Solve Multi-Step Rational Number Problems - 7th Grade Math

1. What is $-18+\left(\frac{3}{4}\times 8\right)$?

   $-12$. Calculate $\frac{3}{4}\times^8=6$, then $-18+6=-12$.

2. What is the total cost of $3$ items at $-2.50 each (a credit per item)?

   $-7.50. Multiply quantity by price: $3\times(-\$2.50)=-\$7.50$.

3. What is the length of each side piece if $27.5$ inches is centered with a $9.75$ inch bar?

   $8.875$ inches. Remaining space divided by $2$: $(27.5 - 9.75) ÷ 2 = 17.75 ÷ 2 = 8.875$.

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

   $\frac{11}{2}$. Subtracting negative is adding: $4 + \frac{3}{2} = \frac{8}{2} + \frac{3}{2} = \frac{11}{2}$.

5. What is the decimal form of the fraction $\frac{3}{4}$?

   $0.75$. Divide numerator by denominator: $3 ÷ 4 = 0.75$.

6. What is $-2(3.5)+4$?

   $-3$. Multiply first: $-2(3.5)=-7$, then add $4$.

7. What is the value of $\frac{5}{6} - \frac{1}{4}$ in simplest form?

   $\frac{7}{12}$. Find common denominator $12$: $\frac{10}{12} - \frac{3}{12} = \frac{7}{12}$.

8. Evaluate $3(-2)+4$.

   $-2$. Multiply first: $-6 + 4 = -2$.

9. What is $(-0.4)\div 0.2$?

   $-2$. Negative divided by positive gives negative: $-0.4÷0.2=-2$.

10. What is $\frac{2}{3}+\frac{1}{6}$ in simplest form?

    $\frac{5}{6}$. Find common denominator 6: $\frac{4}{6} + \frac{1}{6} = \frac{5}{6}$.

11. What is the value of $\left(-0.5\right)\left(-0.8\right)$?

    $0.4$. Two negatives multiply to give positive: $(-0.5)(-0.8) = 0.4$.

12. What is $0.08$ written as a percent?

    $8\%$. Move decimal two places right to convert decimal to percent.

13. What is $6-(-\frac{3}{4})$?

    $6\frac{3}{4}$. Subtracting negative equals adding positive: $6 + \frac{3}{4}$.

14. What is $\frac{3}{4}$ written as a decimal?

    $0.75$. Divide numerator by denominator: $3 ÷ 4 = 0.75$.

15. What is $1\frac{1}{2}-\frac{2}{3}$ in simplest form?

    $\frac{5}{6}$. Convert to improper: $\frac{3}{2} - \frac{2}{3} = \frac{9}{6} - \frac{4}{6} = \frac{5}{6}$.

16. What is $\left(2.4-3\right)\div(-0.3)$?

    $2$. Simplify: $(2.4-3)=-0.6$, then $-0.6÷(-0.3)=2$.

17. State the order of operations used to evaluate expressions with rational numbers.

    Parentheses, exponents, multiply/divide, add/subtract (left to right). PEMDAS ensures consistent evaluation of complex expressions.

18. What is $\frac{2}{5}\times(-15)$?

    $-6$. Multiply: $\frac{2}{5}\times(-15)=\frac{-30}{5}=-6$.

19. What is $\frac{1}{2}-\left(-\frac{1}{8}\right)$ in simplest form?

    $\frac{5}{8}$. Subtracting negative is addition: $\frac{1}{2}+\frac{1}{8}=\frac{5}{8}$.

20. What is a quick estimate for $\frac{27.5 - 9.75}{2}$ to check reasonableness?

    $\approx 9$. Round to estimate: $(28 - 10) ÷ 2 = 18 ÷ 2 = 9$.

21. What is the final price after $20\%$ off a $35 item?

    $28. Subtract discount from original: $\$35-\$7=\$28$.

22. A price is $80$ dollars after a 20\text{%} discount. What was the original price?

    $100$ dollars. If $80 is 80% of original, divide: $80 ÷ 0.8 = 100$.

23. What is the change in temperature from $-4^\circ$ to $7^\circ$ (final minus initial)?

    $11^\circ$. Final minus initial: $7 - (-4) = 7 + 4 = 11$.

24. What is $-\frac{2}{3}+\frac{5}{6}$ in simplest form?

    $\frac{1}{6}$. Find common denominator $6$: $-\frac{4}{6}+\frac{5}{6}=\frac{1}{6}$.

25. What property is shown by $6\left(2+3\right)=6\cdot 2+6\cdot 3$?

    Distributive property. Shows multiplication distributed over addition.

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

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

27. What is the decimal form of the fraction $\frac{3}{4}$?

    $0.75$. Divide numerator by denominator: $3 ÷ 4 = 0.75$.

28. What is $\frac{7}{10}+\left(-\frac{9}{20}\right)$ in simplest form?

    $\frac{1}{4}$. Common denominator $20$: $\frac{14}{20}-\frac{9}{20}=\frac{5}{20}=\frac{1}{4}$.

29. What is the discount price of $40$ dollars after a $15\%$ discount?

    $34$. $15\%$ of $40$ is $6$, so discounted price is $40 - 6 = 34$.

