Ratio

[jose.mario.amaya]

In the first M games of a team´s season, the ratio of the team´s wins to its losses was 1:2. In the subsequent N games, the ratio of the team´s wins to losses was 1:3. If M:N = 4:5, what was the ratio of the team´s wins to its losses for all M+N games?

a) 7:18
b) 9:23
c) 11:27
d) 23:54
e) 31:77
[/spoiler]e

[Brent@GMATPrepNow]

In the first M games of a team´s season, the ratio of the team´s wins to its losses was 1:2. In the subsequent N games, the ratio of the team´s wins to losses was 1:3. If M:N = 4:5, what was the ratio of the team´s wins to its losses for all M+N games?

a) 7:18
b) 9:23
c) 11:27
d) 23:54
e) 37:77

Given:
First M games, the team won 1/3 of its games
Next N games, the team won 1/4 of its games
If M:N = 4:5, we know that, out of every 9 games, 4 are from the first set of M games, and 5 are from the set of N games.
In other words, 4/9 of the games are from set M, and 5/9 are from set N.

When we combine all games, we have a weighted average of the two sets of games. So, we'll use this formula:
Weighted average = (group A proportion)(group A average) + (group B proportion)(group B average) + (group C proportion)(group C average) + ...

So, weighted average = (group M proportion)(group M average) + (group N proportion)(group N average)
= (4/9)(1/3) + (5/9)(1/4)
= 4/27 + 5/36
= 16/108 + 15/108
= 31/108
So, in total, the team won 31/108 of it's games.
This means it lost 108-31 games. So, it lost 77 games.

So, for all M+N games, the win:loss ratio = [spoiler]31:77[/spoiler]

Cheers,
Brent

For more information on weighted averages, you can watch this free GMAT Prep Now video: https://www.gmatprepnow.com/module/gmat- ... ics?id=805

[GMATGuruNY]

In the first M games of a team´s season, the ratio of the team´s wins to its losses was 1:2. In the subsequent N games, the ratio of the team´s wins to losses was 1:3. If M:N = 4:5, what was the ratio of the team´s wins to its losses for all M+N games?

a) 7:18
b) 9:23
c) 11:27
d) 23:54
e) 31:77

M:
Since wins to losses = 1:2, of every 3 games, 1 game is won.
Thus, wins/total = 1/3.

N:
Since wins to losses = 1:3, of every 4 games, 1 game is won.
Thus, wins/total = 1/4.

We can plug in for M and N.
To make the math easy, multiply M:N = 4:5 by the denominators in red above:
M:N = (4*3*4) : (5*3*4) = 48:60.

Let M = 48 games and N = 60 games.
Since 1/3 of the M games are won, M games won = (1/3)48 = 16, implying that M games lost = 48-16 = 32.
Since 1/4 of the N games are won, N games won = (1/4)60 = 15, implying that N games lost = 60-15 = 45.
Thus:
(total won) : (total lost) = (16+15) : (32+45) = 31:77.

The correct answer is E.

[fifafreak]

In the first M games of a team´s season, the ratio of the team´s wins to its losses was 1:2. In the subsequent N games, the ratio of the team´s wins to losses was 1:3. If M:N = 4:5, what was the ratio of the team´s wins to its losses for all M+N games?

a) 7:18
b) 9:23
c) 11:27
d) 23:54
e) 31:77
[/spoiler]e

M matches - M/3 wins, 2M/3 losses
N matches - N/4 wins, 3N/4 losses

M+N matches - [ M/3 + N/4 ] wins, [ 2M/3 + 3N/4 ] losses

M:N = 4:5 => Put M = 4, N=5 above.

Wins 31/12 ; Losses 77/12 => W:L = 31:77

[Mo2men]

In the first M games of a team´s season, the ratio of the team´s wins to its losses was 1:2. In the subsequent N games, the ratio of the team´s wins to losses was 1:3. If M:N = 4:5, what was the ratio of the team´s wins to its losses for all M+N games?

a) 7:18
b) 9:23
c) 11:27
d) 23:54
e) 31:77

M:
Since wins to losses = 1:2, of every 3 games, 1 game is won.
Thus, wins/total = 1/3.

N:
Since wins to losses = 1:3, of every 4 games, 1 game is won.
Thus, wins/total = 1/4.

