without calculating which of the following quotients can you determine will be positive.
A: -725 divide (-29)
B: 1,001 divide (-143)
C: -111 divide 37
D: 657 divide (-23)
To determine if a quotient will be positive or negative, we need to consider the signs of the dividend and divisor.
For a quotient to be positive:
- The dividend and divisor can both be positive, or
- The dividend and divisor can both be negative.
Based on this, we can determine that the following quotients will be positive without calculating them:
A: -725 divided by -29
C: -111 divided by 37
To determine which of the given quotients will be positive without calculating, we can use the following logic:
A quotient will be positive if:
- Both the numerator and denominator are positive.
- Both the numerator and denominator are negative.
A quotient will be negative if:
- The numerator is negative and the denominator is positive.
- The numerator is positive and the denominator is negative.
Applying this logic to the given options:
A: -725 divide (-29) -> Both the numerator (-725) and the denominator (-29) are negative, so this quotient will be positive.
B: 1,001 divide (-143) -> The numerator (1,001) is positive, but the denominator (-143) is negative, so this quotient will be negative.
C: -111 divide 37 -> The numerator (-111) is negative, but the denominator (37) is positive, so this quotient will be negative.
D: 657 divide (-23) -> The numerator (657) is positive, but the denominator (-23) is negative, so this quotient will be negative.
Therefore, the only quotient that can be determined to be positive without calculation is option A: -725 divide (-29).
To determine which of the given quotients will be positive without actually calculating them, you need to understand a few basic rules of dividing numbers.
1. Dividing a positive number by a negative number: The quotient will be negative.
2. Dividing a negative number by a positive number: The quotient will be negative.
3. Dividing a negative number by a negative number: The quotient will be positive.
Now let's apply these rules to each option:
A: -725 divided by (-29)
-725 is negative, and -29 is also negative. According to rule 3, when dividing two negative numbers, the quotient will be positive. Therefore, the quotient for option A will be positive.
B: 1,001 divided by (-143)
1,001 is positive, and -143 is negative. According to rule 2, when dividing a positive number by a negative number, the quotient will be negative. Therefore, the quotient for option B will be negative.
C: -111 divided by 37
-111 is negative, and 37 is positive. According to rule 2, when dividing a negative number by a positive number, the quotient will be negative. Therefore, the quotient for option C will be negative.
D: 657 divided by (-23)
657 is positive, and -23 is negative. According to rule 2, when dividing a positive number by a negative number, the quotient will be negative. Therefore, the quotient for option D will be negative.
So, out of the four options, only option A: -725 divided by (-29) will have a positive quotient.