30. Identify the best estimate for $\frac{19.8}{4.9}$ using compatible numbers.

    $\frac{20}{5}=4$. Round to compatible numbers: $20 \div 5 = 4$.

31. What is the discount amount for $20\%$ off a $35 item?

    $7. Calculate $20\%$ of $35: $0.20\times35=7$.

32. A bank balance is $-12.50$ dollars and you deposit $20$ dollars. What is the new balance?

    $7.50$ dollars. Add deposit to negative balance: $-12.50 + 20 = 7.50$.

33. What is the fraction form of the decimal $0.6$ in simplest form?

    $\frac{3}{5}$. Write as $\frac{6}{10}$ and simplify by dividing by $2$.

34. What is the new hourly wage after a $10\%$ raise on $25 per hour?

    $27.50. $10\%$ of $25 is $2.50, so new wage is $\$25+\$2.50$.

35. What is the total cost of $3$ items at $-2.50$ dollars each (a refund per item)?

    $-7.50$. Multiply quantity by unit price: $3 \times (-2.50) = -7.50$.

36. What is the value of $-2\left(\frac{3}{5}\right) + \frac{1}{5}$ in simplest form?

    $-1$. Multiply first: $-\frac{6}{5} + \frac{1}{5} = -\frac{5}{5} = -1$.

37. What is the value of $\frac{2}{3} + \frac{1}{6} - \frac{1}{2}$?

    $\frac{1}{3}$. Common denominator $6$: $\frac{4}{6} + \frac{1}{6} - \frac{3}{6} = \frac{2}{6} = \frac{1}{3}$.

38. What is the new hourly pay if $25$ dollars increases by $10\%$?

    $27.50$. $10\%$ of $25$ is $2.50$, so new pay is $25 + 2.50 = 27.50$.

39. What is the value of $18 - 3\left(2 - 5\right)$?

    $27$. Evaluate parentheses first: $18 - 3(-3) = 18 + 9 = 27$.

40. What is the product $(-3)\times\frac{1}{2}$?

    $-\frac{3}{2}$. Negative times positive gives negative; $3 \times \frac{1}{2} = \frac{3}{2}$.

41. A door is $27\frac{1}{2}$ inches wide and a centered bar is $9\frac{3}{4}$ inches long. What is the distance from each edge?

    $8\frac{7}{8}$ inches. Subtract bar from door, divide by 2: $(27.5 - 9.75) \div 2 = 8.875$

42. A worker earns $25$ dollars per hour and gets a $10\%$ raise. What is the new hourly wage?

    $27.50$ dollars per hour. $10\%$ of $25$ is $2.50$; add to get $27.50.

43. What is $-3.5-(-1.2)$?

    $-2.3$. Subtracting a negative is addition: $-3.5+1.2=-2.3$.

44. What is the value of $-3.5 + 2.1$?

    $-1.4$. Add a negative and positive: $-3.5 + 2.1 = -1.4$.

45. What is $15\%$ written as a decimal?

    $0.15$. Move decimal point two places left to convert percent to decimal.

46. What is a quick estimate for each edge distance: door $27\frac{1}{2}$ in, bar $9\frac{3}{4}$ in?

    $\text{About }9\text{ in}$. Round: door $≈28$ in, bar $≈10$ in, so $(28-10)÷2≈9$ in.

47. What is $-2.4+1.7$?

    $-0.7$. Add by moving right on number line: $-2.4 + 1.7 = -0.7$.

48. Evaluate $\frac{1}{2}(8-14)$.

    $\!-3$. Evaluate parentheses first: $\frac{1}{2}(-6) = -3$.

49. What is $\left(-\frac{3}{4}\right)\div\left(\frac{3}{8}\right)$?

    $-2$. Multiply by reciprocal: $-\frac{3}{4}\times\frac{8}{3}=-2$.

50. What is 15\text{%} written as a decimal?

    $0.15$. Move decimal two places left to convert percent to decimal.

51. What is the total change after $-2.5$ then $+1.75$?

    $-0.75$. Add the changes: $-2.5+1.75=-0.75$.

52. What property justifies rewriting $a(b+c)$ as $ab+ac$?

    Distributive property. Allows multiplication to be distributed over addition.

53. What is the value of $\left(-12\right) \div 3$?

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

54. What is $0.6$ written as a fraction in simplest form?

    $\frac{3}{5}$. Write as $\frac{6}{10}$ and simplify by dividing by 2.

55. What is the exact distance from each door edge if centered: door $27\frac{1}{2}$ in, bar $9\frac{3}{4}$ in?

    $8\frac{7}{8}\text{ in}$. $(27\frac{1}{2}-9\frac{3}{4})÷2=17\frac{3}{4}÷2=8\frac{7}{8}$ inches.

56. What is the fraction form of the decimal $0.6$ in simplest terms?

    $\frac{3}{5}$. Write $0.6$ as $\frac{6}{10}$ and simplify by dividing by $2$.

57. What is $3\left(\frac{2}{3}-\frac{1}{6}\right)$ in simplest form?

    $\frac{3}{2}$. Simplify inside first: $\frac{4}{6}-\frac{1}{6}=\frac{3}{6}=\frac{1}{2}$, then multiply.

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