To determine whether a quotient is positive or negative without calculating each one, we need to consider the signs of the numerator and the denominator in each case. Specifically, a quotient is positive if both the numerator and denominator are either both positive or both negative. Conversely, a quotient is negative if one is positive and the other is negative.
Let’s analyze each option:
A. \( 657 / -23 \)
- Numerator: \( 657 \) (positive)
- Denominator: \( -23 \) (negative)
- Sign: Negative (since one is positive and the other is negative)
B. \( 1001 / -143 \)
- Numerator: \( 1001 \) (positive)
- Denominator: \( -143 \) (negative)
- Sign: Negative (since one is positive and the other is negative)
C. \( -725 / -29 \)
- Numerator: \( -725 \) (negative)
- Denominator: \( -29 \) (negative)
- Sign: Positive (since both are negative)
D. \( -111 / 37 \)
- Numerator: \( -111 \) (negative)
- Denominator: \( 37 \) (positive)
- Sign: Negative (since one is negative and the other is positive)
Based on this analysis, we can conclude that the only quotient that will be positive is \( C. -725 / -29 \).