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:
$$
= -\frac{2}{4} - \frac{3}{4} = -\frac{5}{4}
$$

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

✔ Answer: $-\frac{3}{4}$

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Question 10:

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

Step 1: Add left to right:
$$
-1.3 + 4.5 = 3.2
$$
$$
3.2 + (-0.7) = 3.2 - 0.7 = 2.5
$$

✔ 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:
$-1\frac{2}{5} = -\frac{7}{5}$

Now evaluate each group:

Group 1:
$$
-\frac{7}{5} + \frac{2}{3}
$$
Common denominator = 15:
$$
= -\frac{21}{15} + \frac{10}{15} = -\frac{11}{15}
$$

Group 2:
$$
\frac{1}{3} + \left(-\frac{3}{5}\right) = \frac{1}{3} - \frac{3}{5}
$$
Common denominator = 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:
$$
\left(-4\frac{1}{5} - 10.5\right) + 0.5
$$

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:
$$
\frac{1}{6} - 7.25 = \frac{1}{6} - \frac{29}{4} \quad \text{(since } 7.25 = \frac{29}{4})
$$
But easier in decimals:
$$
\frac{1}{6} \approx 0.1667,\quad 0.1667 - 7.25 = -7.0833
$$

But let's do exact fractions.

Convert all to fractions:

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

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

Simplify inside:
$$
\frac{1}{6} - \frac{29}{4} = \frac{2}{12} - \frac{87}{12} = -\frac{85}{12}
$$

Now:
$$
\frac{69}{4} + \frac{5}{6} - \frac{85}{12}
$$

Find common denominator = 12:

- $\frac{69}{4} = \frac{207}{12}$
- $\frac{5}{6} = \frac{10}{12}$
- $-\frac{85}{12}$

Add:
$$
\frac{207}{12} + \frac{10}{12} - \frac{85}{12} = \frac{132}{12} = 11
$$

✔ Answer: $11$

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Question 16:

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

Step 1: Simplify inside first parentheses:
$$
-6 + \frac{1}{3} = -\frac{18}{3} + \frac{1}{3} = -\frac{17}{3}
$$

Now:
$$
-\frac{17}{3} + \left(-\frac{1}{3}\right) = -\frac{18}{3} = -6
$$

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 | $-2$ |

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