Subtract Rational Numbers Lesson Plans & Worksheets - Free Printable

It seems you've uploaded an image containing a math worksheet focused on adding and subtracting rational numbers. Below, I will solve the problems step by step and explain the solutions.

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Problem 1:

Simplify each expression.

#### (a) \( \frac{3}{4} + \left(-\frac{1}{4}\right) \)

- Step 1: Identify the fractions.
- The fractions are \( \frac{3}{4} \) and \( -\frac{1}{4} \).
- Both have the same denominator (4).

- Step 2: Add the numerators.
- Numerator: \( 3 + (-1) = 3 - 1 = 2 \).
- Denominator remains the same: 4.

- Step 3: Write the result.
- \( \frac{3}{4} + \left(-\frac{1}{4}\right) = \frac{2}{4} \).

- Step 4: Simplify the fraction.
- \( \frac{2}{4} = \frac{1}{2} \).

- Final Answer:
\[
\boxed{\frac{1}{2}}
\]

#### (b) \( -\frac{5}{6} + \frac{1}{6} \)

- Step 1: Identify the fractions.
- The fractions are \( -\frac{5}{6} \) and \( \frac{1}{6} \).
- Both have the same denominator (6).

- Step 2: Add the numerators.
- Numerator: \( -5 + 1 = -4 \).
- Denominator remains the same: 6.

- Step 3: Write the result.
- \( -\frac{5}{6} + \frac{1}{6} = -\frac{4}{6} \).

- Step 4: Simplify the fraction.
- \( -\frac{4}{6} = -\frac{2}{3} \).

- Final Answer:
\[
\boxed{-\frac{2}{3}}
\]

#### (c) \( \frac{7}{8} + \left(-\frac{3}{8}\right) \)

- Step 1: Identify the fractions.
- The fractions are \( \frac{7}{8} \) and \( -\frac{3}{8} \).
- Both have the same denominator (8).

- Step 2: Add the numerators.
- Numerator: \( 7 + (-3) = 7 - 3 = 4 \).
- Denominator remains the same: 8.

- Step 3: Write the result.
- \( \frac{7}{8} + \left(-\frac{3}{8}\right) = \frac{4}{8} \).

- Step 4: Simplify the fraction.
- \( \frac{4}{8} = \frac{1}{2} \).

- Final Answer:
\[
\boxed{\frac{1}{2}}
\]

#### (d) \( -\frac{2}{5} + \frac{3}{5} \)

- Step 1: Identify the fractions.
- The fractions are \( -\frac{2}{5} \) and \( \frac{3}{5} \).
- Both have the same denominator (5).

- Step 2: Add the numerators.
- Numerator: \( -2 + 3 = 1 \).
- Denominator remains the same: 5.

- Step 3: Write the result.
- \( -\frac{2}{5} + \frac{3}{5} = \frac{1}{5} \).

- Final Answer:
\[
\boxed{\frac{1}{5}}
\]

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Problem 2:

Evaluate each expression.

#### (a) \( -7 + (-3) \)

- Step 1: Add the two negative numbers.
- When adding two negative numbers, add their absolute values and keep the negative sign.
- Absolute values: \( | -7 | = 7 \) and \( | -3 | = 3 \).
- Sum of absolute values: \( 7 + 3 = 10 \).

- Step 2: Apply the negative sign.
- \( -7 + (-3) = -10 \).

- Final Answer:
\[
\boxed{-10}
\]

#### (b) \( -12 + 18 \)

- Step 1: Add the two numbers.
- One number is negative (\( -12 \)), and the other is positive (\( 18 \)).
- Subtract the smaller absolute value from the larger absolute value and keep the sign of the larger number.
- Absolute values: \( | -12 | = 12 \) and \( | 18 | = 18 \).
- Difference: \( 18 - 12 = 6 \).
- The larger number is positive, so the result is positive.

- Step 2: Write the result.
- \( -12 + 18 = 6 \).

- Final Answer:
\[
\boxed{6}
\]

#### (c) \( -25 + (-15) \)

- Step 1: Add the two negative numbers.
- When adding two negative numbers, add their absolute values and keep the negative sign.
- Absolute values: \( | -25 | = 25 \) and \( | -15 | = 15 \).
- Sum of absolute values: \( 25 + 15 = 40 \).

- Step 2: Apply the negative sign.
- \( -25 + (-15) = -40 \).

- Final Answer:
\[
\boxed{-40}
\]

#### (d) \( -32 + 45 \)

- Step 1: Add the two numbers.
- One number is negative (\( -32 \)), and the other is positive (\( 45 \)).
- Subtract the smaller absolute value from the larger absolute value and keep the sign of the larger number.
- Absolute values: \( | -32 | = 32 \) and \( | 45 | = 45 \).
- Difference: \( 45 - 32 = 13 \).
- The larger number is positive, so the result is positive.

- Step 2: Write the result.
- \( -32 + 45 = 13 \).

- Final Answer:
\[
\boxed{13}
\]

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Final Answers:

1. (a) \( \boxed{\frac{1}{2}} \)
(b) \( \boxed{-\frac{2}{3}} \)
(c) \( \boxed{\frac{1}{2}} \)
(d) \( \boxed{\frac{1}{5}} \)

2. (a) \( \boxed{-10} \)
(b) \( \boxed{6} \)
(c) \( \boxed{-40} \)
(d) \( \boxed{13} \)

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If you have more questions or need further clarification, feel free to ask!