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If a,b, and c are real numbers, which of the following must be true?
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24 Nov 2025, 01:25
24 Nov 2025, 01:25
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Question Stats:
83% (00:53) correct
16% (01:40) wrong based on 6 sessions
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If \(a\),\(b\), and \(c\) are real numbers, which of the following must be true?
(A). \(a+(b+c)=(a+b)+c\)
(B). \(a-(b-c)=(a-b)-c\)
(C). \(a*(b*c)=(a*b)*c\)
(D). \(\frac{a}{\frac{b}{c}}=\frac{\frac{a}{b}}{c}\)
(A). \(a+(b+c)=(a+b)+c\)
(B). \(a-(b-c)=(a-b)-c\)
(C). \(a*(b*c)=(a*b)*c\)
(D). \(\frac{a}{\frac{b}{c}}=\frac{\frac{a}{b}}{c}\)
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Re: If a,b, and c are real numbers, which of the following must be true?
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11 Dec 2025, 02:09
11 Dec 2025, 02:09
Expert Reply
The properties that must always be true for real numbers are (A) and (C).
(A) $\(a+(b+c)=(a+b)+c\)$
This is the associative law of addition, which holds for all real numbers.
(B) $\(a-(b-c)=(a-b)-c\)$
This is not always true.
Example: let $\(a=1, b=2, c=3\)$.
Left side: $\(1-(2-3)=1-(-1)=2\)$.
Right side: $\((1-2)-3=-1-3=-4\)$.
So (B) fails.
(C) $\(a(b c)=(a b) c\)$
This is the associative law of multiplication, which holds for all real numbers.
(D) $\(\frac{a}{\frac{b}{c}}=\frac{\frac{a}{b}}{c}\)$
Rewrite left side: $\(\frac{a}{b / c}=a \cdot \frac{c}{b}=\frac{a c}{b}\)$.
Right side: $\(\frac{a / b}{c}=\frac{a}{b c}\)$.
These are equal only in special cases (e.g., $\(c^2=1\)$ ), so ( D ) is not always true.
Correct answers: (A) and (C).
(A) $\(a+(b+c)=(a+b)+c\)$
This is the associative law of addition, which holds for all real numbers.
(B) $\(a-(b-c)=(a-b)-c\)$
This is not always true.
Example: let $\(a=1, b=2, c=3\)$.
Left side: $\(1-(2-3)=1-(-1)=2\)$.
Right side: $\((1-2)-3=-1-3=-4\)$.
So (B) fails.
(C) $\(a(b c)=(a b) c\)$
This is the associative law of multiplication, which holds for all real numbers.
(D) $\(\frac{a}{\frac{b}{c}}=\frac{\frac{a}{b}}{c}\)$
Rewrite left side: $\(\frac{a}{b / c}=a \cdot \frac{c}{b}=\frac{a c}{b}\)$.
Right side: $\(\frac{a / b}{c}=\frac{a}{b c}\)$.
These are equal only in special cases (e.g., $\(c^2=1\)$ ), so ( D ) is not always true.
Correct answers: (A) and (C).
gmatclubot
Re: If a,b, and c are real numbers, which of the following must be true? [#permalink]
11 Dec 2025, 02:09
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