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805+ (Hard)|   Math Related|         
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Bunuel
In the town of Xionia, elections were held every 5 years, between the year 2000 to the year 2020. During the 5 elections held in this period, the number of female voters increased while the number of male voters decreased from 3300 in the year 2000.

In the given expressions, F and M represent the percentage change in the number of female and male voters, respectively, over the 5 elections, and R represents the ratio of female to male voters in the year 2000. The percentage change in voters is calculated as \(\frac{\text{number of new voters – number of old voters}}{\text{number of old voters}}* 100\).

Select the expression that represents the Number of female voters in the year 2000, and select the expression that represents the Ratio of female to male voters in the year 2020. Make only two selections, one in each column.
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Answer : 3300*R ; R* (100+F)/(100-M)

Explanation:

1) R= # of females in 2000 / 3300

#of females in 2000= 3300*R
2) ratio of # of females in 2020/ # of males in 2020 = (1+F/100) * 3300*R/ (3300 * (1-R/100))=
R*(100+F)/(100-R)

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Quote:
In the town of Xionia, elections were held every 5 years, between the year 2000 to the year 2020. During the 5 elections held in this period, the number of female voters increased while the number of male voters decreased from 3300 in the year 2000.

In the given expressions, F and M represent the percentage change in the number of female and male voters, respectively, over the 5 elections, and R represents the ratio of female to male voters in the year 2000. The percentage change in voters is calculated as number of new voters – number of old votersnumber of old voters∗100
number of new voters – number of old voters
number of old voters
∗
100
.

Select the expression that represents the Number of female voters in the year 2000, and select the expression that represents the Ratio of female to male voters in the year 2020. Make only two selections, one in each column.
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Let x be number of male voters in 2000, and y be that of female voters in 2000.

x = 3300
It is given that y/x = R, hence y = xR = 3300R -> Answer to the first part

Male voters in 2020 = 3300 - 3300M/100 (Subtracting since number of men voters decreased)

Female voters in 2020 = 3300R + 3300RF/100 (Adding since number of women voters increased)

Ration of F to M = (3300R + 33RF)/(3300 - 33M) = R * (100 + F)/(100 - M)
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Year 2000:
Female : Male (3300) = R -> Female = 3300R

Year 2020:
(New Fe-3300R)/3300R*100% =F→New Fe=F*33R+3300R
(New Ma-3300)/3300*100%= M →New Ma=M*33+3300
(New Fe)/(New Ma)=R*(F+100)/(M+100)
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The number of female voters in the year 2000 can be calculated using the ratio 𝑅 = f/m (where, m = 3300 in year 2000)
Number of female voters in 2000 = 3300𝑅

F represents the percentage increase in female voters, so the number of female voters in 2020 is
3300𝑅×(1+F/100)

M represents the percentage decrease in male voters, so the number of male voters in 2020 is
3300×(1-M/100)
therefore, the ratio of female to male voters in 2020 is given by: (Number of male voters in 2020/Number of female voters in 2020)
=> R*(100+F)/(100-M)
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Bunuel
In the town of Xionia, elections were held every 5 years, between the year 2000 to the year 2020. During the 5 elections held in this period, the number of female voters increased while the number of male voters decreased from 3300 in the year 2000.

In the given expressions, F and M represent the percentage change in the number of female and male voters, respectively, over the 5 elections, and R represents the ratio of female to male voters in the year 2000. The percentage change in voters is calculated as \(\frac{\text{number of new voters – number of old voters}}{\text{number of old voters}}* 100\).

Select the expression that represents the Number of female voters in the year 2000, and select the expression that represents the Ratio of female to male voters in the year 2020. Make only two selections, one in each column.
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­Given,
F/100 = [f(2020) - f(2000)] / f(2000) ....(1) (increased)
M/100 = [m(2000) - m(2020)] / m(2000) ....(2) (decreased)
R = f(2000)/m(2000)
m(2000) = 3300

Now, q1. f(2000) = R * m(2000) = 3300R

q2. f(2020)/m(2020) = ?

