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Intern  Joined: 18 Feb 2010
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In a 200 member association consisting of men and women  [#permalink]

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Difficulty:   65% (hard)

Question Stats: 63% (02:23) correct 37% (02:31) wrong based on 1008 sessions

### HideShow timer Statistics In a 200 member association consisting of men and women, exactly 20% of men and exactly 25 % women are homeowners. What is the least number of members who are homeowners?

A. 49
B. 47
C. 45
D. 43
E. 41
Math Expert V
Joined: 02 Sep 2009
Posts: 56270
Re: Least number of homeowners  [#permalink]

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13
19
In a 200 member association consisting of men and women, exactly 20% of men and exactly 25 % women are homeowners. What is the least number of members who are homeowners?

A) 49
B) 47
c) 45
D) 43
E) 41

Thanks

Let the # of women be $$w$$, then # of men will be $$200-w$$. We want to minimize $$0.25w+0.2(200-w)$$ --> $$0.25w+0.2(200-w)=0.05w+40=\frac{w}{20}+40$$, so we should minimize $$w$$, but also we should make sure that $$\frac{w}{20}+40$$ remains an integer (as it represent # of people). Min value of $$w$$ for which w/20 is an integer is for $$w=20$$ --> $$\frac{w}{20}+40=1+40=41$$.

Or: as there are bigger percentage of homeowner women then we should minimize # of women, but we should ensure that $$\frac{1}{4}*w$$ and $$\frac{1}{5}*(200-w)$$ are integers. So $$w$$ should be min multiple of 4 for which $$200-w$$ is a multiple of 5 (basically w should be min positive multiple of 20), which turns out to be for $$w=20$$.

Hope it's clear.
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Intern  Joined: 18 Feb 2010
Posts: 22
Re: Least number of homeowners  [#permalink]

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Great! Thanks for the quick reply. Did not think it would be so straightforward

You rock!
Intern  Joined: 30 Nov 2010
Posts: 23
Location: Boston
Schools: Boston College, MIT, BU, IIM, UCLA, Babson, Brown
Re: Least number of homeowners  [#permalink]

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awesome explanation. thanks!
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Re: Least number of homeowners  [#permalink]

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Great explanation.
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Re: Least number of homeowners  [#permalink]

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Nice Question and Great explaination by Bunuel .. Thanks a lot
Intern  Joined: 20 Jun 2011
Posts: 44
Re: Least number of homeowners  [#permalink]

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Bunuel wrote:
In a 200 member association consisting of men and women, exactly 20% of men and exactly 25 % women are homeowners. What is the least number of members who are homeowners?

A) 49
B) 47
c) 45
D) 43
E) 41

Thanks

Let the # of women be $$w$$, then # of men will be $$200-w$$. We want to minimize $$0.25w+0.2(200-w)$$ --> $$0.25w+0.2(200-w)=0.05w+40=\frac{w}{20}+40$$, so we should minimize $$w$$, but also we should make sure that $$\frac{w}{20}+40$$ remains an integer (as it represent # of people). Min value of $$w$$ for which w/20 is an integer is for $$w=20$$ --> $$\frac{w}{20}+40=1+40=41$$.

Or: as there are bigger percentage of homeowner women then we should minimize # of women, but we should ensure that $$\frac{1}{4}*w$$ and $$\frac{1}{5}*(200-w)$$ are integers. So $$w$$ should be min multiple of 4 for which $$200-w$$ is a multiple of 5 (basically w should be min positive multiple of 20), which turns out to be for $$w=20$$.

Hope it's clear.

Could you explain this one; $$=0.05w+40=\frac{w}{20}+40$$ ? How did you get $$\frac{w}{20}$$ ?
Math Expert V
Joined: 02 Sep 2009
Posts: 56270
Re: Least number of homeowners  [#permalink]

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2
superpus07 wrote:
Bunuel wrote:
In a 200 member association consisting of men and women, exactly 20% of men and exactly 25 % women are homeowners. What is the least number of members who are homeowners?

A) 49
B) 47
c) 45
D) 43
E) 41

Thanks

Let the # of women be $$w$$, then # of men will be $$200-w$$. We want to minimize $$0.25w+0.2(200-w)$$ --> $$0.25w+0.2(200-w)=0.05w+40=\frac{w}{20}+40$$, so we should minimize $$w$$, but also we should make sure that $$\frac{w}{20}+40$$ remains an integer (as it represent # of people). Min value of $$w$$ for which w/20 is an integer is for $$w=20$$ --> $$\frac{w}{20}+40=1+40=41$$.

Or: as there are bigger percentage of homeowner women then we should minimize # of women, but we should ensure that $$\frac{1}{4}*w$$ and $$\frac{1}{5}*(200-w)$$ are integers. So $$w$$ should be min multiple of 4 for which $$200-w$$ is a multiple of 5 (basically w should be min positive multiple of 20), which turns out to be for $$w=20$$.

Hope it's clear.

