Divide Rational Numbers - 7th Grade Math

1. Find the value of $-\left(\frac{7}{5}\right)$.

   $-\frac{7}{5}$. Negating a positive fraction makes it negative.

2. What is $\frac{0}{q}$ for any integer $q$ with $q \ne 0$?

   $0$. Zero divided by any nonzero number equals zero.

3. What is the meaning of $-20 \div 4$ in context of temperature change per hour?

   A decrease of $5$ degrees per hour. $-20$ degrees over $4$ hours means $5$ degrees drop per hour.

4. What is the quotient $-24 \div 6$?

   $-4$. Negative divided by positive gives negative result.

5. What is the value of $\frac{-p}{-q}$ in simplest sign form, assuming $q \ne 0$?

   $\frac{p}{q}$. Two negatives cancel out to give positive.

6. What identity shows a negative sign can be placed outside or in the numerator: $-\left(\frac{p}{q}\right)$?

   $-\left(\frac{p}{q}\right) = \frac{-p}{q}$. The negative can be moved from outside to the numerator.

7. What quotient represents "$-18$ dollars shared equally among $3$ people"?

   $-18 \div 3$. Division models sharing equally, including debt (negative values).

8. What is the result of dividing any nonzero integer by $0$, such as $\frac{5}{0}$?

   Undefined (division by $0$ is not allowed). Division by zero violates the fundamental rule of division.

9. Identify the value: $(-12) \div (-3)$.

   $4$. Negative divided by negative gives positive: $(-12) \div (-3) = 4$.

10. What is the quotient $\frac{5}{6} \div \left(-\frac{1}{3}\right)$? (Simplify.)

    $-\frac{5}{2}$. $\frac{5}{6} \div (-\frac{1}{3}) = \frac{5}{6} \times (-3) = -\frac{15}{6} = -\frac{5}{2}$.

11. Identify the value: $7 \div (-1)$.

    $-7$. Dividing by $-1$ changes the sign: $7 \div (-1) = -7$.

12. Identify whether $\frac{5}{0}$ is defined or undefined.

    Undefined.

13. What is the quotient $-\frac{3}{4} \div 6$? (Write as a simplified fraction.)

    $-\frac{1}{8}$. $-\frac{3}{4} \div 6 = -\frac{3}{4} \times \frac{1}{6} = -\frac{3}{24} = -\frac{1}{8}$

14. What is the quotient $24 \div (-6)$?

    $-4$. Positive divided by negative gives negative result.

15. What is the sign rule for integer division when the signs are different?

    Different signs: the quotient is negative.

16. Which is equivalent to $-\left(\frac{8}{5}\right)$: $\frac{-8}{5}$ or $\frac{8}{-5}$?

    Both: $\frac{-8}{5}$ and $\frac{8}{-5}$. By the identity, negative can be in numerator or denominator.

17. Identify the value: $(-12) \div 3$.

    $-4$. Negative divided by positive gives negative: $-12 \div 3 = -4$.

18. Identify the value: $(-15) \div 5$.

    $-3$. Different signs yield negative: $(-15) \div 5 = -3$.

19. Find the sign of the quotient without computing: $(-24) \div (-6)$.

    Positive. Same signs (negative ÷ negative) yield positive quotient.

20. What is the quotient $-24 \div (-6)$?

    $4$. Two negatives make a positive: $(-24) \div (-6) = 4$.

21. What is the simplified form of $\frac{-10}{15}$ in lowest terms?

    $-\frac{2}{3}$.

22. What quotient represents splitting a debt of $24 equally among $6$ people?

    $\frac{-24}{6}$.

23. Identify the rational number written as an integer: $\frac{12}{-3}$.

    $-4$. $12 \div (-3) = -4$ since positive/negative = negative.

24. Find the value of $\frac{-7}{5}$.

    $-\frac{7}{5}$. Negative numerator makes the fraction negative.

25. What is the sign of $\frac{p}{q}$ when $p>0$ and $q>0$?

    Positive. Positive divided by positive yields positive.