Solving (1)
=> F/100 = f(2020)/f(2000) - 1
=> (F+100)/100 = f(2020)/f(2000)
=> (F+100)*f(2000)/100 = f(2020) ....(3)

Solving (2)
=> M/100 = 1 - m(2020)/m(2000)
=> (100-M)/100 = m(2020)/m(2000)
=> (100-M)*m(2000)/100 = m(2020) ....(4)

Now dividing (3) by (4) we get,

=> f(2020)/m(2020) = (F+100)*f(2000)/(100-M)*m(2000)
=> f(2020)/m(2020) = R*(F+100)/(100-M)

Answers: 3300R, R*(F+100)/(100-M)
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­Number of female voters in 2000 = f2000
Number of female voters in 2020 = f2020
Number of male voters in 2000 = m2000
Number of male voters in 2000 = m2020


m2000 = 3300

R = f2000 / m2000 = f2000 / 3300

So f2000 = 3300 * R


F = (f2020 - f2000) * 100 / f2000

So f2020 = f2000 * F/100 + f2000 = f2000 (1 + F/100) = f2000 (100 + F) / 100


M = (m2020 - m2000) * 100 / m2000 (Note that in this case M is negative)

So m2020 = m2000 * M/100 + m2000 = m2000 (1 + M/100) = m2000 (100 + M) / 100


f2020/m2020 = f2000 (100 + F) / m2000 (100 + M)

And as R = f2000 / m2000

f2020/m2020 = R * (100 + F) / (100 + M)


IMO 3300R and R * (100 + F) / (100 + M)
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R represents the ratio of female to male voters in the year 2000.
R = no. of female voters / no of male voters
Hence , the no. of female voters = 3300*R

Ratio of female to male voters in the year 2020 = 3300 R * ( 1 + F/100) / 3300* ( 1 - M/100)

Because no of female voters in 2020 = 3300 R * ( 1 + F/100) ; F = percentage change.
no of male voters in 2020 = 3300* ( 1 - M/100)

Simplifying , R( 100 + F) * 100 / 100 * (100 - M) =R * (100+ F) / 100 - M => Answer
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Number of female voters in the year 2000 3300/R Ratio of female to male voters in the year 2020 -3300/R
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men@2000=3300

F=fem@2020/fem@2000*100-100

M=men@2020/men@2000*100-100

R=fem@2000/men@2000

1) 3300*R=men@2000*fem@2000/men@2000=fem@2000

2) R*(100+F)/(100+M)=fem@2000/men@2000*(100+fem@2020/fem@2000*100-100)/(100+men@2020/men@2000*100-100)=(fem@2000*fem@2020/fem@2000*100)/(men@2000*men@2020/men@2000*100)=fem@2020/men@2020
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let initial female voters = f
given, R = f / 3300
hence f = 3300R

now, female voters increased by F%, so 3300R(1+(F/100))
similarly male voters = 3300(1-(M/100))
ratio would be R∗(100+F)/100−M
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In the town of Xionia, elections were held every 5 years, between the year 2000 to the year 2020. During the 5 elections held in this period, the number of female voters increased while the number of male voters decreased from 3300 in the year 2000.

In the given expressions, F and M represent the percentage change in the number of female and male voters, respectively, over the 5 elections, and R represents the ratio of female to male voters in the year 2000. The percentage change in voters is calculated as number of new voters – number of old votersnumber of old voters∗100number of new voters – number of old votersnumber of old voters∗100.

Select the expression that represents the Number of female voters in the year 2000, and select the expression that represents the Ratio of female to male voters in the year 2020. Make only two selections, one in each column.

Solution: The number of men in 2000 is 3,300. Since R is the ratio of women to men in 2000, the number of women in 2000 is 3,300R. 

To find the ratio of women to men in 2020, we first need to find the number of men and the number of women in terms of F, M, and R. The number of women in 2020 is: \( \frac{F}{100 } \times 3300R +3300R = 33RF +3300R \). ­Similarly, the number of men in 2020 is \( 33M+3300 \). The ratio of women to men in 2020 is thus \( \frac{ 33RF + 3300R }{ 33M +3300 } = R \times \frac{ F + 100 }{ M+100 } \). ­
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Let's consider that :
F2000 : Number of female in the year of 2000
F2020 : Number of female in the year of 2020
M2000 = Number of male in the year of 2000 =3300
M2020 : Number of male in the year of 2020