Could you explain this one; $$=0.05w+40=\frac{w}{20}+40$$ ? How did you get $$\frac{w}{20}$$ ?

$$0.05w=\frac{5}{100}*w=\frac{1}{20}*w=\frac{w}{20}$$.

Hope it's clear.
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Intern  Joined: 11 Jul 2012
Posts: 40
Re: In a 200 member association consisting of men and women  [#permalink]

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1
I got E too (41)
X: number of men and 200-X: number of women
0.20*X +0.25*(200-X) = 50 - 0.05*X = 50 - (1/20)*X To minimize this, we should maximize X (i.e X = 180, since there must be some women and X should be a multiple of 20)
50 - (1/20)*180 = 41
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Re: In a 200 member association consisting of men and women  [#permalink]

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Hi Bunuel

I have reached to the answer but the solution is little weird after I saw yours.
Please guide me whether we can go by this

Question
In a 200 member association consisting of men and women, exactly 20% of men and exactly 25 % women are homeowners. What is the least number of members who are homeowners?

A. 49
B. 47
C. 45
D. 43
E. 41

Solution simple

Out of 200 20% are male i.e 40 and 25% are female i.e 50 , so total homeowner is 90.

Now min number homeowner is 40 and max is 90 so question ask us to find least and 41 has least value among all option.

So ans is 41.

Rgds
Prasannajeet
Math Expert V
Joined: 02 Sep 2009
Posts: 56270
Re: In a 200 member association consisting of men and women  [#permalink]

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prasannajeet wrote:
Hi Bunuel

I have reached to the answer but the solution is little weird after I saw yours.
Please guide me whether we can go by this

Question
In a 200 member association consisting of men and women, exactly 20% of men and exactly 25 % women are homeowners. What is the least number of members who are homeowners?

A. 49
B. 47
C. 45
D. 43
E. 41

Solution simple

Out of 200 20% are male i.e 40 and 25% are female i.e 50 , so total homeowner is 90.

Now min number homeowner is 40 and max is 90 so question ask us to find least and 41 has least value among all option.

So ans is 41.

Rgds
Prasannajeet

We are told that exactly 20% of men and exactly 25 % women are homeowners, not that 20% of 200 are males (homeowners)...

If there are 40 males and 50 females who are the remaining 200-(90)=110 people?
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Joined: 17 Dec 2012
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Location: India
Re: In a 200 member association consisting of men and women  [#permalink]

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7
1
1. The percentage of women who are homemakers is higher than the percentage of men who are homemakers. So the least number of homemakers would be, when the number of women is the lowest possible
2. The number of women should be a multiple of 4 and the number of men should be a multiple of 5
3. Given our objective in (1) and the constraint in (2), the minimum number of women is 20 and the corresponding number of men is 180
4. So 25% of 20 is 5 and 20% of 180 is 36. The least number of homemakers is 5+36= 41.

Therefore the answer is choice E.
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Intern  Joined: 28 Dec 2012
Posts: 9
Re: In a 200 member association consisting of men and women  [#permalink]

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Total numbers = 200. Men = X, Women = Y.
We are told that exactly 20% of men are homeowner => X is a multiple of 5 => X=5a (a>=1). Similarly, Y = 4b (b>=1). Find min(a+b)

X + Y = 200 <=> 5a + 4b = 200. a and b are all integers.
When a = 36, b=5 => (a+b)--->min.

E.
Intern  Joined: 29 Aug 2012
Posts: 29
WE: General Management (Consulting)
Re: In a 200 member association consisting of men and women  [#permalink]

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1
I didn't set up any algebra equations.

Basically, we know that we have 1/5 of the men and 1/4 of the women. So both, men and women should be a multiple of 20 (lcm of 5 and 4).

Now, in order to minimize, we want the maximum multiple for men (because is the least fraction) and the minimun (dintinct to zero) for women.

So we have 180 men and 20 women. Now the solution is clear. 180*(1/5)+20(1/4)=41
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Re: Least number of homeowners  [#permalink]

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Bunuel wrote:
In a 200 member association consisting of men and women, exactly 20% of men and exactly 25 % women are homeowners. What is the least number of members who are homeowners?

A) 49
B) 47
c) 45
D) 43
E) 41

Thanks

Let the # of women be $$w$$, then # of men will be $$200-w$$. We want to minimize $$0.25w+0.2(200-w)$$ --> $$0.25w+0.2(200-w)=0.05w+40=\frac{w}{20}+40$$, so we should minimize $$w$$, but also we should make sure that $$\frac{w}{20}+40$$ remains an integer (as it represent # of people). Min value of $$w$$ for which w/20 is an integer is for $$w=20$$ --> $$\frac{w}{20}+40=1+40=41$$.

Or: as there are bigger percentage of homeowner women then we should minimize # of women, but we should ensure that $$\frac{1}{4}*w$$ and $$\frac{1}{5}*(200-w)$$ are integers. So $$w$$ should be min multiple of 4 for which $$200-w$$ is a multiple of 5 (basically w should be min positive multiple of 20), which turns out to be for $$w=20$$.

Hope it's clear.

