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Manager  Joined: 02 Dec 2012
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In a certain district, the ratio of the number of registered  [#permalink]

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Question Stats: 71% (02:15) correct 29% (02:29) wrong based on 1156 sessions

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In a certain district, the ratio of the number of registered Republicans to the number of registered Democrats was 3/5. After 600 additional Republicans and 500 additional Democrats registered, the ratio was 4/5. After these registrations, there were how many more voters in the district registered as Democrats than as Republicans?

(A) 100
(B) 300
(C) 400
(D) 1,000
(E) 2,500
Math Expert V
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In a certain district, the ratio of the number of registered Republicans to the number of registered Democrats was 3/5. After 600 additional Republicans and 500 additional Democrats registered, the ratio was 4/5. After these registrations, there were how many more voters in the district registered as Democrats than as Republicans?

(A) 100
(B) 300
(C) 400
(D) 1,000
(E) 2,500

Old ratio: $$\frac{republicans}{democrats}=\frac{3x}{5x}$$.

New ratio: $$\frac{3x+600}{5x+500}=\frac{4}{5}$$ --> $$x=200$$.

Current difference is $$(5x+500)-(3x+600)=2x-100=300$$.

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Re: In a certain district, the ratio of the number of registered  [#permalink]

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If ever you're not seeing the math, you know one of the answers has to be correct, so you can just backsolve. Looking at the increase in numbers, it would make sense that the number of democrats MORE than republicans will probably be in the hundreds, so you'd likely start with B or C.

If you start with C: 400 is the difference between D's and R's in the new ratio: So there are 4x400 = 1600 Republicans and 5x400 = 2000 Democrats. If you subtract 600 from R and 500 from D, you get 1000 Republicans and 1500 Democrats, a 2/3 ratio. This isn't 3/5, but it's not far (66.7% vs 60%). The next attempt should be B since it's close but slightly too big:

B: 300 is the difference between D's and R's in the new ratio: So there are 4x300 = 1200 Republicans and 5x300 = 1500 Democrats. If you subtract 600 from R and 500 from D, you get 600 Republicans and 1000 Democrats, a 3/5 ratio. Bingo.

If you started with B, you got to the right answer quickly. If you started with A, D or E, your answer will be way off. While this is not a better solution than the algebraic one, sometimes it is easier to see.

Hope this helps!
-Ron
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Re: In a certain district, the ratio of the number of registered  [#permalink]

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gmatser1 wrote:
Bunuel wrote:
mjb2 wrote:
sorry for violating the rules.. I didn't really know them as this was my first post. Anyway, I still don't understand why we can not take the multiplier and multiply it by the new ration 4/5 (200*5)-(200*4)= 200 difference.

Sorry but I don't understand the logic behind your approach.

It's a shame this wasn't addressed, Bunuel. Let me follow up this question with the logic and you can let me know why it's wrong.

After we find x = 200 as the multiplier, why can't we say that since AFTER these registrations the ratio is 4/5, then 4x/5x. Since x=200, 4x equals 800 registered Republicans, and 5x equals 1000 registered Democrats. Thats means that the difference after the registrations is 1000 Democrats - 800 Republicans is 200!

Because x = 200 is for the old ratio and you cannot use it for the new one.
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Re: In a certain district, the ratio of the number of registered  [#permalink]

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The difference of 400 is before the new registrations. But the problems asks for "After these (new) registrations..." So the new number of registered Republicans is 600 + 600 = 1200, and the new number of registered Democrats is 1000 + 500 = 1500. So the difference is 300.

Does that make sense?
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Math Expert V
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Re: In a certain district, the ratio of the number of registered  [#permalink]

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Hi, just wondering what is wrong with my approach as follows:

3/5 republican voters = 0.6*x
Another 600 register as republican, with the total amount being 80% republican (4/5)
Total new voters is 1,100 (500+600)

So, 0.6x + 600 = 0.8(x+1,100)

0.6x + 600 = 0.8x + 880

However, this is where I go awry.

Could someone pls explain why I can't use the approach noted above?

The ratio of Republicans to Democrats was 3:5, which means that Republicans were 3/(3+5)=3/8 of the total number (not 3/5 of the total number)

After 600 additional Republicans and 500 additional Democrats registered, the ratio was 4:5, which means that that Republicans were 4/(4+5)=4/9 of the total number (not 4/5 of the total number).

