Question

In a cricket match Dhoni and Dravid scored 185 runs. If the score was in the ratio \[18:19\]. How many runs did each of them score?

Answer 1

Hint: Here we assume the score made by Dhoni and Dravid as separate variables. Then using the information in the question we form an equation of sum of scores. Also, using the concept of ratio, we write the scores made by Dhoni and Dravid in the ratio form and equate it to the given ratio.
* Ratio \[m:n\] can be written in the form of fraction as \[\dfrac{m}{n}\].

Complete step-by-step answer:
Let us assume a score made by Dhoni as x and score made by Dravid as y.
Then from the statement of question we know that Dhoni and Dravid scored 185 runs.
So we can write the sum of scores made by Dhoni and Dravid equal to 185.
\[ \Rightarrow x + y = 185\] … (1)
Now we are given that the scores are in the ratio \[18:19\].
So we can write that the ratio of scores made by Dhoni to Dravid is \[18:19\].
Substitute the values of scores we assumed in the beginning of the solution.
\[ \Rightarrow x:y = 18:19\] … (2)
Now since we know we can convert the ratio \[a:b\]into a form of fraction as \[\dfrac{a}{b}\].
Therefore, we can write the equation (2) as
\[ \Rightarrow \dfrac{x}{y} = \dfrac{{18}}{{19}}\]
Multiply both sides of the equation by y
\[ \Rightarrow \dfrac{x}{y} \times y = \dfrac{{18}}{{19}} \times y\]
Cancel the same terms from numerator and denominator.
\[ \Rightarrow x = \dfrac{{18}}{{19}}y\] … (3)
Substitute the value of x from equation (3) in equation (1).
\[ \Rightarrow \dfrac{{18y}}{{19}} + y = 185\]
Take LCM on the left hand side of the equation.
\[ \Rightarrow \dfrac{{18y + 19y}}{{19}} = 185\]
Calculate the sum in the numerator.
\[ \Rightarrow \dfrac{{37y}}{{19}} = 185\]
Multiply both sides by \[\dfrac{{19}}{{37}}\]
\[ \Rightarrow \dfrac{{37y}}{{19}} \times \dfrac{{19}}{{37}} = 185 \times \dfrac{{19}}{{37}}\]
Cancel out the same terms from numerator and denominator.
\[ \Rightarrow y = 5 \times 19\]
Calculate the product.
\[ \Rightarrow y = 95\]
Now we substitute the value of y in equation (1) to calculate the value of x.
\[ \Rightarrow x + 95 = 185\]
Shift all constants to one side of the equation.
\[ \Rightarrow x = 185 - 95\]
Calculate the value on RHS.
\[ \Rightarrow x = 90\]

So, the score made by Dhoni is 90 runs and the score made by Dravid is 95 runs.

Note: Students many times make the equation formed by the ratio as a complex equation when they cross multiply the values to both sides and then solve. Always try to keep that value on one side of the equation which can later be directly substituted in another equation. Also, keep in mind ratio should always be in simplest form i.e. there should not be any common factor between numerator and denominator.