Then,
(1) F = (F2020 - F2000)/ F2000 *100 and (2)M = (M2020 - 3300)/ 3300 *100 and (3)R =F2000/3300

From (3) we can conclude that : F2000 = 3300R

From (1) we can conclude that : F2020 = (F2000 * F)/100 +3300R => F2020 = (3300R * F)/100 +3300R=>F2020 = 3300R (F/100+1) (4)
From (2) we can conclude that : M2020 =( 3300* M)/100 +3300 =>M2020 =3300(M/100 +1) (5)

Then the Ratio of female to male voters in the year 2020 = (4)/(5) = 3300R(F/100+1) / 3300(M/100+1) = R*(F+100)/(M+100)­
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1) F/M=R in 2000
2) M=3300
gives F/3300=R
=>F=3300R
3) in 2020, F becomes F(1+f/100) &
M becomes M(1-m/100)
Hence F/M in 2020 is R(1+f/100)/(1-m/100)
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Number of female voters in the year 2000 = A 3300R
Ratio of female to male voters in the year 2020 = E

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To solve this problem, we need to determine two things:

The number of female voters in the year 2000.
The ratio of female to male voters in the year 2020.
Number of female voters in the year 2000:
Let's denote:
F as the percentage change in the number of female voters over the 5 elections.
M as the percentage change in the number of male voters over the 5 elections.
R as the ratio of female to male voters in the year 2000.
f1 as the number of female voters in the year 2000.
​m1 as the number of male voters in the year 2000.
We are given that the number of male voters in the year 2000 was 3300 (=m1). Therefore, we can express
using the ratio
R = f1/m1 ⟹ R = f1/3300 => f1 = 3300R
So, the expression for the number of female voters in the year 2000 is: 3300R

Ratio of female to male voters in the year 2020:
We need to determine the new ratio of female to male voters in the year 2020.
Let No. of Female and male voters be f5 & m5 respectively in the year 2020.

Given:
The number of female voters in the year 2000 was 3300R.
The number of male voters in the year 2000 was 3300.

We can rearrange the given formula to find the new number of voters:

number of new voters=number of old voters×(1+ percentage change/100)

Using this formula:
The number of female voters in 2020: f5 = f1 ( 1 + F/100 ) = 3300R {(100+F)/100}

The number of male voters in 2020: m5 = m1 ( 1+ M/100 ) = 3300 {(100+M)/100}

The new ratio of female to male voters in 2020 is:
f5/m5 = R [(100+F)/(100+M)]


Selections:
Number of female voters in the year 2000: 3300R
Ratio of female to male voters in the year 2020: R× [(100+F)/(100+M)]
​
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Bunuel
In the town of Xionia, elections were held every 5 years, between the year 2000 to the year 2020. During the 5 elections held in this period, the number of female voters increased while the number of male voters decreased from 3300 in the year 2000.

In the given expressions, F and M represent the percentage change in the number of female and male voters, respectively, over the 5 elections, and R represents the ratio of female to male voters in the year 2000. The percentage change in voters is calculated as \(\frac{\text{number of new voters – number of old voters}}{\text{number of old voters}}* 100\).

Select the expression that represents the Number of female voters in the year 2000, and select the expression that represents the Ratio of female to male voters in the year 2020. Make only two selections, one in each column.
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­in 2000 m=3300 
R= w/m
=> W = mR
in 2000 women = 3300R ( A)
CHANGE in women
F= w1-3300R/3300R * 100 => w1 = 33R(F+100) (women in 2020)

Change in Men
M = 3300-m1/3300  *100=> m1 = 33 ( 100 - M) (

w1/m1 = R (F+100)/100-M (E)
Answer A,E

 
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­In the year 2000, 

male voters = 3300
female voters = f

We are given R = 3300/f or f= 3300R …(1)

Therefore, female voters in year 2000 are 3300R

Also, M = (Decrease in male voters since the year 2000)/ 3300 *100 (or) Decrease in male voters since 2000= 33M. Note that we are given that male voters decreased since 2000. Hence M is a decrease in % of male voters.

