Question

Participation in team sports in a certain college was up 25 percent at the end of last year, but participation in individual sports like tennis was down 25 percent. The ratio of students participating in team sports to students taking part in individual sports was how many times greater at the end of the year than the ratio at the beginning of the year?
 A.$\dfrac{2}{3}$
B. $\dfrac{5}{3}$
C. $\dfrac{5}{6}$
D.None

Answer 1

Hint: We can take the number of students in team sports and individual sports at the beginning of the year as 2 variables. Then we can take their ratio. Using the decrement and increment percentage, we can find the number of students in team sports and individual sports at the end of the year. Using these we can find the new ratio. Then we can divide the two ratios to get the required ratio.

Complete step-by-step answer:
Let x be the number of students participating in team sports and y be the number of students participating in individual sports at the beginning of the year.
So the ratio of students participating in team sports to students taking part in individual sports at the beginning of the year is given by
 ${R_b} = \dfrac{x}{y}$
At the end of the year,
The number up students in team sports increases by 25 percent. So, x becomes.
 $x' = x + 0.25x$
 $ \Rightarrow x' = 1.25x$
It is given that the participation in individual sports decreased by 25 percent. So y becomes,
 $y' = y - 0.25y$
 $ \Rightarrow y' = 0.75y$
Now we can take the ratio of students participating in team sports to students taking part in individual sports at the end of the year.
 $ \Rightarrow {R_2} = \dfrac{{x'}}{{y'}}$
On substituting the values, we get,
 $ \Rightarrow {R_2} = \dfrac{{1.25x}}{{0.75y}}$
Now we need to find the ratio of the ratio at the end of the year to the ratio of beginning of the year. So we can divide the new ratio with old ratio to get the required ratio.
 $ \Rightarrow R = \dfrac{{{R_2}}}{{{R_1}}}$
On substituting the values, we get,
 $ \Rightarrow R = \dfrac{{\dfrac{{1.25x}}{{0.75y}}}}{{\dfrac{x}{y}}}$
On cancelling the common terms, we get,
 $ \Rightarrow R = \dfrac{{125}}{{75}}$
On further simplification, we get,
 $ \Rightarrow R = \dfrac{5}{3}$
So the required ratio is $\dfrac{5}{3}$ .
Therefore, the correct answer is option B.

Note: Ratios are used to express the quantity of one object compared to another. In this problem we took the ratio of the ratios. While taking the ratio, the order is important. So we need to read the question carefully and do the divisions accordingly. As the questions come with options, even if the order is reversed, we can find the correct answer by checking with the options. We must assign the variables to the number of students who participate at the beginning of the year and find the participation in the end of the year using the given condition.