Adding and Subtracting Rational Numbers using properties worksheet ... - Free Printable

Let's solve each expression step by step, using properties of operations (like the associative and commutative properties of addition, and converting fractions to common denominators or decimals as needed).

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Question 9:

Evaluate:
$$
\frac{1}{2} + \left(-\frac{1}{2} + \left(-\frac{3}{4}\right)\right)
$$

Step 1: Simplify inside the parentheses:
$$
-\frac{1}{2} + \left(-\frac{3}{4}\right) = -\frac{1}{2} - \frac{3}{4}
$$
Convert to common denominator (4):
$$
= -\frac{2}{4} - \frac{3}{4} = -\frac{5}{4}
$$

Step 2: Add to $\frac{1}{2}$:
$$
\frac{1}{2} + \left(-\frac{5}{4}\right) = \frac{1}{2} - \frac{5}{4}
$$
Convert $\frac{1}{2}$ to fourths:
$$
= \frac{2}{4} - \frac{5}{4} = -\frac{3}{4}
$$

✔ Answer: $-\frac{3}{4}$

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Question 10:

Evaluate:
$$
-1.3 + 4.5 + (-0.7)
$$

Step 1: Combine the decimals:
$$
-1.3 + 4.5 = 3.2
$$
Then:
$$
3.2 + (-0.7) = 3.2 - 0.7 = 2.5
$$

✔ Answer: $2.5$

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Question 11:

Evaluate:
$$
\left(-1\frac{2}{5} + \frac{2}{3}\right) + \left(\frac{1}{3} + \left(-\frac{3}{5}\right)\right)
$$

Step 1: Convert mixed number:
$$
-1\frac{2}{5} = -\frac{7}{5}
$$

So first group:
$$
-\frac{7}{5} + \frac{2}{3}
$$
Find LCD of 5 and 3 → 15:
$$
= -\frac{21}{15} + \frac{10}{15} = -\frac{11}{15}
$$

Second group:
$$
\frac{1}{3} + \left(-\frac{3}{5}\right) = \frac{1}{3} - \frac{3}{5}
$$
LCD = 15:
$$
= \frac{5}{15} - \frac{9}{15} = -\frac{4}{15}
$$

Now add both results:
$$
-\frac{11}{15} + \left(-\frac{4}{15}\right) = -\frac{15}{15} = -1
$$

✔ Answer: $-1$

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Question 12:

Evaluate:
$$
11.2 + (-1.2 + 7.5)
$$

Step 1: Simplify inside parentheses:
$$
-1.2 + 7.5 = 6.3
$$

Step 2: Add to 11.2:
$$
11.2 + 6.3 = 17.5
$$

✔ Answer: $17.5$

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Question 13:

Evaluate:
$$
-2.7 + \frac{1}{2} + (-1.3) + \left(-\frac{1}{2}\right)
$$

Step 1: Group like terms:
- Decimals: $-2.7 + (-1.3) = -4.0$
- Fractions: $\frac{1}{2} + \left(-\frac{1}{2}\right) = 0$

So total:
$$
-4.0 + 0 = -4
$$

✔ Answer: $-4$

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Question 14:

Evaluate:
$$
\left(-4\frac{1}{5} - 10.5\right) + 0.5
$$

Step 1: Convert $-4\frac{1}{5}$ to improper fraction or decimal:
$$
-4\frac{1}{5} = -4.2
$$

Now:
$$
-4.2 - 10.5 = -14.7
$$

Then:
$$
-14.7 + 0.5 = -14.2
$$

✔ Answer: $-14.2$

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Question 15:

Evaluate:
$$
17.25 + \frac{5}{6} + \left(\frac{1}{6} - 7.25\right)
$$

Step 1: Simplify inside parentheses:
$$
\frac{1}{6} - 7.25 = -7.25 + \frac{1}{6}
$$

But better to combine all terms.

Note: $\frac{5}{6} + \frac{1}{6} = 1$, so:

$$
17.25 + \frac{5}{6} + \frac{1}{6} - 7.25 = (17.25 - 7.25) + \left(\frac{5}{6} + \frac{1}{6}\right)
$$

$$
= 10 + 1 = 11
$$

✔ Answer: $11$

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Question 16:

Evaluate:
$$
\left(-6 + \frac{1}{3}\right) + \left(-\frac{1}{3}\right) + 4
$$

Step 1: Simplify:
$$
-6 + \frac{1}{3} = -5\frac{2}{3}
$$

Now:
$$
-5\frac{2}{3} + \left(-\frac{1}{3}\right) = -5\frac{2}{3} - \frac{1}{3} = -6
$$

Then:
$$
-6 + 4 = -2
$$

✔ Answer: $-2$

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✔ Final Answers:

| Question | Answer |
|---------|----------------|
| 9 | $-\frac{3}{4}$ |
| 10 | $2.5$ |
| 11 | $-1$ |
| 12 | $17.5$ |
| 13 | $-4$ |
| 14 | $-14.2$ |
| 15 | $11$ |
| 16 | $-2$ |

Let me know if you'd like these boxed for submission!