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GMAT Club Olympics 2024 (Day 1): In the town of Xionia, elections were

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08 Jul 2024, 20:35

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R = Number of Female voters/ Number of male voters
in year 2000
Male = 3300
for female voters = R * 3300

from 2000 to 2020
female voters have increased and male have decreased
R∗(100+F/100−M)

option correct
3300R & R∗((100+F)/(100−M))

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for the female voters in 2000, we needed to note that male = 3300, and R = F/M, so the number of F was 3300M.
For the percent change of female and men, E was the only option in which F increased and M decreased.

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Number of female voters in the year 2000: 3300R

Ratio of female to male voters in the year 2020: R∗100+F100−M

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08 Jul 2024, 21:56

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$\frac{R=300}{f00}$
To find female population in 2020, use the equation of percent change:
$\frac{f20 - f00 * 100 }{ f00 = F}$
$\frac{m20 - m00 * 100 }{ m00 = M}$
Rearrange the above equations to get values of f20 and m20 and the Ratio of 2020 will equal:
$\frac{f20}{m20} = \frac{f00}{m00}*\frac{F+100}{M+100}$

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F Percentage change in female voters,
M Percentage change in male voters.
Let F0 represent the number of female voters in the year 2000. Since R is the ratio of female to male voters in the year 2000, we have:
R= F0/3300
so. F0=R×3300
So, Number of Female Voters in 2000 = R×3300.

F2020 = number of female and male voters in 2020,
M2020 = number of male voters in 2020,
F2020=F0×(1+F/100)
M2020=3300×(1+M/100)F2020/M2020=F0×(1+F/100)/ 3300×(1+M/100)

We can subsitute F0 as R×3300

F2020/M2020=R×3300×(1+F/100)/ 3300×(1+M/100)
F2020/M2020=R×(1+F/100)/(1+M/100)
Upon simiplification, we get the answer for Female to Male Voters in the Year 2020.
F2020/M2020=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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# of male voters in 2000 = 3300 voters
# of female voters in 2000 = $F_i$ voters
Ratio of female to male voters:$ R = \frac{F_i}{3300}$ so $F_i = 3300R$ -> answer for number of female voters

Let F = % change in the number of female voters
Let M = % change in the number of male voters

Number of female voters in 2020 = $F_i * (1 + \frac{F}{100})$
Number of female voters in 2020 = $3300R* (1 + \frac{F}{100})$

Number of male voters in 2020 = $3300* (1 - \frac{M}{100})$ (note: subtraction due to decrease in male voters)

Ratio of female to male voters in 2020 = $3300R* (1 + \frac{F}{100}) / 3300* (1 - \frac{M}{100})$

Ratio of female to male voters in 2020 = $R* (1 + \frac{F}{100}) / (1 - \frac{M}{100})$ -> answer for ratio of female to male voters in 2020

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1. Number of female voters in the year 2000: 3300R
This formula calculates how many female voters there were in 2000. We start with the number of male voters, which is 3300, and then multiply by the ratio of female to male voters, represented by R. This way, we find out how many female voters there were based on the number of male voters.

2.*Ratio of female to male voters in the year 2020: R * 100 + F / 100 + M**
This formula shows the ratio of female to male voters in 2020. It starts with the initial ratio of female to male voters (R), then adds the percentage changes in female voters (F) and male voters (M) over five elections. By doing this, the formula accounts for the changes in voter numbers and gives us the ratio of female to male voters in 2020.

