Question

x, y, z are three sums of money such that y is the simple interest on x, and z is the simple interest on y for the same time and same rate. The relation between x, y, and z is
A) ${x^2} = yz$
B) ${z^2} = xy$
C) ${y^2} = xy$
D) $xyz = 1$

Answer 1

Hint: First assume the rate of interest and time. Then, find the simple interest on x and equate it to y. After that find the simple interest on y and equate to z. Then, find the ratio of y and z. After that do simplification to get the desired result.

Complete step-by-step answer:
Let the rate of interest be R% and the time be T years.
Simple interest is a quick and easy method of calculating the interest charge on a loan. Simple interest is determined by multiplying the daily interest rate by the principal by the number of days that elapse between payments. Mathematically it is given by the formula:
$S.I. = \dfrac{{P \times R \times T}}{{100}}$
Where P = Principal
R = Rate of interest
T = Time
Simple interest is calculated on the principal portion of a loan or the original contribution to a savings account. Simple interest does not compound, meaning that an account holder will only gain interest on the principal, and a borrower will never have to pay interest on interest already occurred.
Find the simple interest on x,
$ \Rightarrow S.I. = \dfrac{{x \times R \times T}}{{100}}$
Now, equate it to y,
$ \Rightarrow y = \dfrac{{x \times R \times T}}{{100}}$...........….. (1)
Find the simple interest on y,
$ \Rightarrow S.I. = \dfrac{{y \times R \times T}}{{100}}$
Now, equate it to z,
$ \Rightarrow z = \dfrac{{y \times R \times T}}{{100}}$..........….. (2)
Divide equation (1) by (2),
$ \Rightarrow \dfrac{y}{z} = \dfrac{{\dfrac{{x \times R \times T}}{{100}}}}{{\dfrac{{y \times R \times T}}{{100}}}}$
Cancel out the common factors,
$ \Rightarrow \dfrac{y}{z} = \dfrac{x}{y}$
Cross-multiply the terms,
$\therefore {y^2} = xz$

Hence, option (C) is correct.

Note: In this question, use the concept of Simple interest which is a quick and easy method of calculating the interest charge on a loan. Simple interest is determined by multiplying the daily interest rate by the principal by the number of days that elapse between payments.
Simple interest is based on the principal amount. Compound interest is based on the principal amount as well as the interest.