Allegation Section A has Boys : girls ratio 1:2..

Question

Hi this is allegation question

Section A has boys : girls ratio 1:2
Section B has boys : girls ratio 3:1

The boys : girls ratio for 2 sections combined is 1:1. What was the ratio of students in the two sections ?

Source: expert global
a)3:2

b)2:3

Ans: 3:2

I know method of solving it via Allegation

My doubt is is if we take common value as 10 and multiply all ratio it should all tally?

Section A should have 1×10 boys = 10 boys and 2 × 10 =20 girls

Total of section A = 30

Section B should be 3× 10 = 30 boys
 
 1 * 10 = 10 girls

Total of section B: 40

Now if we combine the two, as per question we should have ratio of 1:1 of boys and girls but i combine section A boys with section B boys I get :10 + 30 = 40 boys

For girls combined it should be: 20 + 10 = 30

Ratio of boys and girls now combined is : 4:3, this is not 1:1 as given in question.

I don't get ratio as 1:1 as given in question of combined boys and girls of section A and Section B.

Is it because we take numbers of boys and girls from section a and b in such a way to make new ratio of 1:1 ? we are not taking total boys and total girls and combining ? Ty

Posted from my mobile device

AndrewN · Answer

Terminator  - You may always feel free to use an intuitive approach for such questions as well. I find that analytical reasoning is often just as effective as rules-based thinking when it comes to getting the correct answers on this test in an efficient manner. Here, we have two given ratios, 1:2 and 3:1, which result in an overall 1:1 ratio. We could consider the number of  parts  in each ratio to test sensible values. For instance, 1 boy to 2 girls in the first ratio indicates 3  parts , or 3 people. Likewise, 3 boys to 1 girl in the second ratio indicates 4 parts, which we can think of as people. The numbers 3 and 4 overlap at 12 for the first time, so what would happen if we multiplied the first ratio, the 3 total parts, by 4 to get 12 total parts, and then multiplied the second ratio, the 4 total parts, by 3 to also get 12 total parts?

1 * 4: 2 * 4 = 4:8
3 * 3: 1 * 3 = 9:3

When we add up the boys and girls, respectively, we can see that there would be 13 parts (from 4 + 9) representing boys and 11 parts (8 + 3) representing girls. A 13:11 ratio is close, but not quite there. What if we tweaked our multiplication factor with one ratio or the other? How about 2 for the second one (to keep everything in even-numbered units)?

1 * 4: 2 * 4 = 4:8
3 * 2: 1 * 2 = 6:2

Aha! Now we would have 10:10 as a ratio of boys to girls, which would reduce to the 1:1 ratio we are seeking. Adding up the parts of the first ratio and the parts of the second, we get 4 + 8 = 12 parts to 6 + 2 = 8 parts, or a 12:8 ratio, which reduces to 3:2. From the other side, starting with the second section, since the question does not specify which section needs to come first in the ratio, we would simply reverse the order, 8:12, which reduces to 2:3. We have our answers, and it did not require a rigorous framework to get there.

Remember, this test rewards analytical ability, whether that is based on mathematical prowess or not. Sometimes a less refined approach will do.

Good luck with your studies.

- Andrew

GMATPanther · Answer

hey ty for your detailed explanation

so after our LCM method did not give us correct ratio of 1:1, we need to edit one of the ratio to make sure we get 1:1 ? tx

AndrewN · Answer

Anant770  - I used a least common multiple out of number sense. There are really only two types of ratio problems, those that are part-to-part, as we have here, or those that are part-to-whole, in which you can directly relate a part to an actual number (e.g., a number of boys, or vehicles, or trees). I always prefer to work with smaller numbers than larger ones if I can help it, so I just explored what  would  happen if we set the total number of parts from each ratio equal to each other. What we saw was that we derived an answer pretty close the desired one, with a 13:11 ratio--note the difference of 2 between the numbers. So why alter the second ratio and not the first? Once again, the reason has to do with number sense more than anything. That is, in a 3:1 ratio, that 3 will grow much faster than the 1, so it is not hard to see that multiplying that second ratio by even a number such as 4 will create a larger divide in our target 1:1 ratio. To illustrate,

1 * 4: 2 * 4 = 4:8
3 * 4: 1 * 4 = 12:4

And we now have a 16:12 ratio, with a gap of 4 between the two numbers. In other words, we just doubled the amount of "space" between our values, from 2 to 4. If we still doubted what would happen if we increased the multiplier on the second ratio, we could test it:

1 * 4: 2 * 4 = 4:8
3 * 5: 1 * 5 = 15:5

And now the ratio would be 19:13, with a gap of 6. Notice a pattern here? It seems that for every increase by 1 of the multiplier in the second ratio, we increase the gap between the numbers by 2. There is no reason to believe that going in reverse would not produce the opposite result, and, as shown in my original post, that is exactly what happens.

To be honest, I am something of an undisciplined math guy. I eschew formulaic thinking and prize creativity, so my approach was just an extension of that type of reasoning. I would be happy to discuss this question further if you would like.

- Andrew

Allegation Section A has Boys : girls ratio 1:2..

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