We can plug in for M and N.
To make the math easy, multiply M:N = 4:5 by the denominators in red above:
M:N = (4*3*4) : (5*3*4) = 48:60.

Let M = 48 games and N = 60 games.
Since 1/3 of the M games are won, M games won = (1/3)48 = 16, implying that M games lost = 48-16 = 32.
Since 1/4 of the N games are won, N games won = (1/4)60 = 15, implying that N games lost = 60-15 = 45.
Thus:
(total won) : (total lost) = (16+15) : (32+45) = 31:77.

The correct answer is E.

Dear GMATGuru,
While I followed your method in my original solutions, I tried to use the alligation method. I treated both M & N like a mix of water and alcohol.

I used the 'win' part of each game of M & N

Win games of M = 1/3 = 4/12
Win games of N = 1/4= 3/12

M................Mix................N
4/12...................................3/12

So difference = 1/12

However, I stumbled in the above..............How can I continue?

Can you help please? Thanks

[GMATGuruNY]

Dear GMATGuru,
While I followed your method in my original solutions, I tried to use the alligation method. I treated both M & N like a mix of water and alcohol.

I used the 'win' part of each game of M & N

Win games of M = 1/3 = 4/12
Win games of N = 1/4= 3/12

M................Mix................N
4/12...................................3/12

So difference = 1/12

However, I stumbled in the above..............How can I continue?

Can you help please? Thanks

Let S = the entire season (in other words, the mixture of M and N).
Alligation can be performed only with percentages or fractions.

Step 1: Convert the ratios to FRACTIONS.
M:
Since win:losses = 1:2, and 1+2=3, wins/total = 1/3.
N:
Since wins:losses = 1:3, and 1+3=4, wins/total = 1/4.

Step 2: Put the fractions over a COMMON DENOMINATOR.
M = 1/3 = 4/12.
N = 1/4 = 3/12.

Step 3: Plot the fractions on a number line, with M and N on the ends and S in the middle.
M 4/12-----------S-----------3/12 M

Step 4: Calculate the distances between M, S and N.
Since M:N = 4:5, the distances on the number line are yielded by the RECIPROCAL of this ratio:
5x:4x.
Plotting the distances on the number line, we get:
M 4/12----5x----S----4x----3/12 N

Step 5: Determine the value of x.
The number line indicates that the total distance (5x+4x) is equal to the difference between 4/12 and 3/12:
5x+4x = 4/12 - 3/12
9x = 1/12
x = 1/108.

Thus:
wins/total in S = (M's fraction) - 5x = 4/12 - 5(1/108) = 36/108 - 5/108 = 31/108.
Implication:
Of every 108 games, 31 games are wins, implying that the remaining 77 games are losses.
Thus in S:
wins:losses = 31:77.

The correct answer is E.

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Hi All,

We're told that a in the first M games of a team´s season, the ratio of the team´s wins to its losses was 1:2 and in the subsequent N games, the ratio of the team´s wins to losses was 1:3. If M:N = 4:5, we're asked for the ratio of the team´s wins to its losses for all M+N games. This question can be approached in a number of different ways, including by TESTing VALUES.

For the first M games, the ratio 1:2 means that for every 1 win, there were 2 losses, so M MUST be a multiple of 3.
For the next N games, the ratio 1:3 means that for every 1 win, there were 3 losses, so N MUST be a multiple of 4.
With the ratio of M:N, M MUST also be a multiple of 4 and N MUST be an equivalent multiple of 5.

All of these rules severely limits the possible values of M and N. Let's start with N, since it has to be a multiple of 4 AND 5. Thus, N could be 20, 40, 60, 80, etc. We'll have to do a little bit of additional work to find a pair of values that fits everything that we're told....

IF...
N = 20, then M:N = 4:5 would mean that M would have to be 16, but that is NOT possible (since M must be a multiple of 3)....
N = 40, then M:N = 4:5 would mean that M would have to be 32, but that is NOT possible (since M must be a multiple of 3)....
N = 60, then M:N = 4:5 would mean that M would have to be 48, and that IS possible.

IF....
M = 48, then the ratio of 1 win to 2 losses means that there were 16 wins and 32 losses in the first 48 games
N = 60, then the ratio of 1 win to 3 losses means that there were 15 wins and 45 losses in the next 60 games
Total wins = 16+15 = 31 and Total losses = 32+45 = 77 losses

Final Answer: E

GMAT assassins aren't born, they're made,
Rich