Solved Exercises on Fractions with Positive and Negative Values in Grade 7

Master fractions with positive and negative values: operations, comparisons, and applications through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Adding Positive and Negative Fractions
Exercise 1
Add: \(\frac{3}{4} + \left(-\frac{2}{3}\right)\)
Definition:

Signed Fractions: Fractions that can have positive or negative values

Addition method for signed fractions:
  1. Find the LCD of the denominators
  2. Convert to equivalent fractions with common denominator
  3. Add numerators while keeping signs
  4. Simplify if possible
Expression
\(\frac{3}{4} + \left(-\frac{2}{3}\right)\)
Find LCD
\(12\)
Convert
\(\frac{9}{12} + \left(-\frac{8}{12}\right)\)
Result
\(\frac{1}{12}\)
Step 1: Identify the denominators

Denominators are 4 and 3

Step 2: Find the LCD

LCM of 4 and 3 is 12

Step 3: Convert to equivalent fractions

\(\frac{3}{4} = \frac{3 \times 3}{4 \times 3} = \frac{9}{12}\)

\(-\frac{2}{3} = -\frac{2 \times 4}{3 \times 4} = -\frac{8}{12}\)

Step 4: Add the numerators

\(\frac{9}{12} + \left(-\frac{8}{12}\right) = \frac{9 + (-8)}{12} = \frac{1}{12}\)

\(\frac{3}{4} + \left(-\frac{2}{3}\right) = \frac{1}{12}\)
Final answer:

\(\frac{3}{4} + \left(-\frac{2}{3}\right) = \frac{1}{12}\)

Applied rules:

• Sign Rules: Adding a negative is the same as subtracting

• Common Denominator: Required for fraction addition/subtraction

2 Subtracting Negative Fractions
Exercise 2
Subtract: \(\frac{5}{6} - \left(-\frac{1}{4}\right)\)
Definition:

Subtracting a Negative: Subtracting a negative number is equivalent to adding the positive value

Original
\(\frac{5}{6} - \left(-\frac{1}{4}\right)\)
Convert
\(\frac{5}{6} + \frac{1}{4}\)
Find LCD
\(12\)
Convert
\(\frac{10}{12} + \frac{3}{12}\)
Result
\(\frac{13}{12}\)
Step 1: Apply the rule for subtracting a negative

\(\frac{5}{6} - \left(-\frac{1}{4}\right) = \frac{5}{6} + \frac{1}{4}\)

Step 2: Find the LCD

LCM of 6 and 4 is 12

Step 3: Convert to equivalent fractions

\(\frac{5}{6} = \frac{5 \times 2}{6 \times 2} = \frac{10}{12}\)

\(\frac{1}{4} = \frac{1 \times 3}{4 \times 3} = \frac{3}{12}\)

Step 4: Add the numerators

\(\frac{10}{12} + \frac{3}{12} = \frac{13}{12}\)

\(\frac{5}{6} - \left(-\frac{1}{4}\right) = \frac{13}{12}\)
Final answer:

\(\frac{5}{6} - \left(-\frac{1}{4}\right) = \frac{13}{12}\)

Applied rules:

• Subtracting Negative: \(a - (-b) = a + b\)

• Common Denominator: Required for fraction addition

3 Comparing Signed Fractions
Exercise 3
Compare: \(-\frac{3}{5}\) and \(-\frac{4}{7}\)
Definition:

Comparing Negative Fractions: The fraction with the larger absolute value is actually smaller when both are negative

Fractions
\(-\frac{3}{5}\) and \(-\frac{4}{7}\)
Find LCD
\(35\)
Convert
\(-\frac{21}{35}\) and \(-\frac{20}{35}\)
Comparison
\(-\frac{21}{35} < -\frac{20}{35}\)
Step 1: Find the LCD

LCM of 5 and 7 is 35

Step 2: Convert to equivalent fractions

\(-\frac{3}{5} = -\frac{3 \times 7}{5 \times 7} = -\frac{21}{35}\)

\(-\frac{4}{7} = -\frac{4 \times 5}{7 \times 5} = -\frac{20}{35}\)