26. What condition must be true about the divisor to divide integers?

    The divisor must be nonzero. Division by zero is undefined in mathematics.

27. What is the equivalent form of $-\left(\frac{p}{q}\right)$ using a negative denominator?

    $\frac{p}{-q}$. The negative sign can be placed on the denominator.

28. Find the sign of the quotient without computing: $(-24) \div 6$.

    Negative. Different signs (negative ÷ positive) yield negative quotient.

29. What is the sign of $\frac{p}{q}$ when $p<0$ and $q>0$?

    Negative. Negative divided by positive yields negative.

30. What type of number is $ p div q $ when $p$ and $q$ are integers and $q \ne 0$?

    A rational number. Any quotient of integers (with nonzero divisor) is rational.

31. What is the sign of $\frac{p}{q}$ when $p<0$ and $q<0$?

    Positive. Two negatives make a positive when dividing.

32. Find the value of $\frac{7}{-5}$.

    $-\frac{7}{5}$. Negative denominator makes the fraction negative.

33. What is the sign rule for dividing two integers with the same sign?

    The quotient is positive. Same signs divide to give a positive result.

34. Identify the correct statement: $\frac{-p}{-q}=\frac{p}{q}$ or $\frac{-p}{-q}=-\frac{p}{q}$.

    $\frac{-p}{-q}=\frac{p}{q}$.

35. What is the definition of a rational number in terms of integers $p$ and $q$?

    A number that can be written as $\frac{p}{q}$ with $p,q$ integers and $q \ne 0$.

36. What is $\frac{p}{0}$ for any integer $p$?

    Undefined (division by $0$ is not allowed). Cannot divide by zero; the operation is undefined.

37. What condition must be true about the divisor to divide integers and get a rational number?

    The divisor must be nonzero: $q \ne 0$. Division by zero is undefined in mathematics.

38. What is the equivalent form of $-\left(\frac{p}{q}\right)$ using a negative numerator?

    $\frac{-p}{q}$. The negative sign can be placed on the numerator.

39. What identity shows a negative sign can be placed in the denominator: $-\left(\frac{p}{q}\right)$?

    $-\left(\frac{p}{q}\right) = \frac{p}{-q}$. The negative can also be placed in the denominator.

40. Which is greater: $\frac{-3}{5}$ or $\frac{3}{-5}$?

    They are equal: $\frac{-3}{5}=\frac{3}{-5}$.

41. What is the definition of a rational number in fraction form using integers?

    A number of the form $\frac{p}{q}$ where $p,q$ are integers and $q \ne 0$. This is the standard definition of a rational number.

42. What is the value of $\frac{0}{q}$ for any integer $q$ with $q \ne 0$?

    $0$. Zero divided by any nonzero number equals zero.

43. What is the simplified form of $\frac{9}{-12}$ in lowest terms?

    $-\frac{3}{4}$.

44. Identify the value: $12 \div (-3)$.

    $-4$. Positive divided by negative gives negative: $12 \div (-3) = -4$.

45. What is $-\left(\frac{p}{q}\right)$ equal to in terms of $p$ and $q$?

    $-\left(\frac{p}{q}\right)=\frac{-p}{q}=\frac{p}{-q}$.

46. What is the sign rule for dividing two integers with different signs?

    The quotient is negative. Different signs divide to give a negative result.

47. What is the sign of $\frac{p}{q}$ when $p>0$ and $q<0$?

    Negative. Positive divided by negative yields negative.

48. What complete identity connects all three equivalent forms of a negative rational quotient?

    $-\left(\frac{p}{q}\right)=\frac{-p}{q}=\frac{p}{-q}$. All three forms represent the same negative rational number.

49. What is the sign rule for integer division when the signs are the same?

    Same signs: the quotient is positive.

50. What is the unit rate if a temperature changes by $-12^\circ\text{C}$ over $ 4 $ hours?

    $\frac{-12}{4} = -3^\circ\text{C}$ per hour.

Study these flashcards