We know Male voters in 2020 = Male voters in 2000 - Decrease in male voters since 2000 = 3300 - 33M

Similarly,

F = Increase in female voters since the year 2000)/ f *100 (or) Increase in female voters since 2000= F.f/100

From (1) substituting for f, the RHS becomes 33FR

We know female voters in 2020 = female voters in 2000 + increase in female voters since 2000 = 3300R + 33FR

Therefore, ratio of female to male voters in 2020 is 3300+33FR/(3300-33M)

Simplifying gives 100R + FR/(100- M) or R(100+F)/(100-M)
 
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­Let's unpack the given information.

                                        Year 2000           Year 2020
No. of female voters                x                      y  (>x)
No. of male voters                  3300                  z  (<3300)

\(R = \frac{x}{3300}\)

Ratio of female to male voters in the year 2020 = Number of female voters in the year 2000 / Number of male voters in the year 2000
\(R = \frac{x}{3300}\)
x = 3300R

Therefore, the updated table is as follows

                                        Year 2000           Year 2020
No. of female voters               3300R                 y (>x)
No. of male voters                  3300                   z  (<3300)

Ratio of female to male voters in the year 2020 = Number of female voters in the year 2020 / Number of male voters in the year 2020 = y/z
The percentage change in voters is calculated as [(number of new voters – number of old voters) /number of old voters]∗100 

Since F and M represent the percentage change in the number of female and male voters, they inherently take care of the positive or negative signs. 

\(M = \frac{z- 3300}{3300} * 100\)

z = 33(M+100)

\(F = \frac{y- 3300R}{3300} * 100\)

y = 33R(F+100)

Therefore, calculating the value of (y/z), we get
\(y/z = \frac{R(F+100)}{(M+100)}\)

The overall updated table looks as below

                                        Year 2000           Year 2020
No. of female voters               3300R             33R(F+100)
No. of male voters                  3300               33(M+100)


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GMAT Club Olympics 2024 (Day 1): In the town of Xionia, elections were 08 Jul 2024, 11:06
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­In 2000, Male voters = 3300

R (Ratio of female voters to male voters) = \(\frac{(Female voters)}{(Male voters)} = \frac{(Female voters)}{3300}\)

[ANSWER 1]
Female voters = \(3300R\) (No of female voters in 2000)

F = Rate of increase of female voters
M= Rate of decrease of male voters

Female voters in 2020 = \((1+\frac{F}{100})*3300R\)
Male voters in 2020 = \((1-\frac{M}{100})*3300\)

[ANSWER 2]
Ratio of female to male voters in the year 2020 =
\({(1+\frac{F}{100})*3300R}/{(1-\frac{M}{100})*3300}\)­
\(=>R*\frac{(100+F)}{(100-M)}\)­
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GMAT Club Olympics 2024 (Day 1): In the town of Xionia, elections were 08 Jul 2024, 11:09
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For the first part, it is stated in the question that R represents the ration of female voters to male voters for the year 2000. Since we know that there were 3300 male voters in the year 2000, the number of female voters for the year 2000 = 3300 x R, which is one of the answers. To be quite frank, I wasn't entirely sure how to calculate this next part so I made an intelligent guess. Since the first answer was 3300R, I knew I could get rid of the first three answers for the second choice. From the 3 remaining answer choices, I relied on the fact that there was an increase in the number of female voters and that there was a decrease in the number of male voters over the 5 elections. Since we were looking at the ratio for 2020, which was the last of the 5 elections, and that the information we were given was from the year 2000, I knew that the ratio would have to demonstrate this positive percentage change in female voters and negative percentage change in male voters. From the three answer choices, only one of them demonstrated a clear increase of female voters and decrease of male voters as (100+F)/(100-M)
Bunuel
In the town of Xionia, elections were held every 5 years, between the year 2000 to the year 2020. During the 5 elections held in this period, the number of female voters increased while the number of male voters decreased from 3300 in the year 2000.

In the given expressions, F and M represent the percentage change in the number of female and male voters, respectively, over the 5 elections, and R represents the ratio of female to male voters in the year 2000. The percentage change in voters is calculated as \(\frac{\text{number of new voters – number of old voters}}{\text{number of old voters}}* 100\).

Select the expression that represents the Number of female voters in the year 2000, and select the expression that represents the Ratio of female to male voters in the year 2020. Make only two selections, one in each column.
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