Step 3: Compare the numerators

Since \(-21 < -20\), we have \(-\frac{21}{35} < -\frac{20}{35}\)

Step 4: State the final comparison

\(-\frac{3}{5} < -\frac{4}{7}\)

\(-\frac{3}{5} < -\frac{4}{7}\)
Final answer:

\(-\frac{3}{5} < -\frac{4}{7}\)

Applied rules:

• Common Denominator: Required for comparing fractions

• Negative Comparison: Larger absolute value means smaller number

Fractions with Positive and Negative Values: Concepts and Methods
\(-\frac{a}{b} = \frac{-a}{b} = \frac{a}{-b}\)
Sign Placement in Fractions
Rule 1
\(\frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd}\)
Adding Fractions
Rule 2
\(\frac{a}{b} - \frac{c}{d} = \frac{ad - bc}{bd}\)
Subtracting Fractions
Rule 3
\(-\frac{a}{b} = -\frac{a}{b}\)
Negative Fraction
Key definitions:

Positive Fraction: A fraction where both numerator and denominator are positive, or both are negative

Negative Fraction: A fraction where the numerator and denominator have opposite signs

Absolute Value: The distance from zero, always positive

Sign Rules: Rules governing how positive and negative signs interact in operations

Complete methodology:
  1. Identify the signs: Determine which fractions are positive and which are negative
  2. Apply sign rules: Convert subtraction of negatives to addition of positives
  3. Find LCD: For operations requiring common denominators
  4. Perform operations: On numerators while preserving signs
  5. Simplify: Reduce to lowest terms if possible
Tip 1: Remember that \(-\frac{a}{b} = \frac{-a}{b} = \frac{a}{-b}\) - the negative sign can be placed in any position.
Tip 2: When comparing negative fractions, the one with the larger absolute value is actually smaller.
Tip 3: Subtracting a negative fraction is the same as adding the positive version.
Tip 4: Always check if your answer makes sense by estimating with benchmarks.
Common errors: Forgetting sign rules, not finding LCD, misplacing negative signs, incorrect comparison of negative values.
Exam preparation: Practice sign placement, master LCD finding, and understand how signs affect operations.
Key rules to remember:

• \(-\frac{a}{b} = \frac{-a}{b} = \frac{a}{-b}\)

• \(a - (-b) = a + b\)

• When comparing negatives, larger absolute value means smaller number

• Common denominator required for addition/subtraction

Solution: Exercises 4 to 5
4 Adding Two Negative Fractions
Exercise 4
Add: \(\left(-\frac{2}{3}\right) + \left(-\frac{1}{4}\right)\)
Definition:

Adding Two Negatives: The result is always negative, with absolute value equal to the sum of absolute values

Expression
\(\left(-\frac{2}{3}\right) + \left(-\frac{1}{4}\right)\)
Find LCD
\(12\)
Convert
\(\left(-\frac{8}{12}\right) + \left(-\frac{3}{12}\right)\)
Result
\(-\frac{11}{12}\)
Step 1: Find the LCD

LCM of 3 and 4 is 12

Step 2: Convert to equivalent fractions

\(-\frac{2}{3} = -\frac{2 \times 4}{3 \times 4} = -\frac{8}{12}\)

\(-\frac{1}{4} = -\frac{1 \times 3}{4 \times 3} = -\frac{3}{12}\)

Step 3: Add the numerators

\(\left(-\frac{8}{12}\right) + \left(-\frac{3}{12}\right) = \frac{-8 + (-3)}{12} = \frac{-11}{12}\)

Step 4: Simplify the sign

\(\frac{-11}{12} = -\frac{11}{12}\)

\(\left(-\frac{2}{3}\right) + \left(-\frac{1}{4}\right) = -\frac{11}{12}\)
Final answer:

\(\left(-\frac{2}{3}\right) + \left(-\frac{1}{4}\right) = -\frac{11}{12}\)

Applied rules:

• Adding Negatives: Result is negative

• Common Denominator: Required for fraction addition

5 Absolute Value of Signed Fractions
Exercise 5
Find: \(\left|\frac{-3}{4}\right|\) and \(\left|-\frac{5}{6}\right|\)
Definition:

Absolute Value: The distance from zero on the number line, always non-negative

Expression 1
\(\left|\frac{-3}{4}\right|\)
Result 1
\(\frac{3}{4}\)
Expression 2
\(\left|-\frac{5}{6}\right|\)
Result 2
\(\frac{5}{6}\)
Step 1: Apply absolute value definition

\(\left|\frac{-3}{4}\right| = \frac{|-3|}{|4|} = \frac{3}{4}\)

Step 2: Apply to second expression

\(\left|-\frac{5}{6}\right| = \frac{|-5|}{|6|} = \frac{5}{6}\)

Step 3: Verify the results

Absolute values are always positive or zero

\(\left|\frac{-3}{4}\right| = \frac{3}{4}\) and \(\left|-\frac{5}{6}\right| = \frac{5}{6}\)
Final answer:

\(\left|\frac{-3}{4}\right| = \frac{3}{4}\) and \(\left|-\frac{5}{6}\right| = \frac{5}{6}\)

Applied rules:

• Absolute Value: Makes any number non-negative

• Fraction Property: \(\left|\frac{a}{b}\right| = \frac{|a|}{|b|}\)

Fractions with Positive and Negative Values: Comprehensive Guide
\(\left|\frac{a}{b}\right| = \frac{|a|}{|b|}\)
Absolute Value of Fractions
Key definitions:

Positive Fraction: A fraction with a positive value (both numerator and denominator have the same sign)

Negative Fraction: A fraction with a negative value (numerator and denominator have opposite signs)

Absolute Value: The magnitude of a number without regard to its sign

Opposite: The number that, when added to the original, gives zero

Sign Rules: Rules that govern how positive and negative numbers interact in operations

Complete methodology:
  1. Identify the signs: Determine if fractions are positive or negative
  2. Apply sign rules: Convert operations as needed (subtracting a negative becomes adding)
  3. Find LCD: For operations requiring common denominators
  4. Perform operations: On numerators while preserving signs
  5. Simplify: Reduce to lowest terms and ensure correct sign
Tip 1: When adding fractions with different signs, subtract the absolute values and keep the sign of the fraction with the larger absolute value.
Tip 2: The negative sign can be placed in front of the fraction, in the numerator, or in the denominator - the value remains the same.
Tip 3: When comparing negative fractions, convert to a common denominator and compare numerators.
Tip 4: Always verify your answer by estimating with benchmark fractions like 0, 1/2, and 1.
Common errors: Forgetting to change signs when subtracting a negative, incorrectly placing the negative sign, not finding LCD, comparing absolute values instead of actual values.
Exam preparation: Master sign rules, practice operations with mixed signs, and understand the relationship between fractions and their absolute values.
Key rules to remember:

• Sign placement: \(-\frac{a}{b} = \frac{-a}{b} = \frac{a}{-b}\)

• Subtracting negative: \(a - (-b) = a + b\)

• Adding like signs: Add absolute values, keep the sign

• Adding unlike signs: Subtract absolute values, keep the sign of the larger

• Absolute value: \(\left|\frac{a}{b}\right| = \frac{|a|}{|b|}\)

Exercise with Visualization: Number Line for Signed Fractions
Exercise 6: Visualizing Signed Fractions on Number Line
Consider the following signed fractions on a number line:
\(\frac{1}{2}, -\frac{2}{3}, \frac{3}{4}, -\frac{1}{4}\)
Step 1: Convert to common denominator
Step 2: Plot on number line
Step 3: Order from least to greatest

Analysis: The chart shows how signed fractions are positioned on the number line.

