Question

$\dfrac{{15}}{\square }$ is a fraction that lies between $\dfrac{1}{7}$ and $\dfrac{1}{8}$. What is the missing whole number in the box?
$A)112$
$B)56$
$C)32$
$D)65$

Answer 1

Hint: First, we need to know about the concept of fractions. A fraction simply tells us how many parts of the whole number are there in reference to a given problem. Which can be expressed as numerator and denominator and the type are proper fractions, improper fractions, and mixed fractions.

Complete step-by-step solution:
Since from the given, we need to find the unknown value of $\dfrac{{15}}{\square }$. So, let us fix the unknown term to be $x$.
Also given that the value lies between the two values $\dfrac{1}{7}$ and $\dfrac{1}{8}$. Which can be represented mathematically as $\dfrac{1}{8} < \dfrac{{15}}{x} < \dfrac{1}{7}$ because between means, in between the two values, where $x$ is the unknown variable.
Now just divide all the values with the number $15$ then we have the relation as $\dfrac{1}{{8 \times 15}} < \dfrac{{15}}{{x \times 15}} < \dfrac{1}{{7 \times 15}}$
Now use the division operation and we have the relation as $\dfrac{1}{{8 \times 15}} < \dfrac{1}{x} < \dfrac{1}{{7 \times 15}}$
Now use the multiplication operation we have the relation as $\dfrac{1}{{120}} < \dfrac{1}{x} < \dfrac{1}{{105}}$
Hence finally taking the reciprocal of the values we get $120 > x > 105$
Hence the unknown value can be lie in between the whole numbers $106,107,108,109,110,111,112,113,114,115,116,117,118,119$
Now to check the answers given
Take the option $B)56$ which does not lie between the founded whole number and hence it is wrong
Take the option $C)32$ which does not lie between the founded whole number and hence it is wrong
Take the option $D)65$ which does not lie between the founded whole number and hence it is wrong
Now take the option $A)112$ which is lying between $106,107,108,109,110,111,112,113,114,115,116,117,118,119$ and hence which is the required answer. Thus $x = 112$ is the answer and the option $A)112$ is correct.

Note: After converting fractions to decimals
we know that $\dfrac{3}{4} = 0.75$ in this case, the dividend is exactly divisible after a few steps;
While in the process we get the remainder is zero, such decimal numbers are known as terminating decimals.
Now, look at this $\dfrac{2}{3} = 0.6666......$in some fractions the division does not stop and obtain a certain block of digits which is repeated over and over again. Such kind of decimals numbers are called recurring decimals.
The numbers which are all fall between integers and non-integers are called decimals
The ratio of two numbers and different ways to represent division is called Fractions