\( -2\frac{1}{2} = -\frac{5}{2} \)
\( -1.75 = -\frac{7}{4} \)
LCM of Denominators: 4
Final: \( -\frac{10}{4} + \left(-\frac{7}{4}\right) = -\frac{17}{4} = -4\frac{1}{4} \)
Teacher Answer Key & Solved Guide
Page 2 of 2
3 MULTIPLICATION OF RATIONALS
SOLVED KEY
STEPS
Step 1: Predict the Sign
Determine the final sign first: \((-)\cdot(-) = (+)\) or \((-)\cdot(+) = (-)\).
Negative \(\times\) Negative = Positive (+).
Step 2: Choose Your Format
Convert both numbers to fractions or decimals so they match.
Convert both to fraction formats.
Step 3: Simplify & Multiply
Multiply straight across. Reduce or cross-simplify before multiplying!
Simplify diagonals: 6 and 3 divide by 3.
Completed Workspace \( -1.2 \times \left(-2\frac{1}{3}\right) \)
Predict Sign:
Negative \(\times\) Negative = Positive (+) [ + ] Circled
Convert to Improper Fractions:
\( -1.2 = -\frac{12}{10} = -\frac{6}{5} \)
\( -2\frac{1}{3} = -\frac{7}{3} \)
Diagonally Cross-Simplify & Multiply:
\( \left(-\frac{6}{5}\right) \times \left(-\frac{7}{3}\right) \;\rightarrow\; \) \( \frac{2}{5} \times \frac{7}{1} \)
Multiply straight across...
Final: \( \frac{14}{5} = 2\frac{4}{5} = 2.8 \)
4 DIVISION OF RATIONALS
SOLVED KEY
STEPS
Step 1: Go Improper!
Mixed fractions must be converted to improper fractions first.
\(1\frac{1}{4} = \frac{5}{4}\).
Step 2: Keep-Change-Flip
Keep 1st fraction, Change \(\div\) to \(\times\), Flip 2nd fraction to reciprocal.
\(-\frac{7}{8} \times \frac{4}{5}\).
Step 3: Cross-Simplify & Solve
Simplify diagonally, multiply straight across, and apply sign rules.
Simplify diagonals: 4 and 8 reduce.
Completed Workspace \( -\frac{7}{8} \div 1\frac{1}{4} \)
Make the divisor improper:
\( 1\frac{1}{4} = \frac{5}{4} \)
Apply Keep-Change-Flip (KCF):
Keep \( -\frac{7}{8} \)
×
Flip \( \frac{4}{5} \)
Diagonals simplified: \( -\frac{7}{2} \times \frac{1}{5} \)
Final: \( -\frac{7}{10} \)
ALGORITHMIC SELF-CHECK Did I double check the sign matrix? Is the fraction in lowest terms?
Teacher Guide & Answer Key
Multiplication Example: \( -\frac{2}{3} \times 1.5 \)
Division Example: \( -\frac{4}{5} \div \left(-\frac{2}{3}\right) \)
PROBLEM 09 Multiplication
\( -2.4 \times (-1.5) \)
Final Sign: [ + / - ]
Answer: _________________
PROBLEM 10 Multiplication
\( 1\frac{2}{3} \times \left(-2\frac{1}{4}\right) \)
Final Sign: [ + / - ]
Answer: _________________
PROBLEM 11 Division
\( -\frac{5}{6} \div \left(-\frac{10}{11}\right) \)
Final Sign: [ + / - ]
Answer: _________________
PROBLEM 12 Division
\( -3\frac{1}{2} \div 0.25 \)
Final Sign: [ + / - ]
Answer: _________________
Part 2: Multiplication & Division (Expansion)
Page 4 of 4
PROBLEM 13 Multiplication
\( -2\frac{2}{5} \times \left(-1\frac{7}{8}\right) \)
Tip: Convert to improper fractions. Check if you can cross-simplify to keep numbers smaller!
Final Sign: [ + / - ]
Answer: _________________
PROBLEM 14 Multiplication
\( \frac{9}{10} \times (-0.4) \)
Tip: Convert \( -0.4 = -\frac{4}{10} = -\frac{2}{5} \), or convert \( \frac{9}{10} = 0.9 \).