  • Step 1: Find LCD of denominators 2, 3, 4 → LCD = 12
  • Step 2: Convert fractions → \(\frac{1}{2} = \frac{6}{12}, -\frac{2}{3} = -\frac{8}{12}, \frac{3}{4} = \frac{9}{12}, -\frac{1}{4} = -\frac{3}{12}\)
  • Step 3: Order → \(-\frac{8}{12} < -\frac{3}{12} < \frac{6}{12} < \frac{9}{12}\)
  • Final order: \(-\frac{2}{3} < -\frac{1}{4} < \frac{1}{2} < \frac{3}{4}\)

Questions & Answers

Question: I'm confused about where to put the negative sign in a fraction. Can it go anywhere?

Answer: Yes, the negative sign can be placed in different positions, and the value remains the same! Here's the rule:

  • \(-\frac{a}{b} = \frac{-a}{b} = \frac{a}{-b}\)

For example:

  • \(-\frac{3}{4} = \frac{-3}{4} = \frac{3}{-4}\)

However, the most common and preferred notation is to place the negative sign in front of the entire fraction: \(-\frac{3}{4}\). This makes it clear that the entire fraction is negative.

Question: How do I compare two negative fractions? Which one is bigger?

Answer: When comparing two negative fractions, the one with the larger absolute value is actually smaller! Here's the process:

To compare \(-\frac{2}{3}\) and \(-\frac{3}{4}\):

  • Find the LCD: LCM of 3 and 4 is 12
  • Convert: \(-\frac{2}{3} = -\frac{8}{12}\) and \(-\frac{3}{4} = -\frac{9}{12}\)
  • Compare numerators: Since \(-9 < -8\), we have \(-\frac{9}{12} < -\frac{8}{12}\)
  • Therefore: \(-\frac{3}{4} < -\frac{2}{3}\)

Remember: On the number line, the fraction farther from zero in the negative direction is smaller.

Question: When adding fractions with different signs, how do I know what sign the answer will have?

Answer: When adding fractions with different signs, the answer has the same sign as the fraction with the larger absolute value. Here's the process:

For example: \(\frac{5}{6} + \left(-\frac{2}{3}\right)\)

  • Find LCD: \(\frac{5}{6} + \left(-\frac{4}{6}\right)\)
  • Compare absolute values: \(\left|\frac{5}{6}\right| = \frac{5}{6}\) and \(\left|-\frac{4}{6}\right| = \frac{4}{6}\)
  • Since \(\frac{5}{6} > \frac{4}{6}\), the answer has the same sign as \(\frac{5}{6}\) (positive)
  • Subtract absolute values: \(\frac{5}{6} - \frac{4}{6} = \frac{1}{6}\)
  • Result: \(\frac{1}{6}\)

The answer is positive because \(\frac{5}{6}\) has the larger absolute value.

Question: What happens when I subtract a negative fraction? Is it the same as regular subtraction?

Answer: No, subtracting a negative fraction is NOT the same as regular subtraction. In fact, subtracting a negative is the same as adding the positive! Here's the rule:

  • \(a - (-b) = a + b\)

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

  • Find LCD: \(\frac{3}{4} + \frac{2}{4} = \frac{5}{4}\)

Think of it this way: subtracting a negative means removing a debt or loss, which effectively adds to your total.

Question: How can I check if my operations with signed fractions are correct?

Answer: Here are several verification methods for operations with signed fractions:

  • Estimation: Round fractions to benchmarks (0, 1/2, 1) and check if your answer is reasonable
  • Number line: Visualize the operation on a number line
  • Sign check: Verify that the sign of your answer makes sense based on the operation
  • Reverse operation: Perform the inverse operation to see if you get back to the original values

For example, if you calculated \(\frac{1}{2} + \left(-\frac{1}{3}\right) = \frac{1}{6}\):

  • Estimate: \(\frac{1}{2} - \frac{1}{3} \approx 0.5 - 0.33 = 0.17\), and \(\frac{1}{6} \approx 0.17\) ✓
  • Sign check: Adding a negative to a positive should give a smaller positive, which it does ✓
  • Reverse: \(\frac{1}{6} - \left(-\frac{1}{3}\right) = \frac{1}{6} + \frac{1}{3} = \frac{1}{6} + \frac{2}{6} = \frac{3}{6} = \frac{1}{2}\) ✓

Always develop a habit of checking your work using at least one verification method.