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 simplifying fractions and decimals carefully.

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

Convert to common denominator (LCM of 2 and 4 is 4):
$$
-\frac{2}{4} - \frac{3}{4} = -\frac{5}{4}
$$

Now add:
$$
\frac{1}{2} + \left(-\frac{5}{4}\right) = \frac{1}{2} - \frac{5}{4}
$$

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

Add step by step:
$$
-1.3 + 4.5 = 3.2
$$
$$
3.2 + (-0.7) = 3.2 - 0.7 = 2.5
$$

✔ Answer: $2.5$

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Question 11:

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

First, convert mixed number:
$$
-1\frac{2}{5} = -\frac{7}{5}
$$

So:
$$
\left(-\frac{7}{5} + \frac{2}{3}\right) + \left(\frac{1}{3} - \frac{3}{5}\right)
$$

Find a common denominator for each pair (LCM of 5 and 3 is 15):

First group:
$$
-\frac{7}{5} = -\frac{21}{15}, \quad \frac{2}{3} = \frac{10}{15}
\Rightarrow -\frac{21}{15} + \frac{10}{15} = -\frac{11}{15}
$$

Second group:
$$
\frac{1}{3} = \frac{5}{15}, \quad -\frac{3}{5} = -\frac{9}{15}
\Rightarrow \frac{5}{15} - \frac{9}{15} = -\frac{4}{15}
$$

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

First simplify inside parentheses:
$$
-1.2 + 7.5 = 6.3
$$

Then:
$$
11.2 + 6.3 = 17.5
$$

✔ Answer: $17.5$

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Question 13:

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

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

Convert mixed number:
$$
-4\frac{1}{5} = -\frac{21}{5} = -4.2
$$

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

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

✔ Answer: $-14.2$

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Question 15:

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

Simplify inside parentheses:
$$
\frac{1}{6} - 7.25 = \frac{1}{6} - \frac{29}{4} \quad (\text{since } 7.25 = \frac{29}{4})
$$

But easier to use decimals:
$$
\frac{1}{6} \approx 0.1667, \quad 7.25 = 7.25
\Rightarrow 0.1667 - 7.25 = -7.0833
$$

Now compute:
$$
17.25 + \frac{5}{6} + (-7.0833)
$$

$\frac{5}{6} \approx 0.8333$

So:
$$
17.25 + 0.8333 = 18.0833
$$
$$
18.0833 - 7.0833 = 11
$$

But let's do it exactly:

Convert all to fractions:

- $17.25 = \frac{69}{4}$
- $\frac{5}{6}$
- $\frac{1}{6}$
- $7.25 = \frac{29}{4}$

So:
$$
\frac{69}{4} + \frac{5}{6} + \left(\frac{1}{6} - \frac{29}{4}\right)
$$

Distribute:
$$
= \frac{69}{4} + \frac{5}{6} + \frac{1}{6} - \frac{29}{4}
$$

Combine like terms:

- Fractions with denominator 4: $\frac{69}{4} - \frac{29}{4} = \frac{40}{4} = 10$
- Fractions with denominator 6: $\frac{5}{6} + \frac{1}{6} = \frac{6}{6} = 1$

Total: $10 + 1 = 11$

✔ Answer: $11$

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Question 16:

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

Simplify step by step:

First: $-6 + \frac{1}{3} = -\frac{18}{3} + \frac{1}{3} = -\frac{17}{3}$

Then add $-\frac{1}{3}$:
$$
-\frac{17}{3} - \frac{1}{3} = -\frac{18}{3} = -6
$$

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

✔ 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 | $-2$ |

You can now fill these into the boxes on your worksheet!