Answer
Verified
425.4k+ views
Hint: From the given data we will form two linear equations in two variables(the two integers), and then solve the linear equations by using the substitution method to find the greatest number.
Complete step by step answer:
Given data: The product of two positive integers is 936.
The ratio of those integers is \[13:18.\]
Now, let us assume that the two integers are a and b, where ${\text{b > a}}$
From the given data, we can say that
$
ab = 936...........(i) \\
\dfrac{a}{b} = \dfrac{{13}}{{18}}..............(ii) \\
a = \dfrac{{13}}{{18}}b.............(iii) \\
$
On substituting the value of ‘a’ in equation (i), we will get
\[
(\dfrac{{13}}{{18}}b)b = 936 \\
\Rightarrow {b^2} = 936\left( {\dfrac{{18}}{{13}}} \right) \\
On{\text{ }}simplifying{\text{ }}we{\text{ }}get, \\
\Rightarrow {b^2} = 72(18) \\
\Rightarrow {b^2} = 1296 \\
On{\text{ }}taking{\text{ }}square{\text{ }}root{\text{ }}we{\text{ }}get,{\text{ }} \\
\Rightarrow b = \pm 36 \\
\]
But it is given that ‘a’ and ‘b’ are positive integers, giving us the value of ‘b’ i.e.
$b = 36$
Now, putting the value of ‘b’ in equation (iii)
$
a = \dfrac{{13}}{{18}}(36) \\
\Rightarrow a = 13(2) \\
\Rightarrow a = 26 \\
$
From the assumption we made it is clear the ‘b’ is the greater integer and ${\text{b = 36}}{\text{.}}$
Hence, option (C) is correct.
Note: While solving for ‘a’ and ‘b’ we can also substitute the value of ‘b’ as is shown below
$
ab = 936, \\
\Rightarrow \dfrac{a}{b} = \dfrac{{13}}{{18}} \\
\Rightarrow a = \dfrac{{13}}{{18}}b \\
\Rightarrow b = \dfrac{{18}}{{13}}a..........(iv) \\
$
On substituting the value of ‘b’ in equation (i), we will get
\[
{\text{a(}}\dfrac{{{\text{18}}}}{{{\text{13}}}}{\text{a) = 936}} \\
\Rightarrow {{\text{a}}^{\text{2}}}{\text{ = 936}}\left( {\dfrac{{{\text{13}}}}{{{\text{18}}}}} \right) \\
{\text{On simplification we get,}} \\
\Rightarrow {{\text{a}}^{\text{2}}}{\text{ = 52(13)}} \\
\Rightarrow {{\text{a}}^{\text{2}}}{\text{ = 676}} \\
{\text{On taking square root, we get,}} \\ \]
$\Rightarrow$ $a$ =$\pm$ $26$
But it is given that ‘a’ and ‘b’ are positive integers, giving us the value of ‘a’ i.e.
${\text{a = 26}}$
Now, putting the value of ‘a’ in equation (iv)
$
{\text{b = }}\dfrac{{{\text{18}}}}{{{\text{13}}}}{\text{(26)}} \\
\Rightarrow {\text{b = 18(2)}} \\
\Rightarrow {\text{b = 36}} \\
$
Additional information: Square root of any number always gives us the answer in pairs either it is of a real number or complex number. Here we also get a pair of values of ‘b’ but it is mentioned that the integers are positive, so we proceed further with the positive value of ‘b’.
Complete step by step answer:
Given data: The product of two positive integers is 936.
The ratio of those integers is \[13:18.\]
Now, let us assume that the two integers are a and b, where ${\text{b > a}}$
From the given data, we can say that
$
ab = 936...........(i) \\
\dfrac{a}{b} = \dfrac{{13}}{{18}}..............(ii) \\
a = \dfrac{{13}}{{18}}b.............(iii) \\
$
On substituting the value of ‘a’ in equation (i), we will get
\[
(\dfrac{{13}}{{18}}b)b = 936 \\
\Rightarrow {b^2} = 936\left( {\dfrac{{18}}{{13}}} \right) \\
On{\text{ }}simplifying{\text{ }}we{\text{ }}get, \\
\Rightarrow {b^2} = 72(18) \\
\Rightarrow {b^2} = 1296 \\
On{\text{ }}taking{\text{ }}square{\text{ }}root{\text{ }}we{\text{ }}get,{\text{ }} \\
\Rightarrow b = \pm 36 \\
\]
But it is given that ‘a’ and ‘b’ are positive integers, giving us the value of ‘b’ i.e.
$b = 36$
Now, putting the value of ‘b’ in equation (iii)
$
a = \dfrac{{13}}{{18}}(36) \\
\Rightarrow a = 13(2) \\
\Rightarrow a = 26 \\
$
From the assumption we made it is clear the ‘b’ is the greater integer and ${\text{b = 36}}{\text{.}}$
Hence, option (C) is correct.
Note: While solving for ‘a’ and ‘b’ we can also substitute the value of ‘b’ as is shown below
$
ab = 936, \\
\Rightarrow \dfrac{a}{b} = \dfrac{{13}}{{18}} \\
\Rightarrow a = \dfrac{{13}}{{18}}b \\
\Rightarrow b = \dfrac{{18}}{{13}}a..........(iv) \\
$
On substituting the value of ‘b’ in equation (i), we will get
\[
{\text{a(}}\dfrac{{{\text{18}}}}{{{\text{13}}}}{\text{a) = 936}} \\
\Rightarrow {{\text{a}}^{\text{2}}}{\text{ = 936}}\left( {\dfrac{{{\text{13}}}}{{{\text{18}}}}} \right) \\
{\text{On simplification we get,}} \\
\Rightarrow {{\text{a}}^{\text{2}}}{\text{ = 52(13)}} \\
\Rightarrow {{\text{a}}^{\text{2}}}{\text{ = 676}} \\
{\text{On taking square root, we get,}} \\ \]
$\Rightarrow$ $a$ =$\pm$ $26$
But it is given that ‘a’ and ‘b’ are positive integers, giving us the value of ‘a’ i.e.
${\text{a = 26}}$
Now, putting the value of ‘a’ in equation (iv)
$
{\text{b = }}\dfrac{{{\text{18}}}}{{{\text{13}}}}{\text{(26)}} \\
\Rightarrow {\text{b = 18(2)}} \\
\Rightarrow {\text{b = 36}} \\
$
Additional information: Square root of any number always gives us the answer in pairs either it is of a real number or complex number. Here we also get a pair of values of ‘b’ but it is mentioned that the integers are positive, so we proceed further with the positive value of ‘b’.
Recently Updated Pages
How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE
Why Are Noble Gases NonReactive class 11 chemistry CBSE
Let X and Y be the sets of all positive divisors of class 11 maths CBSE
Let x and y be 2 real numbers which satisfy the equations class 11 maths CBSE
Let x 4log 2sqrt 9k 1 + 7 and y dfrac132log 2sqrt5 class 11 maths CBSE
Let x22ax+b20 and x22bx+a20 be two equations Then the class 11 maths CBSE
Trending doubts
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
At which age domestication of animals started A Neolithic class 11 social science CBSE
Which are the Top 10 Largest Countries of the World?
Give 10 examples for herbs , shrubs , climbers , creepers
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
Difference Between Plant Cell and Animal Cell
Write a letter to the principal requesting him to grant class 10 english CBSE
Change the following sentences into negative and interrogative class 10 english CBSE
Fill in the blanks A 1 lakh ten thousand B 1 million class 9 maths CBSE