3/8*x+600=4/9(x+1100) --> x=1600 --> after registration = x+1100=2700 --> Republicans=4/9*1200 and Democrats=1500 --> the difference=300.

Hope it helps.
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In a certain district, the ratio of the number of registered  [#permalink]

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In a certain district, the ratio of the number of registered Republicans to the number of registered Democrats was 3/5. After 600 additional R and 500 additional D registered, the ratio was 4/5. After these registrations, the there were how many more voters in the district registered as D than as R?

When you solve the two given equations, you arrive at D = 1000, which is perfectly logical. As R = 3/5 D, R must be 600.

Now comes the thing I don't understand. In the sample solution, the newly registered voters are now added to the above numbers, which results in D = 1000 + 600 = 1500, respectively R = 600 + 600 = 1200. The difference is 300 now, which corresponds to the OA. But isn't it true that the numbers which result from solving the given equations must already be post the additions?
Math Expert V
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Re: In a certain district, the ratio of the number of registered  [#permalink]

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tomtom1610 wrote:
In a certain district, the ratio of the number of registered Republicans to the number of registered Democrats was 3/5. After 600 additional R and 500 additional D registered, the ratio was 4/5. After these registrations, the there were how many more voters in the district registered as D than as R?

When you solve the two given equations, you arrive at D = 1000, which is perfectly logical. As R = 3/5 D, R must be 600.

Now comes the thing I don't understand. In the sample solution, the newly registered voters are now added to the above numbers, which results in D = 1000 + 600 = 1500, respectively R = 600 + 600 = 1200. The difference is 300 now, which corresponds to the OA. But isn't it true that the numbers which result from solving the given equations must already be post the additions?

As for your question: we are asked to find the difference between the numbers of Democrats and Republicans AFTER the registration of 600 additional Republicans and 500 additional Democrats, so it should be 1500-1200=300.

Hope it's clear.
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Re: In a certain district, the ratio of the number of registered  [#permalink]

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why cant we write it down as R/D = 3/5 and go further with (R+600)/(D+500). If we combine them we get R = 600 and D = 1000. Difference = 400. Whats wrong with this logic. Thanks in advance
Math Expert V
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Re: In a certain district, the ratio of the number of registered  [#permalink]

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why cant we write it down as R/D = 3/5 and go further with (R+600)/(D+500). If we combine them we get R = 600 and D = 1000. Difference = 400. Whats wrong with this logic. Thanks in advance

Check my response here: in-a-certain-district-the-ratio-of-the-number-of-registered-143983.html#p1158046
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Re: In a certain district, the ratio of the number of registered  [#permalink]

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Hi, just wondering what is wrong with my approach as follows:

3/5 republican voters = 0.6*x
Another 600 register as republican, with the total amount being 80% republican (4/5)
Total new voters is 1,100 (500+600)

So, 0.6x + 600 = 0.8(x+1,100)

0.6x + 600 = 0.8x + 880

However, this is where I go awry.

Could someone pls explain why I can't use the approach noted above?

Intern  Joined: 03 Mar 2013
Posts: 24
Re: In a certain district, the ratio of the number of registered  [#permalink]

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Bunuel wrote:
Hi, just wondering what is wrong with my approach as follows:

3/5 republican voters = 0.6*x
Another 600 register as republican, with the total amount being 80% republican (4/5)
Total new voters is 1,100 (500+600)

So, 0.6x + 600 = 0.8(x+1,100)

0.6x + 600 = 0.8x + 880

However, this is where I go awry.

Could someone pls explain why I can't use the approach noted above?

The ratio of Republicans to Democrats was 3:5, which means that Republicans were 3/(3+5)=3/8 of the total number (not 3/5 of the total number)

After 600 additional Republicans and 500 additional Democrats registered, the ratio was 4:5, which means that that Republicans were 4/(4+5)=4/9 of the total number (not 4/5 of the total number).

3/8*x+600=4/9(x+1100) --> x=1600 --> after registration = x+1100=2700 --> Republicans=4/9*1200 and Democrats=1500 --> the difference=300.

Hope it helps.

