Question

When \[2\dfrac{1}{2}\] is added to a number and the sum is multiplied by \[4\dfrac{1}{2}\] and then \[3\] is added to the product and then the sum is divided by \[1\dfrac{1}{5}\] , the quotient becomes \[25\]. What is that number?
(A) \[2\dfrac{1}{2}\]
(B) \[3\dfrac{1}{2}\]
(C) \[4\dfrac{1}{2}\]
(D) \[5\dfrac{1}{2}\]

Answer 1

Hint: We use the concepts based on fractions to solve this problem. Proper, improper, and mixed fractions concepts are also used to solve this problem. We also use a variable in this problem, and to find the variable, we use transpositions also.

Complete step-by-step solution:
First of all, we will take a variable \[x\] which is equal to the number that we need to find. And let us form an equation with this variable as given in question.
So, in the first step, this number is added to \[2\dfrac{1}{2}\] .
So, the expression becomes as \[x + 2\dfrac{1}{2}\]
And this sum is multiplied by \[4\dfrac{1}{2}\] in the second step. So, the expression is written as,
\[\left( {x + 2\dfrac{1}{2}} \right) \times 4\dfrac{1}{2}\]
Now, this product is added to \[3\] , so the expression becomes as \[\left( {\left( {x + 2\dfrac{1}{2}} \right) \times 4\dfrac{1}{2}} \right) + 3\]
Now in the last step, this whole expression is divided by \[1\dfrac{1}{5}\] and the quotient is equal to \[25\] .
So, the equation is as follows,
\[\left[ {\left( {\left( {x + 2\dfrac{1}{2}} \right) \times 4\dfrac{1}{2}} \right) + 3} \right] \div 1\dfrac{1}{5} = 25\]
Now, we will convert all the mixed fractions to improper fractions for our convenience.
\[ \Rightarrow \left[ {\left( {\left( {x + \dfrac{5}{2}} \right) \times \dfrac{9}{2}} \right) + 3} \right] \div \dfrac{6}{5} = 25\]
Now, we will transpose every constant to the right hand side, and get the value of \[x\] .
\[ \Rightarrow \left[ {\left( {\left( {x + \dfrac{5}{2}} \right) \times \dfrac{9}{2}} \right) + 3} \right] = 25 \times \dfrac{6}{5}\]
\[ \Rightarrow \left( {\left( {x + \dfrac{5}{2}} \right) \times \dfrac{9}{2}} \right) + 3 = 30\]
After transposing three to right hand side,
\[ \Rightarrow \left( {\left( {x + \dfrac{5}{2}} \right) \times \dfrac{9}{2}} \right) = 30 - 3\]
\[ \Rightarrow \left( {x + \dfrac{5}{2}} \right) \times \dfrac{9}{2} = 27\]
Now, transpose \[\dfrac{9}{2}\]
\[ \Rightarrow \left( {x + \dfrac{5}{2}} \right) = 27 \times \dfrac{2}{9}\]
\[ \Rightarrow x + \dfrac{5}{2} = 6\]
And now, after transposing \[\dfrac{5}{2}\] to the right hand side, we get the final answer.
\[ \Rightarrow x = 6 - \dfrac{5}{2}\]
\[ \Rightarrow x = \dfrac{7}{2}\]
\[\therefore x = 3\dfrac{1}{2}\]
So, option (B) is the correct option.

Note: Make sure that you change all the mixed fractions to improper fractions so that you don’t go wrong at any of the steps. And in transpositions, positive terms get changed into negatives and vice-versa.
And similarly, multiplications change into divisions and vice-versa. Also, powers will change into their respective roots. Changing signs plays a key role in transposition. So, be careful while transposing. And also, don’t forget any steps while forming the equations.