Question

The difference of the two numbers is 4. If the difference of their reciprocal is \[\dfrac{4}{{21}}\], find the numbers.
A. 7 and 3
B. 7 and \[ - 3\]
C. \[ - 7\] and \[ - 3\]
D. Both A and C

Answer 1

Hint: We will first let the two numbers be \[a\] and \[b\]. We are given that the difference between two numbers as 14 so we will form an equation from this. The other equation can be formed as the difference of their reciprocal is given by \[\dfrac{4}{{21}}\]. Then we will simplify the second equation and use the substitution method to simplify the equation and find the values of two numbers.

Complete step by step answer:

We will first consider the two numbers be \[a\] and \[b\]. So, the reciprocal of these two numbers bs \[\dfrac{1}{a}\] and \[\dfrac{1}{b}\].
As the difference of two numbers be given as 14 so, we get,
\[ \Rightarrow a - b = 4\]-(1)
Now, we know that the difference of the reciprocals of two numbers be \[\dfrac{4}{{21}}\] so, we get,
\[
   \Rightarrow \dfrac{1}{b} - \dfrac{1}{a} = \dfrac{4}{{21}} \\
   \Rightarrow \dfrac{{a - b}}{{ab}} = \dfrac{4}{{21}} \\
 \]
Since we have taken \[\dfrac{1}{b} - \dfrac{1}{a}\] as we have consider \[a > b\] so the reciprocal will be opposite of both the numbers that is \[\dfrac{1}{b} > \dfrac{1}{a}\]
Hence, from this we get that \[ab = 21\]-(2)
Now, we will substitute the value of \[b\] from equation (2) into equation (1).
Thus, we get,
\[
   \Rightarrow a - \dfrac{{21}}{a} = 4 \\
   \Rightarrow {a^2} - 4a - 21 = 0 \\
 \]
Now, we will use the middle term splitting method to determine the value of \[a\].
Thus, we get,
\[
   \Rightarrow {a^2} - 7a + 3a - 21 = 0 \\
   \Rightarrow \left( {a - 7} \right) + 3\left( {a - 7} \right) = 0 \\
   \Rightarrow \left( {a - 7} \right)\left( {a + 3} \right) = 0 \\
 \]
Next, we will apply the zero-factor property to determine the values of \[a\].
Thus, we get,
\[ \Rightarrow a - 7 = 0\] and \[a + 3 = 0\]
\[ \Rightarrow a = 7\] and \[a = - 3\]
Thus, we will substitute both the values of \[a\] in equation (2) to determine the values of \[b\].
Thus, we get,
\[
   \Rightarrow b = \dfrac{{21}}{7} \\
   \Rightarrow b = 3 \\
 \] and \[
   \Rightarrow b = \dfrac{{21}}{{ - 3}} \\
   \Rightarrow b = - 7 \\
 \]
Thus, we get the two pairs of the values of \[a\] and \[b\] as one pair is 7 and 3, the other pair is \[ - 3\] and \[ - 7\].
Hence, option D is correct.

Note: We have considered in the second equation that \[ab = 21\] that is the denominator of both the sides, as the numerator is already obtained. Since we are getting the two values for \[a\] so, we will consider both the values as negative value is not ignored. We have used the middle term splitting method to determine the values so we have to remember that method. Do remember that if we have considered \[a > b\] then its reciprocal is \[\dfrac{1}{b} > \dfrac{1}{a}\]. Do the calculation part carefully to avoid calculation errors.