Here are the step-by-step solutions for each problem on the worksheet.
9) Evaluate: $\frac{1}{2} + \left(-\frac{1}{2} + \left(-\frac{3}{4}\right)\right)$
*
Step 1: Look inside the parentheses first. We have $\frac{1}{2}$ and $-\frac{1}{2}$. These are opposites, so they add up to zero.
$$ \frac{1}{2} + (-\frac{1}{2}) = 0 $$
*
Step 2: Now add the remaining number.
$$ 0 + (-\frac{3}{4}) = -\frac{3}{4} $$
10) Evaluate: $-1.3 + 4.5 + (-0.7)$
*
Step 1: It is often easier to group the negative numbers together first.
$$ -1.3 + (-0.7) = -2.0 $$
*
Step 2: Add that result to the positive number.
$$ -2.0 + 4.5 = 2.5 $$
11) Evaluate: $\left(-1\frac{2}{5} + \frac{2}{3}\right) + \left(\frac{1}{3} + \left(-\frac{3}{5}\right)\right)$
*
Step 1: Use the commutative property to rearrange the terms. Group the fractions with denominator 5 together and the fractions with denominator 3 together.
$$ \left(-1\frac{2}{5} + (-\frac{3}{5})\right) + \left(\frac{2}{3} + \frac{1}{3}\right) $$
*
Step 2: Simplify the "fifths" group.
$$ -1\frac{2}{5} - \frac{3}{5} = -1\frac{5}{5} = -2 $$
*
Step 3: Simplify the "thirds" group.
$$ \frac{2}{3} + \frac{1}{3} = \frac{3}{3} = 1 $$
*
Step 4: Add the two results together.
$$ -2 + 1 = -1 $$
12) Evaluate: $11.2 + (-1.2 + 7.5)$
*
Step 1: Remove the parentheses. Since it is addition, we can just list the numbers.
$$ 11.2 - 1.2 + 7.5 $$
*
Step 2: Combine $11.2$ and $-1.2$. They cancel out nicely.
$$ 11.2 - 1.2 = 10 $$
*
Step 3: Add the remaining number.
$$ 10 + 7.5 = 17.5 $$
13) Evaluate: $-2.7 + \frac{1}{2} + (-1.3) + \left(-\frac{1}{2}\right)$
*
Step 1: Group the decimals together and the fractions together.
Decimals: $-2.7 + (-1.3)$
Fractions: $\frac{1}{2} + (-\frac{1}{2})$
*
Step 2: Solve the decimal part.
$$ -2.7 - 1.3 = -4.0 $$
*
Step 3: Solve the fraction part. They are opposites, so they equal zero.
$$ \frac{1}{2} - \frac{1}{2} = 0 $$
*
Step 4: Add the results.
$$ -4 + 0 = -4 $$
14) Evaluate: $\left(-4\frac{1}{5} - 10.5\right) + 0.5$
*
Step 1: Convert the mixed number to a decimal to make it easier. $\frac{1}{5} = 0.2$, so $-4\frac{1}{5} = -4.2$.
$$ (-4.2 - 10.5) + 0.5 $$
*
Step 2: Combine the two negative numbers inside the parentheses.
$$ -4.2 - 10.5 = -14.7 $$
*
Step 3: Add $0.5$ to the result.
$$ -14.7 + 0.5 = -14.2 $$
15) Evaluate: $17.25 + \frac{5}{6} + \left(\frac{1}{6} - 7.25\right)$
*
Step 1: Rearrange the terms to group the decimals and the fractions separately.
Decimals: $17.25 - 7.25$
Fractions: $\frac{5}{6} + \frac{1}{6}$
*
Step 2: Solve the decimal part.
$$ 17.25 - 7.25 = 10 $$
*
Step 3: Solve the fraction part.
$$ \frac{5}{6} + \frac{1}{6} = \frac{6}{6} = 1 $$
*
Step 4: Add the results together.
$$ 10 + 1 = 11 $$
16) Evaluate: $\left(-6 + \frac{1}{3}\right) + \left(-\frac{1}{3}\right) + 4$
*
Step 1: Look for opposites. We have $+\frac{1}{3}$ and $-\frac{1}{3}$.
$$ \frac{1}{3} + (-\frac{1}{3}) = 0 $$
*
Step 2: Now just add the remaining whole numbers.
$$ -6 + 4 = -2 $$
──────────────────────────────────────
Final Answer:
9) $-\frac{3}{4}$
10) $2.5$
11) $-1$
12) $17.5$
13) $-4$
14) $-14.2$
15) $11$
16) $-2$
Parent Tip: Review the logic above to help your child master the concept of operations with rational numbers worksheet.
