Question

An English word consists of 9 alphabets. The sum of twice the number of vowels and three times the number of consonants present in the word is equal to four more than four times the total number of vowels in the English alphabets. The product of the number of vowels and consonants present in the word is
A) 9
B) 20
C) 18
D) 14

Answer 1

Hint: Here we will assume the number of vowels as x and the number of consonants as y and then we will form the linear equations in two variables using the given information in the question and then solve them to get the value of x and y and then finally multiply them to get the desired answer.

Complete step-by-step answer:
Let the number of vowels be x
Let the number of consonants be y.
Now it is given that there are total 9 alphabets
Therefore, \[x + y = 9\]………………….. (1)
Now it is given that the sum of twice the number of vowels and three times the number of consonants present in the word is equal to four more than four times the total number of vowels in the English alphabets.
We know that there are 5 vowels in English alphabets.
Hence, forming the required linear equation using the above information we get:-
\[2x + 3y = 4 + \left( {4 \times 5} \right)\]
Simplifying it further we get:-
$ \Rightarrow$\[2x + 3y = 4 + 20\]
\[ \Rightarrow 2x + 3y = 24\]………………………… (2)
Now we will solve the equations 1 and 2 to get the values of x and y.
Multiplying equation1 by 2 we get:-
$ \Rightarrow$\[2x + 2y = 18\]…………………………. (3)
Now subtracting equation 3 from equation 2 we get:-
$ \Rightarrow$\[2x + 3y - 2x - 2y = 24 - 18\]
Simplifying it further we get:-
$ \Rightarrow$\[y = 6\]
Putting this value in equation 1 we get:-
$ \Rightarrow$\[x + 6 = 9\]
Solving for the value of x we get:-
$ \Rightarrow$\[x = 3\]
Now we have to find the product of the number of vowels and consonants present in the word.
Hence, we need to multiply the values of x and y to get the desired value.
\[xy = 3 \times 6\]
\[ \Rightarrow xy = 18\]

Hence, option C is the correct option.

Note: Students should note that the linear equation in one variable has only one variable with highest power 1 while the linear equation in two variables has two variables each with highest power as 1.