Compound Interest Relationships Explained
The question asks us to determine the correct mathematical connection between three amounts: \(x\), \(y\), and \(z\). We are given that \(y\) is the compound interest calculated on \(x\), and \(z\) is the compound interest calculated on \(y\). A key detail is that the annual interest rate and the time period (in years) are the same for both calculations.
Compound Interest Formula Basics
To solve this, we first recall the formula for compound interest. If \(P\) is the principal amount, \(r\) is the annual interest rate, and \(t\) is the time period in years, the compound interest (\(CI\)) is given by:
\(CI = P \left[ \left(1 + r\right)^t - 1 \right]\)
Here, \(P \left(1 + r\right)^t\) represents the total amount (principal + interest) after time \(t\). The term \(\left[ \left(1 + r\right)^t - 1 \right]\) is a multiplier that depends only on the rate and time.
Deriving x, y, z Relationship
Let's use the given information and the compound interest formula. Let the common rate be \(r\) and the common time period be \(t\). We can represent the term \(\left(1 + r\right)^t - 1\) as a constant factor, let's call it \(k\). So, \(k = \left(1 + r\right)^t - 1\). This factor \(k\) is the same in both interest calculations because the rate and time are the same.
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Amount y Calculation:
We are told \(y\) is the compound interest on \(x\). Here, \(x\) acts as the principal (\(P=x\)). Applying the formula:
\(y = x \times k\)
This means \(y\) is \(x\) multiplied by the interest factor \(k\).
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Amount z Calculation:
We are told \(z\) is the compound interest on \(y\). Here, \(y\) acts as the principal (\(P=y\)). Applying the formula again with the same factor \(k\):
\(z = y \times k\)
This means \(z\) is \(y\) multiplied by the same interest factor \(k\).
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Finding the Relationship:
We now have two equations:
- Equation 1: \(y = xk\)
- Equation 2: \(z = yk\)
Our goal is to find a relationship between \(x\), \(y\), and \(z\) that does not involve \(k\). We can do this by expressing \(k\) from each equation and setting them equal.
From Equation 1, if \(x\) is not zero, we get:
\(k = \frac{y}{x}\)
From Equation 2, if \(y\) is not zero, we get:
\(k = \frac{z}{y}\)
Since both expressions equal \(k\), they must be equal to each other:
\(\frac{y}{x} = \frac{z}{y}\)
To simplify this proportion, we cross-multiply:
\(y \times y = z \times x\)
Which simplifies to:
\(y^2 = zx\)
This derived relationship \(y^2 = zx\) shows how the amounts are connected when the compound interest is calculated successively under the same rate and time conditions.
Similar Questions
A train of certain length takes time t to pass completely through a station of length x. The same train with the same speed takes time 2t to pass completely through another station of length y. What is the time taken by the train to pass completely through a station of length (x + y)?
How much will ₹10,000 amount to in one year's time at 4% rate of interest per annum if the interest is compounded once in every three months? (take approximate value)
- An amount of ₹10,000 is borrowed at 10% per annum on compound interest for 3 years, compounded annually, and paid back in 3 equal annual installments during these years. What is the amount of each installment (approximately)?
Important Questions from Compound Interest
A person borrowed Rs. 10000 on compound interest at the rate of 40 percent per annum. If the interest is compounded half yearly, then what will be the amount to be paid after 1.5 years?
The difference between the compound interest (compounding annually) and the simple interest on a sum of money at the rate of 40 per cent per annum for 2 years is Rs. 2400. What is the amount?
In how many years will a sum of Rs.1875 amount to Rs.2187 at 8 percent p.a. compound interest?
A sum of money has increased by 45% in 9 years at simple interest. What will be the compound interest of Rs. 12,000 after 3 years at the same rate?
At a certain rate of compound interest a certain sum amounts to Rs. 64800 in 4 years and Rs. 93312 in 6 years. What is the compound interest earned in fifth year?
