇʀғᴜʟ ғᴏʀᴄᴇs ɪɴ ᴛʜᴇ ᴜɴɪᴠᴇʀsᴇ.