FDP Problem

[yuvarajait]

Question : Last year, all registered voters in Kumannia voted either for the Revolutionary Party or for
the Status Quo Party. This year, the number of Revolutionary voters increased 10%, while
the number of Status Quo voters increased 5%. No other votes were cast. If the number of
total voters increased 8%, what fraction of voters voted Revolutionary this year?

The answer says 11/18. I couldn't catch the way its derived. Can someone help explaining this?

[Night reader]

increase in the number of total voters --> 1.08x
increase in the number voted for Revolutionary --> 1.1y
increase in the number voted for Status Quo --> 1.05z

Question: find Revolutionary/total voters?

solution: 1.08x=1.1y+1.05z; last year x=y+z; 1.08x=1.08y+1.08z OR 1.08y+1.08z=1.1y+1.05z <> 0.03z=0.02y OR 3z=2y, z=2y/3. Now rewriting x=y+z OR x=y+2y/3, x=5y/3 AND 1.1y/1.08x --> 1.1y : ((5y/3)*1.08)= 11y/10 : (5.4y/3)=
=(11y/10) * (3/5.4y)= 33y/54y= 11/18

Question : Last year, all registered voters in Kumannia voted either for the Revolutionary Party or for
the Status Quo Party. This year, the number of Revolutionary voters increased 10%, while
the number of Status Quo voters increased 5%. No other votes were cast. If the number of
total voters increased 8%, what fraction of voters voted Revolutionary this year?

The answer says 11/18. I couldn't catch the way its derived. Can someone help explaining this?

[Anurag@Gurome]

Question : Last year, all registered voters in Kumannia voted either for the Revolutionary Party or for
the Status Quo Party. This year, the number of Revolutionary voters increased 10%, while
the number of Status Quo voters increased 5%. No other votes were cast. If the number of
total voters increased 8%, what fraction of voters voted Revolutionary this year?

The answer says 11/18. I couldn't catch the way its derived. Can someone help explaining this?

Solution:
Let the number of revolutionary voters be 'r' and the number of Status Quo voters be 's'.
So, increase in the number of revolutionary voters is (10/100) * r = 0.1r.
Increase in number of Status Quo voters is (5/100) * s = 0.05s.
This is equal to (8/100) * (r+s) = 0.08(r+s).
So, (10/100) * r + (5/100) * s = (8/100) * (r+s).
Or, 10r + 5s = 8r + 8s.
2r = 3s.
Or s/r = 2/3.
Or r/(r+s) = 3/5.
We need to know 1.1r/{1.08(r+s)} which is (110/108) * r/(r+s) = (110/108) * 3/5 = 11/18.

[GMATGuruNY]

Question : Last year, all registered voters in Kumannia voted either for the Revolutionary Party or for
the Status Quo Party. This year, the number of Revolutionary voters increased 10%, while
the number of Status Quo voters increased 5%. No other votes were cast. If the number of
total voters increased 8%, what fraction of voters voted Revolutionary this year?

The answer says 11/18. I couldn't catch the way its derived. Can someone help explaining this?

This is a weighted average question.
Let R = Revolutionary voters and S = Status Quo voters.

A 10% increase in R is being combined with a 5% increase in S to yield a mixture with an increase of 8%.

We can use alligation, which dictates the following:

The proportion of each element in the mixture is equal to the distance between the percentage attributed to the other element in the mixture and the percentage attributed to the mixture.

Proportion of R = |percentage in S - percentage in mixture| = |5-8| = 3.
Proportion of S = |percentage in R - percentage in mixture| = |10-8| = 2.
Thus, last year R:S = 3:2.

To determine the ratio of R to S this year, we can plug in values for R and S.

Let R last year = 30 and S last year = 20.
R increased by 10% = (1.1)*30 = 33.
S increased by 5% = (1.05)*20 = 21.
R/(R+S) = 33/(33+21) = 33/54 = 11/18.

[kamalakarthi]

Mitch,

Thanks for the great idea. However I am still not able to understand this..

The proportion needed of R (10%) = 8-5 = 3.
The proportion needed of S (5%) = 10-8 = 2.



How did u get 8-5 , and same for 10-8...Can you please help us understand this better. Thanks in advance.

[GMATGuruNY]

Mitch,

Thanks for the great idea. However I am still not able to understand this..

The proportion needed of R (10%) = 8-5 = 3.
The proportion needed of S (5%) = 10-8 = 2.



How did u get 8-5 , and same for 10-8...Can you please help us understand this better. Thanks in advance.

The process is called alligation.

When an entity that is X% (a higher percentage) is combined with an entity that is Y% (a lower percentage) to form a mixture that is Z%:

The proportion needed of the X% entity = Z-Y (the distance between the other 2 percentages).
The proportion needed of the Y% entity = X-Z (the distance between the other 2 percentages).

The resulting values give the ratio of X:Y in the mixture.

In the problem above:
R = 10% is combined with S = 5%.
The resulting mixture = 8%.

Thus:
The proportion of R in the population = the distance between 8% and S = 8-5 = 3.
The proportion of S in the population = the distance between R and 8% = 10-8 = 2.
Thus, the ratio of R:S in the population = 3:2.

Here are a few other problems in which I used alligation:

https://www.beatthegmat.com/allegation-t69552.html
https://www.beatthegmat.com/problem-on-m ... 75453.html
https://www.beatthegmat.com/percentage-p ... 70078.html

[nadib002]

@Mitch

I used the "alligation" method as well to set up the proportions.

I was stuck after 3:2

Following was my line of thinking:

3 parts voted for R and 2 parts voted for S

Therefore 3/5 * Total number of voters voted for R and 2/5 * Total number of voters voted for S

But we do-not know the Total number of voters. Can I pick a number, say 30 or 100, for the Total number of voters? For example if i were to pick 100, then 3/5 * 100 = 60 voted for R, but if i do it the way i mention i do not get the correct value.

What I do-not understand is when you picked 30 and 20 for R and S, why did you do the 10% and 5% increase again. Didn't the 3/5 represent the 10% increase already?

What is the flaw in my logic?

Could you please help?

Thank you

[Tani]

Intriguing. I used to do that intuitively but never knew it had a name! Thanks!