Question

Two numbers are in the ratio \[5:6\] .If the sum of numbers is \[66\], What is the value of the larger number?

Answer 1

Hint: We are provided with the ratio of two numbers and their sum. In this question we have to find the largest number. We will assume the common factor of both the numbers to be \[x\]. Therefore, as the two numbers are in the ratio \[5:6\] implies that \[5x\] and \[6x\] are the two numbers. Then, the sum of the two numbers is given as \[66\], so we will form an equation and evaluate the value of \[x\] from that equation. After that we will substitute the value of \[x\] in the two numbers \[5x\] and \[6x\] which will give us the required answer.

Complete step by step solution:
Let us assume the common factor of both the numbers to be \[x\].
Now we have given that the two numbers are in the ratio \[5:6\]
Therefore, the two numbers are represented by \[5x\] and \[6x\] respectively
Now on taking sum of two numbers, we get
\[ \Rightarrow 5x + 6x = 11x\]
Now we have given the sum of numbers is \[66\]
Therefore, on equating we get
\[11x = 66\]
On dividing the above equation by \[11\] on both the sides, we get
\[ \Rightarrow x = 6\]
Now we will find the values of the two numbers by substituting the value of \[x\] in \[5x\] and \[6x\]
Hence, the first number is \[5x = 5 \times 6 = 30\]
And the second number is \[6x = 6 \times 6 = 36\]
Therefore, the two numbers are \[30\] and \[36\]
Now in our question we are asked to find the larger number
Here \[36\] is greater than \[30\]
Hence, the larger number is \[36\].

Note:
The alternate way to solve this question can be as follows:
Let us assume that the two numbers are \[x\] and \[y\] with \[y > x\]
Now we are given that the ratio of the two numbers is \[5:6\]
This means that,
\[\dfrac{x}{y} = \dfrac{5}{6}\]
On cross multiplying we get
\[6x = 5y\]
\[ \Rightarrow 6x - 5y = 0{\text{ }} - - - \left( i \right)\]
We are also given that the sum of two numbers is \[66\]
\[ \Rightarrow x + y = 66{\text{ }} - - - \left( {ii} \right)\]
We will now solve equation \[\left( i \right)\] and \[\left( {ii} \right)\] by using elimination method,
So, multiplying equation \[\left( {ii} \right)\] by \[5\] we get
\[ \Rightarrow 5x + 5y = 330{\text{ }} - - - \left( {iii} \right)\]
Adding equation \[\left( i \right)\] and \[\left( {iii} \right)\] we get
\[11x = 330\]
\[ \Rightarrow x = 30\]
Thus, one number is \[30\]
Now let us back substitute the value of \[x\] in equation \[\left( {ii} \right)\] we get
\[ \Rightarrow 30 + y = 66\]
\[ \Rightarrow y = 36\]
Thus, second number is \[36\]
Therefore, the two numbers are \[30\] and \[36\]
Now in our question we are asked to find the larger number
Here \[36\] is greater than \[30\]
Hence, the larger number is \[36\]