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24 Mar 2024, 06:15
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Two amateur wrestling squads—Squad A and Squad B—competed against each other. Each squad had the same (nonzero) number of its members who were contestants. In the competition, each contestant in Squads A and B wrestled against exactly one contestant in the other squad, and no non-contestant member of either squad wrestled. Let x be the proportion of Squad A's members who were contestants in the competition and let y be the proportion of Squad B’s members who were contestants in the competition. In terms of x and y only, what is the proportion of the total number of members in the two squads who were contestants in the competition?
Divide both the numerator and denominator by a.
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Difficulty:
45%
(medium)
Question Stats:
64% (03:20) correct
36%
(03:32)
wrong
based on 1556
sessions
History
Date
Time
Result
Not Attempted Yet
Which one of the following procedures, when applied to the last mathematical expression in the Concise Solution tab, gives a mathematical expression that answers the question asked in the Problem Statement tab?
| Multiply both the numerator and denominator by a. | |
| Multiply both the numerator and denominator by y. | |
| Divide both the numerator and denominator by a. | |
| Subtract a and then divide the result \(a + \frac{x}{y}a\). | |
| Divide by \(a + \frac{x}{y}a\). and then subtract a from the result. |
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Official Answer
Divide both the numerator and denominator by a.
9: No
10: No
12: Yes
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Difficulty:
95%
(hard)
Question Stats:
40% (02:11) correct
60%
(02:16)
wrong
based on 1583
sessions
History
Date
Time
Result
Not Attempted Yet
A given number is a possible value of a’ if, for each value of x in the Table tab, there is a value of a that is compatible with that value of x and the other information provided. For each of the following numbers, select Yes if, given this condition, it is a possible value of a' Otherwise select No.
| Yes | No | |
| 9 | ||
| 10 | ||
| 12 |
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Official Answer
9: No
10: No
12: Yes
\(a' = yb\): Yes
\(a' = b'\): Yes
\(a + b < xa + yb\): No
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Difficulty:
15%
(low)
Question Stats:
75% (01:05) correct
25%
(01:08)
wrong
based on 1446
sessions
History
Date
Time
Result
Not Attempted Yet
Assume the information provided is true. For each of the following mathematical statements, select Yes if, according to the information provided, the statement is true for the possible values of x and y. Otherwise select No.
| Yes | No | |
| \(a' = yb\) | ||
| \(a' = b'\) | ||
| \(a + b < xa + yb\) |
ShowHide Answer
Official Answer
\(a' = yb\): Yes
\(a' = b'\): Yes
\(a + b < xa + yb\): No
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24 Mar 2024, 09:24
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Question 1
Which one of the following procedures, when applied to the last mathematical expression in the Concise Solution tab, gives a mathematical expression that answers the question asked in the Problem Statement tab?
The equation being talked about is \(\frac{2xa}{a+\frac{x}{y}*a}\)
we want this expression in terms of only x and y. We have an extra in both numerator and denominator => \(\frac{2xa}{a(1+\frac{x}{y})}\)
So, divide numerator and denominator by a to get \(\frac{2x}{1+\frac{x}{y}}\)
Question 2.
A given number is a possible value of [i]a’ if, for each value of x in the Table tab, there is a value of a that is compatible with that value of x and the other information provided. For each of the following numbers, select Yes if, given this condition, it is a possible value of a' Otherwise select No.[/i]
Now x has values 1/3, 1/4 etc, and xa has to be integer as xa is the number of contestants. Thus a has to be a multiple of LCM of denominators of values of x. Thus 12 or multiples of 12.
Hence 9 and 10 are not valid while 12 is.
Question 3.
Assume the information provided is true. For each of the following mathematical statements, select [i]Yes if, according to the information provided, the statement is true for the possible values of x and y. Otherwise select No.[/i]
a'=b'=xa=yb.
Thus, a'=yb is true.
a'=b' is also true.
However, \(a + b < xa + yb\) is not true as xa<a and yb<b, resulting in a+b>xa+yb
Which one of the following procedures, when applied to the last mathematical expression in the Concise Solution tab, gives a mathematical expression that answers the question asked in the Problem Statement tab?
The equation being talked about is \(\frac{2xa}{a+\frac{x}{y}*a}\)
we want this expression in terms of only x and y. We have an extra in both numerator and denominator => \(\frac{2xa}{a(1+\frac{x}{y})}\)
So, divide numerator and denominator by a to get \(\frac{2x}{1+\frac{x}{y}}\)
Question 2.
A given number is a possible value of [i]a’ if, for each value of x in the Table tab, there is a value of a that is compatible with that value of x and the other information provided. For each of the following numbers, select Yes if, given this condition, it is a possible value of a' Otherwise select No.[/i]
Now x has values 1/3, 1/4 etc, and xa has to be integer as xa is the number of contestants. Thus a has to be a multiple of LCM of denominators of values of x. Thus 12 or multiples of 12.
Hence 9 and 10 are not valid while 12 is.
Question 3.
Assume the information provided is true. For each of the following mathematical statements, select [i]Yes if, according to the information provided, the statement is true for the possible values of x and y. Otherwise select No.[/i]
a'=b'=xa=yb.
Thus, a'=yb is true.
a'=b' is also true.
However, \(a + b < xa + yb\) is not true as xa<a and yb<b, resulting in a+b>xa+yb
05 Sep 2024, 07:30
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Dhruv0911
The video solution of this question is given here:
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