Final Sign: [ + / - ]
Answer: _________________
PROBLEM 15 Division
\( -2\frac{1}{4} \div 1\frac{1}{2} \)
Tip: Turn both improper, apply Keep-Change-Flip, and simplify diagonally.
Final Sign: [ + / - ]
Answer: _________________
PROBLEM 16 Division
\( 0.8 \div \left(-\frac{2}{5}\right) \)
Tip: Convert \( 0.8 = \frac{8}{10} = \frac{4}{5} \), then apply Keep-Change-Flip.
Final Sign: [ + / - ]
Answer: _________________
Final Sign: Negative (-)
Answer: \( -6\frac{5}{6} \)
PROBLEM 06 Addition
\( 0.\bar{1} + 1.4 \)
Step 1: Convert Both to Fractions
\( 0.\bar{1} = \frac{1}{9} \)
\( 1.4 = \frac{14}{10} = \frac{7}{5} \)
Step 2: LCD (45) & Add
\( \frac{1}{9} \times \frac{5}{5} = \frac{5}{45} \) \( \frac{7}{5} \times \frac{9}{9} = \frac{63}{45} \)
\( \frac{5}{45} + \frac{63}{45} = \frac{68}{45} = 1\frac{23}{45} \)
Final Sign: Positive (+)
Answer: \( 1\frac{23}{45} \)
PROBLEM 07 Subtraction
\( 5\frac{1}{3} - 6\frac{5}{9} \)
Step 1: Turn Improper
\( 5\frac{1}{3} = \frac{16}{3} \) \( 6\frac{5}{9} = \frac{59}{9} \)
Step 2: LCD (9) & Add Opposite
\( \frac{48}{9} + \left(-\frac{59}{9}\right) = \frac{48 - 59}{9} = -\frac{11}{9} \)
\( = -1\frac{2}{9} \)
Final Sign: Negative (-)
Answer: \( -1\frac{2}{9} \)
PROBLEM 08 Subtraction
\( -4.8 - (-2.75) \)
Step 1: Add the Opposite
\( -4.8 + 2.75 \)
Step 2: Solve & Subtract
Subtract absolute values: \( 4.80 - 2.75 = 2.05 \)
Keep sign of larger: \( -2.05 \)
Fraction equivalent: \( -2\frac{1}{20} \)
Final Sign: Negative (-)
Answer: -2.05 or \( -2\frac{1}{20} \)
Part 2: Multiplication & Division Solutions (9-12)
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MULTIPLICATION & DIVISION REFERENCE GUIDE
Multiplication Example: \( -\frac{2}{3} \times 1.5 \)
Division Example: \( -\frac{4}{5} \div \left(-\frac{2}{3}\right) \)
PROBLEM 09 Multiplication
\( -2.4 \times (-1.5) \)
Step 1: Sign Rule
Negative \(\times\) Negative = Positive (+)
Step 2: Algorithm Execution
Multiply without decimals: \( 24 \times 15 = 360 \)
Add 2 decimal places: \( 3.60 \)
Final Sign: Positive (+)
Answer: 3.6
PROBLEM 10 Multiplication
\( 1\frac{2}{3} \times \left(-2\frac{1}{4}\right) \)
Step 1: Convert to Improper
\( 1\frac{2}{3} = \frac{5}{3} \) \( -2\frac{1}{4} = -\frac{9}{4} \)
Step 2: Simplify & Multiply
\( \frac{5}{3} \times \left(-\frac{9}{4}\right) \rightarrow \text{cross-divide 9 and 3 by 3} \)
\( = \frac{5}{1} \times \left(-\frac{3}{4}\right) = -\frac{15}{4} = -3\frac{3}{4} \)
Final Sign: Negative (-)
Answer: \( -3\frac{3}{4} \) (or -3.75)
PROBLEM 11 Division
\( -\frac{5}{6} \div \left(-\frac{10}{11}\right) \)
Step 1: Keep-Change-Flip (KCF)
\( -\frac{5}{6} \times \left(-\frac{11}{10}\right) \)
Step 2: Cross-Simplify & Multiply
Divide numerator 5 & denominator 10 by 5:
\( = -\frac{1}{6} \times \left(-\frac{11}{2}\right) = \frac{11}{12} \)
(Negative \(\div\) Negative = Positive)
Final Sign: Positive (+)
Answer: \( \frac{11}{12} \)
PROBLEM 12 Division
\( -3\frac{1}{2} \div 0.25 \)
Method: Fraction Conversion
\( -3\frac{1}{2} = -\frac{7}{2} \) \( 0.25 = \frac{1}{4} \)
\( -\frac{7}{2} \div \frac{1}{4} = -\frac{7}{2} \times \frac{4}{1} \)
\( = -\frac{28}{2} = -14 \)
Method: Decimal Route
\( -3.5 \div 0.25 = -14 \)
Final Sign: Negative (-)
Answer: -14
Part 2: Multiplication & Division Solutions (13-16)
Page 4 of 4
PROBLEM 13 Multiplication
\( -2\frac{2}{5} \times \left(-1\frac{7}{8}\right) \)
Step 1: Go Improper
\( -2\frac{2}{5} = -\frac{12}{5} \) \( -1\frac{7}{8} = -\frac{15}{8} \)
Step 2: Cross-Simplify & Multiply
Divide diagonals:
15 and 5 by 5 \(\rightarrow\) 3 and 1
12 and 8 by 4 \(\rightarrow\) 3 and 2
Multiply remaining: \( -\frac{3}{1} \times \left(-\frac{3}{2}\right) = \frac{9}{2} = 4\frac{1}{2} \)
Final Sign: Positive (+)
Answer: \( 4\frac{1}{2} \) or 4.5
PROBLEM 14 Multiplication
\( \frac{9}{10} \times (-0.4) \)
Method A: Decimal Route
Convert to decimal: \( 0.9 \times (-0.4) = -0.36 \)
Method B: Fraction Route
\( \frac{9}{10} \times \left(-\frac{4}{10}\right) = \frac{9}{10} \times \left(-\frac{2}{5}\right) \)
\( = -\frac{18}{50} = -\frac{9}{25} \)
Final Sign: Negative (-)
Answer: -0.36 or \( -\frac{9}{25} \)
PROBLEM 15 Division
\( -2\frac{1}{4} \div 1\frac{1}{2} \)
Step 1: Turn Improper
\( -2\frac{1}{4} = -\frac{9}{4} \) \( 1\frac{1}{2} = \frac{3}{2} \)
Step 2: Keep-Change-Flip (KCF)
\( -\frac{9}{4} \times \frac{2}{3} \rightarrow \text{simplify diagonals} \)
\( = -\frac{3}{2} \times \frac{1}{1} = -\frac{3}{2} = -1\frac{1}{2} \) (or -1.5)
Final Sign: Negative (-)
Answer: \( -1\frac{1}{2} \)
PROBLEM 16 Division
\( 0.8 \div \left(-\frac{2}{5}\right) \)
Method A: Fraction Conversion
\( 0.8 = \frac{8}{10} = \frac{4}{5} \)
\( \frac{4}{5} \div \left(-\frac{2}{5}\right) = \frac{4}{5} \times \left(-\frac{5}{2}\right) = -\frac{20}{10} = -2 \)
Method B: Decimal Route
\( -\frac{2}{5} = -0.4 \)
\( 0.8 \div (-0.4) = -2 \)
Final Sign: Negative (-)
Answer: -2
Simplify & Multiply
\( \left(-\frac{6}{5}\right) \times \left(-\frac{7}{3}\right) \)
Cross-divide 6 and 3 by 3:
\( = \frac{14}{5} = 2\frac{4}{5} = 2.8 \)
Since \(-2\frac{1}{3}\) has a repeating decimal, fraction form is required!
Slide 5 of 6
\( -\frac{7}{8} \div 1\frac{1}{4} \)
STEP 1: Improper Fractions First
Convert Mixed Fraction: \( 1\frac{1}{4} = \frac{5}{4} \)
STEP 2: Keep - Change - Flip (KCF)
Rewrite division as multiplication by reciprocal:
\( -\frac{7}{8} \times \frac{4}{5} \)
Simplify & Solve
Cross-simplify 4 and 8:
\( -\frac{7}{2} \times \frac{1}{5} \)
\( = -\frac{7}{10} \)
Sign Check: Negative \(\div\) Positive = Negative!
Slide 6 of 6
