In a tournament, there are 43 junior level and 51 senior level partici

Question

In a tournament, there are 43 junior level and 51 senior level participants. Each pair of juniors play one match. Each pair of seniors play one match. There is no junior versus senior match. The number of girl versus girl matches in junior level is 153, while the number of boy versus boy matches in senior level is 276. What is the number of matches a boy plays against a girl ?

A. 450
B. 468
C. 998
D. 1098
E. 1208

Are You Up For the Challenge: 700 Level Questions

A. 450
B. 468
C. 998
D. 1098
E. 1208

Are You Up For the Challenge: 700 Level Questions

ArunSharma12 · Answer

Junior level: number of girl vs girl matches = 153 =  G_C_2 
153 =  \frac{G(G-1)}{2} 
G(G-1)=306
G = 18; B = 43-18=25
number of boy vs girl matches = 25*14 = 450

Senior level: number of boy vs boy matches = 276 =  B_C_2 
276 =  \frac{B(B-1)}{2} 
B(B-1)=552
B = 24; G = 51-24=27
number of boy vs girl matches = 24*27 = 648

number of matches a boy plays against a girl = 450 + 648 = 1098
Ans: D

ccooley · Answer

Jot down what they're telling you:

43 juniors = ?? boys/girls
51 seniors = ?? boys/girls

each pair j plays 1 match
each pair s plays 1 match

j: girl vs girl = 153
s: boy vs boy = 276

Total girl vs boy matches?

---

I have two thoughts at this point. One, it seems like I don't have much information to work with. But this is a Problem Solving question, so there HAS to be enough info to narrow it down to one correct answer! Two, what I'd really like to know is the number of girls and boys in each grade. If I did that, I think I could figure out the number of girl vs boy matches, because everybody plays against everybody else.

Okay, let's start with the juniors. I should be able to work with just the juniors first, and then with just the seniors, because the grades never play against each other. So really I'm solving two separate problems, and then just adding the results together at the end.

For just juniors, there are 153 girl vs girl matches. There are 43 students in total.

Because every student plays every other student, every girl must play every other girl at some point. So the number of matches is the same as the number of unique pairs of junior girls.

How many unique pairs are there in a set of people? Well, there's a formula! Suppose the number of junior girls is "g". Then, the number of different pairs of girls would be g(g-1)/2.

Now we have an equation: 153 = g(g-1)/2.

Solve it:

306 = g(g-1)

There are two different ways to tackle this. I could factor it out and then solve the quadratic. Or, I could guess and check and try to come up with some numbers that fit. 306 is a bit bigger than 15^2, which is 225, and a bit smaller than 20^2, which is 400. My numbers should be something between 15 and 20.

16*(16-1) = 16*15 = 4*60 = 240: too small

17*(17-1) = 17*16 = 272: too small

18*(18-1) = 18*17 = 306: perfect.

The number of girls in the junior class is 18 . Therefore,  the number of boys in the junior class is 43-18 = 25 .

Let's figure out the senior class exactly the same way:

(b)(b-1)/2 = 276
b(b-1) = 552

Testing numbers between 20 and 25, since 20^2 = 400 and 25^2 = 625:

21(21-1) = 21(20) = 420: way too small
24(24-1) = 24(23) = 552: perfect!

The number of boys in the senior class is 24 . Therefore,  the number of girls in the senior class is 51-24 = 27 .

Let's draw a chart:

junior girls = 18
junior boys = 25

senior girls = 27
senior boys = 24

The final step is to figure out how many boy vs girl matches there are in each class.

In the junior class, the number of ways to match 1 girl to 1 boy is 18*25.

In the senior class, it's 27*24.

I really don't want to do the math, so let's look at the answer choices and see if I can estimate or eliminate:

A. 450
B. 468
C. 998
D. 1098
E. 1208

My first though is that since both values I'm adding are multiples of 3, the answer has to be a multiple of 3. Eliminate anything that isn't a multiple of 3:

A. 450
B. 468
 C. 998 
D. 1098
 E. 1208

Also, 18*25 is close to 20*20, which is 400. Since the matches for the juniors are already about 400, and I haven't even added the matches for the seniors yet, the right answer must be much bigger than 400.

So, the right answer has to be  D. 1098. . (You can confirm this by doing the multiplication yourself now, but please don't do that on test day!)

ScottTargetTestPrep · Answer

Solution:

Since every 2 players in the same level play exactly one match, the number of matches in the junior level is 43C2 = (43 x 42)/2 = 903. Similarly, the number of matches in the senior level is 51C2 = (51 x 50)/2 = 1275. Therefore, the total number of matches in the whole tournament is 903 + 1275 = 2178. Now, let’s determine the number of girls and boys in each level.

If we let g = the number of girls in the junior level, we can create the equation:

gC2 = 153

g(g - 1)/2 = 153

g(g - 1) = 306

Since 18 x 17 = 306, g = 18, and therefore, there are 43 - 18 = 25 boys in the junior level.

Similarly, if we let b = the number of boys in the senior level, we can create the equation:

bC2 = 276

b(b - 1)/2 = 276

b(b - 1) = 552

Since 24 x 23 = 552, b = 24 and therefore, there are 51 - 24 = 27 girls in the senior level.

Therefore, the number of boy versus boy matches in the junior level is 25C2 = (25 x 24)/2 = 300 and the number of girl versus girl matches in the senior level is 27C2 = (27 x 26)/2 = 351. Since the total number of matches is 2178, the number of boy versus girl matches is therefore:

2178 - (153 + 276 + 300 + 351) = 2178 - 1080 = 1098

Alternate Solution:

If we let g = the number of girls in the junior level, we can create the equation:

gC2 = 153

g(g - 1)/2 = 153

g(g - 1) = 306

Since 18 x 17 = 306, g = 18, and therefore, there are 43 - 18 = 25 boys in the junior level.

Thus, the number of boy-girl matches in the junior level is 18 x 25 = 450.

Similarly, if we let b = the number of boys in the senior level, we can create the equation:

bC2 = 276

b(b - 1)/2 = 276

b(b - 1) = 552

Since 24 x 23 = 552, b = 24 and therefore, there are 51 - 24 = 27 girls in the senior level.

Thus, the number of boy-girl matches in the senior level is 24 x 27 = 648.

In total, 450 + 648 = 1098 boy-girl matches were played.

Answer: D

exc4libur · Answer

junior: 43
total: 43!/2!*41!=903
girls only: 153=(43-b)!/2!( -2)!, 
 306=x(x-1)(x-2)!/(x-2)!, 306=x(x-1), 
 x=18, 43-b=18, b=25, g=18
boys only: 25!/2!23!=300
b&g: 903-(300+153)=450

senior: 51
total: 51!/2!*49!=1275
boys only: 276=(51-g)!/2!( -2)!, 
 276*2=x(x-1)(x-2)!/(x-2)!, 276*2=x(x-1), 
 x=24, 51-g=24, g=27, b=18
girls only: 27!/2!24!=351
b&g: 1275-(351+276)=648

b&g total: 450+648=1098

ans (D)

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In a tournament, there are 43 junior level and 51 senior level partici

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