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# Every person in a certain group is either a Dodgers fan or a Yankees

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Math Expert
Joined: 02 Sep 2009
Posts: 57022
Every person in a certain group is either a Dodgers fan or a Yankees  [#permalink]

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23 Mar 2015, 06:32
00:00

Difficulty:

35% (medium)

Question Stats:

74% (01:57) correct 26% (02:09) wrong based on 197 sessions

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Every person in a certain group is either a Dodgers fan or a Yankees fan, but not both. The ratio of Yankees fans to Dodgers fans is 5 to 3. If 22 Yankees fans change teams to become Dodgers fans, the ratio of Dodgers fans to Yankees fans will be 1 to 1. How many people are in the group?

A. 22
B. 88
C. 128
D. 144
E. 176

Kudos for a correct solution.

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Re: Every person in a certain group is either a Dodgers fan or a Yankees  [#permalink]

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24 Mar 2015, 02:38
7

Let Dodgers fan's = 5x & Yankees fan's = 3x; Total = 8x

$$\frac{5x-22}{3x+22} = \frac{1}{1}$$

x = 22; 8x = 176
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Re: Every person in a certain group is either a Dodgers fan or a Yankees  [#permalink]

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23 Mar 2015, 06:57
5
Bunuel wrote:
Every person in a certain group is either a Dodgers fan or a Yankees fan, but not both. The ratio of Yankees fans to Dodgers fans is 5 to 3. If 22 Yankees fans change teams to become Dodgers fans, the ratio of Dodgers fans to Yankees fans will be 1 to 1. How many people are in the group?

A. 22
B. 88
C. 128
D. 144
E. 176

Kudos for a correct solution.

Since we have a initial ratio of 5:3, the total count should be divisible by 8.

A. Out, since not divisible by 8.
B. 88/8=11, Yankees Fan = 55, Dodgers Fan = 33, Since 55-22 is not equal to 33+22 this as well can not be the answer.
C. 128/8=16, Yankees Fan = 80, Dodgers Fan = 48, Since 80-22 is not equal to 48+22 this as well can not be the answer.
D. 144/8=18, Yankees Fan = 90, Dodgers Fan = 54, Since 90-22 is not equal to 54+22 this as well can not be the answer.
E. 176/8=22, Though no checking required, but just to make sure, Yankees Fan = 110, Dodgers Fan = 66, and 110-22 = 66+22.

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Re: Every person in a certain group is either a Dodgers fan or a Yankees  [#permalink]

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23 Mar 2015, 07:56
1
y/d=5/3 or 3y=5d or d=3y/5------------ 1

y-22/d+22=1/1 or y-22=d+22; y-d=44, putting d=3y/5 in this equation y-d=3y/5=44. therefore y= 110 and d=3y/5= 3*110/5=66

110+66=176 (B)
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Re: Every person in a certain group is either a Dodgers fan or a Yankees  [#permalink]

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23 Mar 2015, 09:24
1
For this kind of question I use matrices for avoiding errors

The equations are
$$\frac{Y}{D}=\frac{5}{3} \Rightarrow 3Y=5D \Rightarrow 3Y-5D=0$$
and
$$\frac{Y-22}{D+22}=\frac{1}{1} \Rightarrow Y-22=D+22 \Rightarrow Y-D=44$$

$$D=\frac{3\cdot 44-0}{3\cdot -1 -1\cdot (-5)}=\frac{132}{2}=66$$
and $$Y=D+44=66+44=110$$

The sum is $$66+110=176$$

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Posts: 57022
Re: Every person in a certain group is either a Dodgers fan or a Yankees  [#permalink]

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30 Mar 2015, 03:08
Bunuel wrote:
Every person in a certain group is either a Dodgers fan or a Yankees fan, but not both. The ratio of Yankees fans to Dodgers fans is 5 to 3. If 22 Yankees fans change teams to become Dodgers fans, the ratio of Dodgers fans to Yankees fans will be 1 to 1. How many people are in the group?

A. 22
B. 88
C. 128
D. 144
E. 176

Kudos for a correct solution.

MAGOOSH OFFICIAL SOLUTION:
Attachment:

baseballfans_text(1).png [ 25.84 KiB | Viewed 4441 times ]

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Re: Every person in a certain group is either a Dodgers fan or a Yankees  [#permalink]

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30 Mar 2015, 10:10
1
Y/D = 5/3, and (Y-22) = (D+22)
solving for Y and D gives Y = 110 and D = 66.
Thus total no of fans = Y+D = 176.
Ans E.
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Re: Every person in a certain group is either a Dodgers fan or a Yankees  [#permalink]

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31 Mar 2015, 04:18
Bunuel wrote:
Every person in a certain group is either a Dodgers fan or a Yankees fan, but not both. The ratio of Yankees fans to Dodgers fans is 5 to 3. If 22 Yankees fans change teams to become Dodgers fans, the ratio of Dodgers fans to Yankees fans will be 1 to 1. How many people are in the group?

A. 22
B. 88
C. 128
D. 144
E. 176

Kudos for a correct solution.

Let Yankees = Y and Dodgers = D
Given that Y/D = 5/3
Also, (Y-22)/(D+22) = 1/1
i.e. Y-22 = D+22
So, Y = D+44

Put Y = D+44 in Y/D = 5/3
So, (D+44)/D = 5/3
So, D = 66.
Hence Y = 110.

Total members = 110 + 66
= 176
Hence option (E).

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Re: Every person in a certain group is either a Dodgers fan or a Yankees  [#permalink]

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08 Sep 2015, 19:57
1
The algebraic approach is fine, but this can be solved in 15 seconds if you understand the concepts.

Initially, the ratio is 5:3. A shift of 22 causes this to go to 1:1, or 4:4.

From here, you should recognize that 1 "unit" in the ratio represents 22 people. There are 8 total "units", so the answer is 8 * 22.

Similar question with a detailed explanation by Brunel can be found here: the-number-of-baseball-cards-that-john-and-bill-had-was-in-the-ratio-o-195786.html
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Re: Every person in a certain group is either a Dodgers fan or a Yankees  [#permalink]

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02 Aug 2019, 10:34
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Re: Every person in a certain group is either a Dodgers fan or a Yankees   [#permalink] 02 Aug 2019, 10:34
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