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08 Jul 2024, 22:16

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2000
M= 3300 , F=X= 3300R

R= X/3300
F=3300R

2020
M=2000= 3300(100+M), F= X(100+F) = 3300R(100+M)

Ratio = X(100+F)/ 3300(100+M)

Ratio= 3300R(100+F) / 3300(100+M)
Ratio = R*(100+F)/(100+M)

IMO
1- 3300R

2- R*(100+F)/(100+M)

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=11pt2000=11pt2020=11ptMale population: 3300 =11ptFrom 2000 to 2020 the number of females grew and the number of males decreased (so proportion of female to male over the latter years increased further) =11ptR = F:M which is R=F/M

=11ptR=F/3300

=11pt3300R = F(number of females in 2000)=11pt%growth for female (1+additional %)

=11pt%growth for males (1-additional %)

=11ptThus

=11ptR=100+F/ 100-M

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Answer:

(i) Given ratio of female to male voters in 2000 is R and no. of male voters is 3300,
Number of female voters in the year 2000 = R*3300

(ii) Let f be the female population in 2020 and m be the male population in 2020,

Percentage change in female voters from 2000 to 2020 F = (f - 3300R)*100/3300R
Percentage change in male voters from 2000 to 2020 M = (m - 3300)*100/3300

f/m = R∗(100+F)/(100+M)

Ratio of female to male voters in the year 2020 = f/m

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08 Jul 2024, 23:58

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Since R represents the ratio of female to male voters in the year 2000, so #female in 2000=R*(#male)=>3300R -------[#x means number of x]
Now regarding the ratio of female to male in 2020, we know that it's not gonna be the one of first three. Let's focus on the last three options:

we know that ratio will increase as the percentage change in the number of female i.e. F will increase so It'll be +F in the numerator. We also know that ratio will increase if the percentage change in males (negative) will increase so in the denominator it has to be -M. Hence option 4 is the correct one.

Option 6 is wrong because as there is a negative sign before M and M is already negative. Hence any big negative change in M will decrease the ratio instead of increasing it.

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Bunuel

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Since ratio of female to male in 2000 is given = R
i.e. F_old / M_old = R

and M_old = 3300 (given)

Therefore, F_old = M_old * R -> F_old = 3300 * R

Given, percentage change of both male and female as M and F respectively.

We can then find out F_new and M_new by using percentage change formula given in question.

Final answer => F_new / M_new = R*(F+100) / (M+100)

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In 2000
Male Voters = 3300
Female Voters = X

In 2020
Male Voters = $\frac{3300* (100+M)}{100}$

Female Voters = $\frac{X* (100+F)}{100}$

Also, R represents the ratio of female to male voters in the year 2000.
R= $\frac{X}{3300}$
X= 3300R

Number of female voters in the year 2000= 3300R

Ratio of female to male voters in the year 2020 = $\frac{3300R*(100+F)}{ 3300(100+M) }$= $R∗\frac{(100+F)}{(100+M)}$

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09 Jul 2024, 03:26

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If $\frac{female}{3300}$ = R
female= 3300R

Female increases: (1 + $\frac{F}{100}$ )*3300R
Male Decreases: (1- $\frac{M}{100}$)*3300

Taking their ratio
R* $\frac{ 100+F }{ 100-M} $

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Question stem looks a little hard to understand and options are big.
Let's assume numbers to get our answers.

We are given that,
population of male in 2000 = $M_o$ = 3300.

Then, assume that
population of female in 2000 = $W_o$ = 6600.

Ratio of female to male in 2000
$R_o$ = $\frac{6600}{3300} = 2$

Answer-1
Number of female voters in the year 2000 = $R*3300$ = $2 * 3300 = 6600$ = $W_o$

Now,
Population of men decreases and women increases from 2000 to 2020.
population of male in 2020 = $M_n$ = 1650.
population of female in 2020 = $W_n$ = 9900.
Ratio of female to male in 2020
$R_n$ = $\frac{9900}{1650} = 6$

And
F = $\frac{number of new female voters – number of old female voters }{ number of old female voters}∗100$ = $ 100 * \frac{(9900 - 6600) }{ 6600} $ = $50$

M = $\frac{number of new male voters – number of old male voters }{ number of old male voters}∗100$ = $ 100 * \frac{(1650-3300) }{ 3300} $ = $-50$

We need new ratio to equal 6.
Using values of F and M into formula only

$R_o∗\frac{100+F}{100+M}$ gives us right answer.

