Question

What is the simplified value of : ${(1.25)^3} - 2.25{(1.25)^3} + 3.75{(0.75)^2} - {(0.75)^3}$
(A) $\dfrac{1}{8}$
(B) $\dfrac{1}{{18}}$
(C) $\dfrac{1}{2}$
(D) None

Answer 1

Hint: In mathematics, we have the integers and the rational numbers. An integer is the type of numbers which can be positive, negative, and even zero, but integers cannot have fractional parts or decimal places. And a rational number is usually a fraction number and a rational number is defined by $\dfrac{p}{q}$ where $p,q$ are both integers and $\gcd \left( {p,q} \right) = 1$ .

Complete step-by-step answer:
Now there is a simple yet effective method to convert a decimal number into a rational number.
To convert a decimal into a rational number or to a fraction, we observe the number of digits occur after the decimal point. For example, in this case, we are given $1.25$. We see there are two digits after the decimal point. Thus we have, $\dfrac{{125}}{{100}}$ as the fraction of $1.25$. Note that for the two digits after the decimal place, we got two zeros after $1$ in the denominator i.e., we got $100$ in the denominator.
Now given, ${(1.25)^3} - 2.25{(1.25)^3} + 3.75{(0.75)^2} - {(0.75)^3}$$ = {\left( {\dfrac{{125}}{{100}}} \right)^3} - \left( {\dfrac{{225}}{{100}}} \right) \times {\left( {\dfrac{{125}}{{100}}} \right)^3} + \left( {\dfrac{{375}}{{100}}} \right) \times {\left( {\dfrac{{75}}{{100}}} \right)^2} - {\left( {\dfrac{{75}}{{100}}} \right)^3}$
$ = {\left( {\dfrac{{125}}{{100}}} \right)^3}\left[ {\dfrac{{100 - 225}}{{100}}} \right] + {\left( {\dfrac{{75}}{{100}}} \right)^2}\left[ {\dfrac{{375 - 75}}{{100}}} \right]$
$ = {\left( {\dfrac{5}{4}} \right)^3}\left[ {\dfrac{{ - 125}}{{100}}} \right] + {\left( {\dfrac{3}{4}} \right)^2}\left[ {\dfrac{{300}}{{100}}} \right]$
$ = \left( {\dfrac{{125}}{{64}}} \right) \times \left( {\dfrac{{ - 5}}{4}} \right) + \left( {\dfrac{9}{{16}}} \right) \times 3$
$ = \dfrac{{ - 625}}{{256}} + \dfrac{{27}}{{16}}$
$ = \dfrac{{ - 625 + \left( {27 \times 16} \right)}}{{256}}$
$ = \dfrac{{ - 625 + 432}}{{256}}$
$ = \dfrac{{ - 193}}{{256}}$ which is not equal to any of the options given.
Therefore option (D) None is correct.
So, the correct answer is “Option D”.

Note: Students should have a proper concept and proper practice of solving these fractional numerical problems. There is a normal and simple way to solve these problems applying the basic algebraic formulae. They should observe the given sum to do this. After observing students can relate the necessary and required formula to simplify the shape of the sum. But in this particular case we are not given the option to apply the formula, so we just solve this in the basic calculation method.