Hi Bunuel,

I solved it with a similar approach but came out with the option A. Can you please explain where am I wrong:

Let males be = m , then women = 200-m

0.2m +0.25(200-m)
we need to minimize m

so solving the equation gives us
50-m/20

For m to be an integer, the least value it can have is 20. So the ans: 50-1= 49.

Where am I wrong please guide.

Thanks.
Manager  Joined: 15 Aug 2013
Posts: 239
Re: Least number of homeowners  [#permalink]

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Bunuel wrote:
In a 200 member association consisting of men and women, exactly 20% of men and exactly 25 % women are homeowners. What is the least number of members who are homeowners?

A) 49
B) 47
c) 45
D) 43
E) 41

Thanks

Let the # of women be $$w$$, then # of men will be $$200-w$$. We want to minimize $$0.25w+0.2(200-w)$$ --> $$0.25w+0.2(200-w)=0.05w+40=\frac{w}{20}+40$$, so we should minimize $$w$$, but also we should make sure that $$\frac{w}{20}+40$$ remains an integer (as it represent # of people). Min value of $$w$$ for which w/20 is an integer is for $$w=20$$ --> $$\frac{w}{20}+40=1+40=41$$.

Or: as there are bigger percentage of homeowner women then we should minimize # of women, but we should ensure that $$\frac{1}{4}*w$$ and $$\frac{1}{5}*(200-w)$$ are integers. So $$w$$ should be min multiple of 4 for which $$200-w$$ is a multiple of 5 (basically w should be min positive multiple of 20), which turns out to be for $$w=20$$.

Hope it's clear.

Hi Bunuel, while solving this, I substituted for males instead of female and my equation looked like this:

.2m+.25(200-m) === 50 - 0.05m

Now, since it's asking me to minimize this value, would I choose the GREATEST possible value for m? How would I go about choose a value? At first, i thought it would be 196 but that doesn't lend itself well to being an integer.

Am I using the right though process here?
Intern  Joined: 17 Jun 2015
Posts: 10
Re: In a 200 member association consisting of men and women  [#permalink]

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good easy solution
Intern  Joined: 08 Mar 2015
Posts: 4
Re: In a 200 member association consisting of men and women  [#permalink]

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Bunuel wrote:
In a 200 member association consisting of men and women, exactly 20% of men and exactly 25 % women are homeowners. What is the least number of members who are homeowners?

A) 49
B) 47
c) 45
D) 43
E) 41

Thanks

brunel:

Let the # of women be $$w$$, then # of men will be $$200-w$$. We want to minimize $$0.25w+0.2(200-w)$$ --> $$0.25w+0.2(200-w)=0.05w+40=\frac{w}{20}+40$$, so we should minimize $$w$$, but also we should make sure that $$\frac{w}{20}+40$$ remains an integer (as it represent # of people). Min value of $$w$$ for which w/20 is an integer is for $$w=20$$ --> $$\frac{w}{20}+40=1+40=41$$.

Or: as there are bigger percentage of homeowner women then we should minimize # of women, but we should ensure that $$\frac{1}{4}*w$$ and $$\frac{1}{5}*(200-w)$$ are integers. So $$w$$ should be min multiple of 4 for which $$200-w$$ is a multiple of 5 (basically w should be min positive multiple of 20), which turns out to be for $$w=20$$.

Hope it's clear.

brunel: If we want to minimize w, why cant we just assume that w =0. Ie. The number of women is 0. The question does not state anywhere that the number of women CANNOT be 0.
Intern  Joined: 26 Feb 2016
Posts: 12
Re: In a 200 member association consisting of men and women  [#permalink]

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Bunuel wrote:
In a 200 member association consisting of men and women, exactly 20% of men and exactly 25 % women are homeowners. What is the least number of members who are homeowners?

A) 49
B) 47
c) 45
D) 43
E) 41

Thanks

i notice if i put w = 200 -m
i get
-m/20 + 50

then it would be completely different and il get min/max wrong way around?
Math Expert V
Joined: 02 Aug 2009
Posts: 7764
In a 200 member association consisting of men and women  [#permalink]

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3
sabxu1 wrote:
Bunuel wrote:
In a 200 member association consisting of men and women, exactly 20% of men and exactly 25 % women are homeowners. What is the least number of members who are homeowners?

A) 49
B) 47
c) 45
D) 43
E) 41

Thanks

i notice if i put w = 200 -m
i get
-m/20 + 50

then it would be completely different and il get min/max wrong way around?

Hi,
no you will still get the correct answer...
Two cases--
1) we maximize m, as the % of m is lower
2) If the answer is in w, we minimize w..

here you have taken your answer in m and you get it as $$50 - \frac{m}{20}$$..
so you have to maximize m..
m has to be a multiple of 20 to get an integer value for m/20..
m cannot be 200, as there are some w...

Next can be 200-20=180..
substitue m as 180, So number =$$50 - \frac{180}{20} = 50 - 9 = 41$$
_________________ In a 200 member association consisting of men and women   [#permalink] 13 Apr 2016, 23:39

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