= $2 * \frac{100 + 50 }{ 100 - 50}$
= $6$

Final answers:
3300R and $R_o∗\frac{100+F}{100+M}$

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09 Jul 2024, 03:36

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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
$\frac{(number of new voters – number of old voters)}{(number of old voters)}$ * 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.

Solution:
In 2000
No. of male voters = 3300
Let No. of female voters = x
Since R is the ratio of No. of Female voters to No. of male voters in 2000
Thus, R = x / 3300
x = 3300R

In 2020
Let No. of male voters = A
Let No. of female voters = B
Also, % change in male voters = M
% change in female voters = F
Since M = $\frac{(number of new voters – number of old voters)}{(number of old voters)}$ * 100
Thus, M = $\frac{(A - 3300) }{ 3300}$ * 100
Since no. of male voters decreased in 2020, we will take into account the sign change in the numerator
Thus, M = $\frac{(3300 - A) }{ 3300}$ * 100
Thus A = 3300 - 33M

Also, F = $\frac{(number of new voters – number of old voters)}{(number of old voters)}$ * 100
Thus, F = $\frac{(B - 3300R) }{ 3300R}$ * 100
B = 33RF + 3300R

We need to find the ratio of female voters to male voters in 2020
i.e., $\frac{B}{A}$ = $\frac{(33RF + 3300R) }{ (3300 - 33M)}$
$\frac{B}{A}$ = $\frac{33R(F + 100) }{ 33(100 - M)}$
$\frac{B}{A}$ = $\frac{R(100 + F) }{ (100 - M)}$

No. of Female voters in 2000 = 3300R
Ratio of Female voters to Male voters in 2020 = $\frac{R(100 + F) }{ (100 - M)}$

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09 Jul 2024, 04:20

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We are given that in 2000, no. of males = 3300. And R = no. of females/ no. males. This gives us R = no. of females/ 3300. Thus, no. of females in 2000 = 3300R. Therefore, 1st question solved.

For the second part we need to find out the ratio of no. of females to that of males in 2020.
Let the no. of females in 2020 be x and males be y. Thus, we need to find out x/y.

Given that F is the % change in females from 2000 to 2020 and M is the % change in the males from 2000 to 2020.

Therefore, F = (x - 3300R)/ 33R and M = (3300 - y)/ 33.

This gives us, x = F(33R) + 3300R and y = 3300 - 33M.

Now, x/y = [F(33R) + 3300R] / (3300 - 33M)
= 33R(F + 100) / 33(100 - M)
= R * (F + 100) / (100 - M).

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09 Jul 2024, 04:44

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Something is wrong with question ?

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Female Male
Year 2000 3300
Year 2020

As per given information, R=Female2000 /Male2000
Female2000 =RxMale2000
Female2000 =Rx(3300)
Number of female voters in the year 2000 = 3300R (Option A)

Now, for part 2:
R2020 = Female2020/Male2020 equation (i)

We know
Female2020= Female2000 + Female2000x%change
Female2020= Female2000 + Female2000xF%
Female2020= Female2000(1 + F%)
Female2020= Female2000(1 + F/100)
Female2020= Female2000(100 + F)/100 equation (ii)

Similarly,
Male2020= Male2000 + Male2000x%change
Male2020= Male2000 + Male2000xM%
Male2020= Male2000(1 + M%)
Male2020= Male2000(1 + M/100)
Male2020= Male2000(100 + M)/100 equation (iii)

By substituting equations (ii) and (iii) in equation (i), we get
R2020 = {Female2000(100 + F)/100} / {Male2000(100 + M)/100}
R2020 = [Female2000(100 + F)]/[Male2000(100 + M)] (Cancelled 100 from numerator and denominator)
R2020 = R(100 + F)/(100 + M) (Substituted R in placce of Female2000 /Male2000 which was calculated above, for part 1)
Ratio of female to male voters in the year 2020 = R(100 + F)/(100 + M) (Option D)

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