Question

If numerator and denominator of a proper fractions are increased by the same quantity, then the resulting factor is then:
a) Always greater than the original fraction.
b) Always less than the original fraction.
c) Always equal to the original fraction.
d) None of these

Answer 1

Hint: To solve this question, we will take up different cases having different proper fractions and we will add constant In both numerator and denominator of that proper fraction and then we will compare the original proper fraction and the new fraction.

Complete step-by-step answer:
Here, we are given that we have to increase the numerator and denominator of a proper fraction. So first we must know what is a proper fraction. A proper fraction is a type of fraction where the numerator i.e. the top number is less than the denominator i.e. the bottom number. So, if a fraction is represented by the numerator $\dfrac{p}{q}$ then q is greater than p.
Now the question demands that we have to add a constant in both the numerator and denominator. Let the constant be ‘a’. now we are going to consider some cases to see what happens when x is added to both the numerator and denominator $\dfrac{p}{q}$:
Case I: let the value of p be 2 and q be 7. Now we consider the value of a=4. Now the initial fraction= $\dfrac{2}{7}$. Now the new fraction becomes= $\dfrac{2+4}{7+4}=\dfrac{6}{11}$. The value of fraction is = $\dfrac{2}{7}=0.285$. The value of the new fraction is = $\dfrac{6}{11}=0.545$.
Case II: let the value of p be 1 and q be 3. Now we consider the value of a=3. Now the initial fraction= \[\dfrac{1}{3}\]. Now the new fraction becomes= \[\dfrac{1+3}{3+3}=\dfrac{4}{6}\]. The values of initial fraction are = $\dfrac{1}{3}=0.33$. The values of the new fraction are= $\dfrac{4}{6}=0.67$.
In both the cases, we can see that the value of the new fraction is greater than the initial fraction. Hence the value of proper fraction increases.
Hence, option (a) is correct.

Note: We cannot say that the original fraction will be smaller than the resulting fraction when the fraction is not proper. In that case the fraction may increase or remain the same.