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If a/b > c/d which of the following statement must be true
1. a/b - c/d > 0
2. ad is less than bc
3. ad is greater than bc
4. All of the above?
1. a/b - c/d > 0
2. ad is less than bc
3. ad is greater than bc
4. All of the above?
Most Upvoted Answer
If a/b > c/d which of the following statement must be true1. a/b - c/d...
**If a/b > c/d, which of the following statements must be true?**
To analyze this problem, let's break it down into three cases based on the signs of a/b and c/d.
Case 1: a/b > 0 and c/d > 0
In this case, both fractions are positive. To find the relationship between them, we can cross-multiply the inequality:
ad > bc
1. a/b - c/d > 0
To determine whether this statement is true, we can simplify the expression:
ad/bd - bc/bd > 0
(ad - bc)/bd > 0
Since ad > bc, we know that ad - bc > 0. And since bd > 0 (since b and d are positive), the inequality holds true. Therefore, statement 1 is true in this case.
2. ad > bc
This inequality is given in the problem statement, so it must be true.
3. All of the above
Since both statement 1 and 2 are true in this case, the "All of the above" option is also true.
Case 2: a/b < 0="" and="" c/d="" />< />
In this case, both fractions are negative. Again, let's cross-multiply the inequality:
ad < />
1. a/b - c/d > 0
Simplifying the expression:
ad/bd - bc/bd > 0
(ad - bc)/bd > 0
Since ad < bc,="" we="" know="" that="" ad="" -="" bc="" />< 0.="" however,="" since="" bd="" />< 0="" (since="" b="" and="" d="" are="" negative),="" the="" inequality="" is="" flipped="" when="" dividing="" by="" a="" negative="" number.="" thus,="" (ad="" -="" bc)/bd="" /> 0 becomes (ad - bc)/bd < 0.="" therefore,="" statement="" 1="" is="" false="" in="" this="" />
2. ad < />
This inequality is given in the problem statement, so it must be true.
3. All of the above
Since statement 1 is false in this case, the "All of the above" option is false.
Case 3: a/b < 0="" and="" c/d="" /> 0 or a/b > 0 and c/d < />
In this case, one fraction is negative and the other is positive. Let's cross-multiply the inequality:
ad < />
1. a/b - c/d > 0
Simplifying the expression:
ad/bd - bc/bd > 0
(ad - bc)/bd > 0
Since ad < bc,="" we="" know="" that="" ad="" -="" bc="" />< 0.="" however,="" since="" bd="" /> 0 (since b and d have different signs), the inequality is preserved when dividing by a positive number. Thus, (ad - bc)/bd > 0 remains (ad - bc)/bd > 0. Therefore, statement 1 is true in this case.
2. ad < />
This inequality is given in the problem statement, so it must be true.
3. All of the above
Since both statement 1 and 2 are true in this case, the "All of the above" option is also true.
In summary, the only statement that must be true in all cases is statement 2: ad > bc.
To analyze this problem, let's break it down into three cases based on the signs of a/b and c/d.
Case 1: a/b > 0 and c/d > 0
In this case, both fractions are positive. To find the relationship between them, we can cross-multiply the inequality:
ad > bc
1. a/b - c/d > 0
To determine whether this statement is true, we can simplify the expression:
ad/bd - bc/bd > 0
(ad - bc)/bd > 0
Since ad > bc, we know that ad - bc > 0. And since bd > 0 (since b and d are positive), the inequality holds true. Therefore, statement 1 is true in this case.
2. ad > bc
This inequality is given in the problem statement, so it must be true.
3. All of the above
Since both statement 1 and 2 are true in this case, the "All of the above" option is also true.
Case 2: a/b < 0="" and="" c/d="" />< />
In this case, both fractions are negative. Again, let's cross-multiply the inequality:
ad < />
1. a/b - c/d > 0
Simplifying the expression:
ad/bd - bc/bd > 0
(ad - bc)/bd > 0
Since ad < bc,="" we="" know="" that="" ad="" -="" bc="" />< 0.="" however,="" since="" bd="" />< 0="" (since="" b="" and="" d="" are="" negative),="" the="" inequality="" is="" flipped="" when="" dividing="" by="" a="" negative="" number.="" thus,="" (ad="" -="" bc)/bd="" /> 0 becomes (ad - bc)/bd < 0.="" therefore,="" statement="" 1="" is="" false="" in="" this="" />
2. ad < />
This inequality is given in the problem statement, so it must be true.
3. All of the above
Since statement 1 is false in this case, the "All of the above" option is false.
Case 3: a/b < 0="" and="" c/d="" /> 0 or a/b > 0 and c/d < />
In this case, one fraction is negative and the other is positive. Let's cross-multiply the inequality:
ad < />
1. a/b - c/d > 0
Simplifying the expression:
ad/bd - bc/bd > 0
(ad - bc)/bd > 0
Since ad < bc,="" we="" know="" that="" ad="" -="" bc="" />< 0.="" however,="" since="" bd="" /> 0 (since b and d have different signs), the inequality is preserved when dividing by a positive number. Thus, (ad - bc)/bd > 0 remains (ad - bc)/bd > 0. Therefore, statement 1 is true in this case.
2. ad < />
This inequality is given in the problem statement, so it must be true.
3. All of the above
Since both statement 1 and 2 are true in this case, the "All of the above" option is also true.
In summary, the only statement that must be true in all cases is statement 2: ad > bc.
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If a/b > c/d which of the following statement must be true1. a/b - c/d > 02. ad is less than bc3. ad is greater than bc4. All of the above?
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