Thanks a lot, appreciate your help!
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Official guide problem solving Q 113  [#permalink]

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In a certain district, the ratio of the number of registered Republicans to the number of registered Democrats was 3/5. After 600 additional Republicans and 500 additional Democrats registered, the ratio was 4/5. After these registrations, there were how many more voters in the district registered as Democrats than as Republicans?

(A) 100
(B) 300
(C) 400
(D) 1,000
(E) 2,500

_________

My question is, why can't we take the multiplier which is 200, and multiply the new ratio 4/5 to get a difference of 200? Why is that wrong? that what was done In question 105 in the official guide. Thank you
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Re: Official guide problem solving Q 113  [#permalink]

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mjb2 wrote:
In a certain district, the ratio of the number of registered Republicans to the number of registered Democrats was 3/5. After 600 additional Republicans and 500 additional Democrats registered, the ratio was 4/5. After these registrations, there were how many more voters in the district registered as Democrats than as Republicans?

(A) 100
(B) 300
(C) 400
(D) 1,000
(E) 2,500

_________

My question is, why can't we take the multiplier which is 200, and multiply the new ration 4/5 to get a difference of 200? Why is that wrong? In question 105 in the official guide that what was done. Thank you

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Math Expert V
Joined: 02 Sep 2009
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Re: Official guide problem solving Q 113  [#permalink]

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mjb2 wrote:
In a certain district, the ratio of the number of registered Republicans to the number of registered Democrats was 3/5. After 600 additional Republicans and 500 additional Democrats registered, the ratio was 4/5. After these registrations, there were how many more voters in the district registered as Democrats than as Republicans?

(A) 100
(B) 300
(C) 400
(D) 1,000
(E) 2,500

_________

My question is, why can't we take the multiplier which is 200, and multiply the new ratio 4/5 to get a difference of 200? Why is that wrong? that what was done In question 105 in the official guide. Thank you

Merging similar topics. Please refer to the solutions above and ask if anything is unclear.

P.S. All OG13 questions with solutions are here: the-official-guide-quantitative-question-directory-143450.html
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Re: In a certain district, the ratio of the number of registered  [#permalink]

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sorry for violating the rules.. I didn't really know them as this was my first post. Anyway, I still don't understand why we can not take the multiplier and multiply it by the new ration 4/5 (200*5)-(200*4)= 200 difference.
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Re: In a certain district, the ratio of the number of registered  [#permalink]

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mjb2 wrote:
sorry for violating the rules.. I didn't really know them as this was my first post. Anyway, I still don't understand why we can not take the multiplier and multiply it by the new ration 4/5 (200*5)-(200*4)= 200 difference.

Sorry but I don't understand the logic behind your approach.
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Re: In a certain district, the ratio of the number of registered  [#permalink]

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R/D = 3x/5x

R/D = 3x+600/(5x+500) = 4/5
Solve for x to get x=200.
Now put in original equation to get total Republicans as 1200, and the total number of democrats as 1500.
difference= 300
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Re: In a certain district, the ratio of the number of registered  [#permalink]

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Bunuel wrote:
mjb2 wrote:
sorry for violating the rules.. I didn't really know them as this was my first post. Anyway, I still don't understand why we can not take the multiplier and multiply it by the new ration 4/5 (200*5)-(200*4)= 200 difference.

Sorry but I don't understand the logic behind your approach.

It's a shame this wasn't addressed, Bunuel. Let me follow up this question with the logic and you can let me know why it's wrong.

After we find x = 200 as the multiplier, why can't we say that since AFTER these registrations the ratio is 4/5, then 4x/5x. Since x=200, 4x equals 800 registered Republicans, and 5x equals 1000 registered Democrats. Thats means that the difference after the registrations is 1000 Democrats - 800 Republicans is 200!
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Re: In a certain district, the ratio of the number of registered  [#permalink]

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Can we solve this somehow with the "change in ratio" method?

Starting ratio:
5R=3D
Ending ratio:
5R=4D

Now we have a change in ratio of 1 unit in democrats corresponding to 500 democrats. This means, 1 unit in 5:4 represents 500. However I believe this just holds true as long as only one number changes. Since we also have changing numbers of republicans, we can not calculate it like that. Can anyone agree/disagree? Re: In a certain district, the ratio of the number of registered   [#permalink] 08 Sep 2015, 03:43

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