Solve Problems With Rational Numbers - 7th Grade Math

1. What is the net gain/loss for $4$ deposits of $\frac{3}{2}$ and $3$ withdrawals of $\frac{5}{2}$?

   $-\frac{3}{2}$. Total: $4(\frac{3}{2})-3(\frac{5}{2})=6-\frac{15}{2}=\frac{12}{2}-\frac{15}{2}=-\frac{3}{2}$.

2. What is the sign of the quotient of a positive rational number and a negative rational number?

   Negative. Positive divided by negative equals negative.

3. What is the total change after $-2.3$ and then $+1.8$?

   $-0.5$. Add changes: $-2.3+1.8=-0.5$.

4. What is the rule for the sign of a quotient $\frac{a}{b}$ when $a$ and $b$ have different signs?

   The quotient is negative.

5. What is the value of $-1.2\div(-0.3)$?

   $4$. Negative divided by negative equals positive.

6. What is the rule for the sign of a product $(a)(b)$ when $a$ and $b$ have the same sign?

   The product is positive.

7. What is the reciprocal of a nonzero rational number $\frac{a}{b}$?

   $\frac{b}{a}$

8. What is the value of $\left(-\frac{3}{2}\right)^2$?

   $\frac{9}{4}$. Square of negative fraction: $\frac{(-3)^2}{2^2}=\frac{9}{4}$.

9. What is $-3+7$?

   $4$. Start at $-3$, move right $7$ units on number line.

10. What is the average of $-2$, $3$, and $-4$?

    $-1$. Sum and divide by count: $\frac{-2+3+(-4)}{3}=\frac{-3}{3}=-1$.

11. What is $\frac{7}{8}\div\left(-\frac{7}{4}\right)$ in simplest form?

    $-\frac{1}{2}$. Multiply by reciprocal: $\frac{7}{8} \cdot \left(-\frac{4}{7}\right)=-\frac{28}{56}=-\frac{1}{2}$.

12. What is the value of $-\left(-\frac{7}{3}\right)$?

    $\frac{7}{3}$. Negative of negative equals positive.

13. What is the value of $-\frac{2}{3}+\frac{5}{6}$?

    $\frac{1}{6}$. Convert to sixths: $-\frac{4}{6}+\frac{5}{6}=\frac{1}{6}$.

14. What is the sign of the quotient $\frac{-a}{-b}$ when $a>0$ and $b>0$?

    Positive. Negative divided by negative gives positive.

15. What is the value of $\frac{2}{3}+\frac{1}{9}+\left(-\frac{4}{9}\right)$?

    $\frac{1}{3}$. Common denominator: $\frac{6}{9}+\frac{1}{9}-\frac{4}{9}=\frac{3}{9}=\frac{1}{3}$.

16. What is $-8-(-5)$?

    $-3$. Subtracting negative is same as adding positive: $-8+5=-3$.

17. What is the value of $\left(-\frac{5}{6}\right)\div\left(\frac{10}{3}\right)$?

    $-\frac{1}{4}$. Multiply by reciprocal: $\frac{-5}{6} \times \frac{3}{10}=\frac{-15}{60}=\frac{-1}{4}$.

18. What is the value of $\frac{3}{5}\div\left(-\frac{9}{10}\right)$?

    $-\frac{2}{3}$. Multiply by reciprocal: $\frac{3}{5} \times \frac{-10}{9}=\frac{-30}{45}=\frac{-2}{3}$.

19. What is the additive inverse of a rational number $r$?

    $-r$

20. What is the change in temperature if it goes from $-3^\circ\text{C}$ to $5^\circ\text{C}$?

    $8^\circ\text{C}$. Final minus initial: $5-(-3)=5+3=8$.

21. What is the total cost of $3$ items at $-2.40 each (a refund per item)?

    $-7.20. Multiply: $3 \times (-2.40)=-7.20$.

