Answer

Verified

457.8k+ views

Hint: Make the fraction \[\dfrac{a}{b}\text{ and }\dfrac{c}{d}\] minimum by taking 2 smallest and 2 largest numbers from the set. We have to select a,b,c and d in such a format that numerator terms like a and c should be smaller and denominator terms like b and d should be bigger to find the minimum value.

We are given the set of numbers {1, 2, 3, ….., 9}. If a, b, c, d are four distinct numbers chosen from this set, then we have to find the minimum value of

\[L=\dfrac{a}{b}+\dfrac{c}{d}\]

To find the minimum value of \[\dfrac{a}{b}+\dfrac{c}{d}\], we have to choose 4 numbers such that \[\left( \dfrac{a}{b} \right)\text{ and }\left( \dfrac{c}{d} \right)\] have minimum values individually and hence \[\dfrac{a}{b}+\dfrac{c}{d}\] would also have minimum value.

Now, we know that if we take any fraction say, \[\dfrac{N}{D}\] where N is numerator and D is denominator and want to make it minimum, then we have to select the smallest possible number as N and biggest possible number as D.

Hence, to get minimum values of fractions \[\dfrac{a}{b}\text{ and }\dfrac{c}{d}\], we will select two largest numbers from the set {1, 2, 3…..9} and two smallest numbers from set {1, 2, 3…..9}

So, the two largest numbers are 8 and 9 and two smallest numbers are 1 and 2 from the set.

Since, we know that for \[\left( \dfrac{a}{b} \right)\text{ and }\left( \dfrac{c}{d} \right)\] to be minimum, a and c must be taken as numbers 1 and 2, while b and d must be taken as 8 and 9.

Now, let us put a = 1 and c = 2. Therefore, we get

\[L=\dfrac{a}{b}+\dfrac{c}{d}=\dfrac{1}{b}+\dfrac{2}{d}\]

Now if b = 9 and d = 8, we get,

\[\begin{align}

& L=\dfrac{1}{9}+\dfrac{2}{8} \\

& =\dfrac{8+18}{72} \\

& =\dfrac{26}{72} \\

& =\dfrac{13}{36} \\

& =0.36111 \\

\end{align}\]

Therefore, we get L = 0.36111 (Approx)

Now, if b= 8 and d = 9. We get,

\[L=\dfrac{1}{8}+\dfrac{2}{9}\]

\[\begin{align}

& =\dfrac{9+16}{72} \\

& =\dfrac{25}{72} \\

& =0.34722 \\

\end{align}\]

Therefore, in this case we get L = 0.34722 (Approx)

As we can see that,

\[0.36111>0.34722\]

Or, \[\dfrac{13}{36}>\dfrac{25}{72}\]

Therefore we get minimum values of \[\dfrac{a}{b}+\dfrac{c}{d}\] as \[\dfrac{25}{72}\].

Hence, option (d) is correct.

Note: Here, some students take \[\dfrac{a}{b}\text{ as }\dfrac{1}{9}\] and \[\dfrac{c}{d}\text{ as }\dfrac{2}{8}\] and get the wrong answer \[\dfrac{13}{36}\] which is option (c). But they must keep in mind that we not only have to make \[\dfrac{a}{b}\text{ and }\dfrac{c}{d}\] minimum but we also need to make \[\left( \dfrac{a}{b}+\dfrac{c}{d} \right)\] minimum. Therefore, we take \[\dfrac{a}{b}=\dfrac{2}{9}\text{ and }\dfrac{c}{d}=\dfrac{1}{8}\] which makes \[\dfrac{a}{b}+\dfrac{c}{d}\] minimum.

__Complete step-by-step answer:__We are given the set of numbers {1, 2, 3, ….., 9}. If a, b, c, d are four distinct numbers chosen from this set, then we have to find the minimum value of

\[L=\dfrac{a}{b}+\dfrac{c}{d}\]

To find the minimum value of \[\dfrac{a}{b}+\dfrac{c}{d}\], we have to choose 4 numbers such that \[\left( \dfrac{a}{b} \right)\text{ and }\left( \dfrac{c}{d} \right)\] have minimum values individually and hence \[\dfrac{a}{b}+\dfrac{c}{d}\] would also have minimum value.

Now, we know that if we take any fraction say, \[\dfrac{N}{D}\] where N is numerator and D is denominator and want to make it minimum, then we have to select the smallest possible number as N and biggest possible number as D.

Hence, to get minimum values of fractions \[\dfrac{a}{b}\text{ and }\dfrac{c}{d}\], we will select two largest numbers from the set {1, 2, 3…..9} and two smallest numbers from set {1, 2, 3…..9}

So, the two largest numbers are 8 and 9 and two smallest numbers are 1 and 2 from the set.

Since, we know that for \[\left( \dfrac{a}{b} \right)\text{ and }\left( \dfrac{c}{d} \right)\] to be minimum, a and c must be taken as numbers 1 and 2, while b and d must be taken as 8 and 9.

Now, let us put a = 1 and c = 2. Therefore, we get

\[L=\dfrac{a}{b}+\dfrac{c}{d}=\dfrac{1}{b}+\dfrac{2}{d}\]

Now if b = 9 and d = 8, we get,

\[\begin{align}

& L=\dfrac{1}{9}+\dfrac{2}{8} \\

& =\dfrac{8+18}{72} \\

& =\dfrac{26}{72} \\

& =\dfrac{13}{36} \\

& =0.36111 \\

\end{align}\]

Therefore, we get L = 0.36111 (Approx)

Now, if b= 8 and d = 9. We get,

\[L=\dfrac{1}{8}+\dfrac{2}{9}\]

\[\begin{align}

& =\dfrac{9+16}{72} \\

& =\dfrac{25}{72} \\

& =0.34722 \\

\end{align}\]

Therefore, in this case we get L = 0.34722 (Approx)

As we can see that,

\[0.36111>0.34722\]

Or, \[\dfrac{13}{36}>\dfrac{25}{72}\]

Therefore we get minimum values of \[\dfrac{a}{b}+\dfrac{c}{d}\] as \[\dfrac{25}{72}\].

Hence, option (d) is correct.

Note: Here, some students take \[\dfrac{a}{b}\text{ as }\dfrac{1}{9}\] and \[\dfrac{c}{d}\text{ as }\dfrac{2}{8}\] and get the wrong answer \[\dfrac{13}{36}\] which is option (c). But they must keep in mind that we not only have to make \[\dfrac{a}{b}\text{ and }\dfrac{c}{d}\] minimum but we also need to make \[\left( \dfrac{a}{b}+\dfrac{c}{d} \right)\] minimum. Therefore, we take \[\dfrac{a}{b}=\dfrac{2}{9}\text{ and }\dfrac{c}{d}=\dfrac{1}{8}\] which makes \[\dfrac{a}{b}+\dfrac{c}{d}\] minimum.

Recently Updated Pages

When people say No pun intended what does that mea class 8 english CBSE

Name the states which share their boundary with Indias class 9 social science CBSE

Give an account of the Northern Plains of India class 9 social science CBSE

Change the following sentences into negative and interrogative class 10 english CBSE

Advantages and disadvantages of science

10 examples of friction in our daily life

Trending doubts

Which are the Top 10 Largest Countries of the World?

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE

The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths

Change the following sentences into negative and interrogative class 10 english CBSE

Difference Between Plant Cell and Animal Cell

Write a letter to the principal requesting him to grant class 10 english CBSE