22. What is $\left(-4\right)\left(-\frac{3}{5}\right)$ in simplest form?

    $\frac{12}{5}$. Negative times negative is positive: $4 \cdot \frac{3}{5}=\frac{12}{5}$.

23. What is the value of $-0.75\times 0.4$?

    $-0.3$. Negative times positive equals negative.

24. What is the reciprocal of a nonzero rational number $r$?

    $\frac{1}{r}$. The number that multiplies with $r$ to give $1$.

25. What operation turns division by a nonzero rational into multiplication?

    Multiply by the reciprocal. Division by $r$ equals multiplication by $\frac{1}{r}$.

26. What is $\left| -\frac{11}{6} \right|$?

    $\frac{11}{6}$. Absolute value removes negative sign.

27. What is the value of $-4.2+1.7$?

    $-2.5$. Add the values: $-4.2+1.7=-2.5$.

28. What is the reciprocal of $-\frac{3}{5}$?

    $-\frac{5}{3}$. Flip the fraction and keep the negative sign.

29. What is the sign of the product of two negative rational numbers?

    Positive. Negative times negative equals positive.

30. What is $6-\frac{9}{4}$ in simplest form?

    $\frac{15}{4}$. Convert: $6=\frac{24}{4}$, so $\frac{24}{4}-\frac{9}{4}=\frac{15}{4}$.

31. What is the value of $(-3)^2$?

    $9$. Parentheses mean square the negative: $(-3)(-3)=9$.

32. What is the value of $-3.6-(-1.2)$?

    $-2.4$. Subtracting negative is adding: $-3.6+1.2=-2.4$.

33. What is $\left(-\frac{2}{3}\right)\left(\frac{9}{4}\right)$ in simplest form?

    $-\frac{3}{2}$. Multiply numerators and denominators: $\frac{-2 \cdot 9}{3 \cdot 4}=-\frac{18}{12}=-\frac{3}{2}$.

34. What is the value of $\left(-\frac{3}{4}\right)\times\left(\frac{8}{9}\right)$?

    $-\frac{2}{3}$. Multiply: $\frac{-3 \times 8}{4 \times 9}=\frac{-24}{36}=\frac{-2}{3}$.

35. What is $\frac{3}{4}+\left(-\frac{1}{2}\right)$ in simplest form?

    $\frac{1}{4}$. Convert to common denominator: $\frac{3}{4}-\frac{2}{4}=\frac{1}{4}$.

36. What is the value of $-3^2$ (no parentheses around $-3$)?

    $-9$. Order of operations: $-3^2=-(3^2)=-9$.

37. What is $-\frac{5}{6}+\frac{1}{3}$ in simplest form?

    $-\frac{1}{2}$. Common denominator: $-\frac{5}{6}+\frac{2}{6}=-\frac{3}{6}=-\frac{1}{2}$.

38. What is the sign of the product $(-a)(b)$ when $a>0$ and $b>0$?

    Negative. Negative times positive gives negative.

39. What is $-2.5\div 0.5$?

    $-5$. Negative divided by positive gives negative: $-2.5 \div 0.5=-5$.

40. What is the account balance after a deposit of $12.50 to a balance of $-\$8.75$?

    $3.75. Add deposit: $-8.75+12.50=3.75$.

41. What is the value of $\frac{7}{8}-\left(-\frac{1}{4}\right)$?

    $\frac{9}{8}$. Subtracting negative is adding: $\frac{7}{8}+\frac{1}{4}=\frac{7}{8}+\frac{2}{8}$.

42. What is the value of $\left| -\frac{11}{6} \right|$?

    $\frac{11}{6}$. Absolute value removes the negative sign.

43. What is the value of $\left(-2\frac{1}{2}\right)\times\left(\frac{4}{5}\right)$?

    $-2$. Convert: $-\frac{5}{2} \times \frac{4}{5}=\frac{-20}{10}=-2$.

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