Simplify, Add, Subtract, Multiply, Divide, Mixed Numbers, and Converting to Decimals

Answer: $$\frac{3}{5}$$

Step 1: Find the GCF of 6 and 10:

• Factors of 6: {1, 2, 3, 6}
• Factors of 10: {1, 2, 5, 10}

=> GCF(6, 10) = 2.

Step 2: Divide numerator and denominator by 2:

• New numerator = $$6 \div 2 = 3$$
• New denominator $$= 10 \div 2 = 5$$

=> Correct Answer: A ($$\frac{3}{5}$$)

Answer: $$\frac{2}{3}$$

Step 1: Find GCF using prime factorization:

• $$24 = 2^3 \times 3$$
• $$36 = 2^2 \times 3^2$$

=> GCF(24, 36) $$= 2^2 \times 3 = 12$$.

Step 2: Divide numerator and denominator by 12:

• Numerator: $$24 \div 12 = 2$$
• Denominator: $$36 \div 12 = 3$$

=> Correct Answer: A ($$\frac{2}{3}$$)

Answer: $$\frac{7}{9}$$

Step 1: Find GCF(84, 108) using the Euclidean Algorithm:

• $$108 = 84 \times 1 + 24$$
• $$84 = 24 \times 3 + 12$$
• $$24 = 12 \times 2 + 0$$ => GCF is 12.

Step 2: Simplify the fraction:

• Numerator: $$84 \div 12 = 7$$
• Denominator: $$108 \div 12 = 9$$

=> Correct Answer: B ($$\frac{7}{9}$$)

Answer: $$\frac{5}{7}$$

Step 1: Observe both fractions already share the same denominator (7).

Step 2: Add numerators and keep the common denominator:

• New numerator = $$2 + 3 = 5$$
• Denominator = 7

=> Resulting fraction: $$\frac{5}{7}$$.

=> Correct Answer: B ($$\frac{5}{7}$$)

Answer: $$\frac{7}{12}$$

Step 1: Find the Least Common Denominator: LCD(4, 6) = 12.

Step 2: Convert to equivalent fractions:

• For $$\frac{3}{4}$$: multiplier $$= 12 \div 4 = 3 \Rightarrow \frac{(3 \times 3)}{(4 \times 3)} = \frac{9}{12}$$
• For $$\frac{1}{6}$$: multiplier $$= 12 \div 6 = 2 \Rightarrow \frac{(1 \times 2)}{(6 \times 2)} = \frac{2}{12}$$

Step 3: Subtract numerators:

• $$\frac{9}{12} - \frac{2}{12} = \frac{(9 - 2)}{12} = \frac{7}{12}$$.

=> Correct Answer: A ($$\frac{7}{12}$$)

Answer: $$\frac{13}{24}$$

Step 1: Find LCD of all 3 denominators (8, 6, 4):

• $$8 = 2^3$$, $$6 = 2 \times 3$$, $$4 = 2^2 \Rightarrow \text{LCD}(8, 6, 4) = 24$$.

Step 2: Convert all fractions to denominator 24:

• $$\frac{5}{8} = \frac{(5 \times 3)}{24} = \frac{15}{24}$$
• $$\frac{1}{6} = \frac{(1 \times 4)}{24} = \frac{4}{24}$$
• $$\frac{1}{4} = \frac{(1 \times 6)}{24} = \frac{6}{24}$$

Step 3: Combine numerators from left to right:

• $$\frac{(15 + 4 - 6)}{24} = \frac{13}{24}$$.

=> Correct Answer: A ($$\frac{13}{24}$$)

Answer: $$\frac{2}{15}$$

Step 1: Multiply the numerators:

• New numerator = $$1 \times 2 = 2$$

Step 2: Multiply the denominators:

• New denominator $$= 3 \times 5 = 15$$

Step 3: Check simplification: GCF(2, 15) = 1, so $$\frac{2}{15}$$ is already simplified.

=> Correct Answer: A ($$\frac{2}{15}$$)

Answer: $$\frac{1}{6}$$

Method 1 (Multiply then simplify):

• Numerator: $$4 \times 3 = 12$$
• Denominator: $$9 \times 8 = 72$$
• Simplify $$\frac{12}{72}$$ by dividing by $$12 \Rightarrow \frac{1}{6}$$.

Method 2 (Cross-cancel first):

• Cancel 4 and 8 (divide by 4): 4 becomes 1, 8 becomes 2
• Cancel 3 and 9 (divide by 3): 3 becomes 1, 9 becomes 3
• Result: $$(\frac{1}{3}) \times (\frac{1}{2}) = \frac{1}{6}$$.

=> Correct Answer: A ($$\frac{1}{6}$$)

Answer: $$\frac{1}{5}$$

Step 1: Express as a single large fraction:

• Numerator = $$5 \times 8 \times 9$$
• Denominator $$= 12 \times 15 \times 10$$

Step 2: Cross-cancel factors:

• Cancel 5 with 15 (divide by 5): 5 -> 1, 15 -> 3
• Cancel 9 with 3 (divide by 3): 9 -> 3
• Cancel 3 with 12 (divide by 3): 12 -> 4
• Cancel 8 with 4 (divide by 4): 8 -> 2
• Cancel 2 with 10 (divide by 2): 10 -> 5

Step 3: Multiply remaining terms: $$\frac{1}{5}$$.

=> Correct Answer: B ($$\frac{1}{5}$$)

Answer: $$\frac{3}{2}$$

Step 1: Keep the first fraction: $$\frac{1}{2}$$.

Step 2: Change ÷ to × and flip $$\frac{1}{3}$$ to $$\frac{3}{1}$$:

• $$\frac{1}{2} \div \frac{1}{3} = \frac{1}{2} \times \frac{3}{1}$$

Step 3: Multiply numerators and denominators:

• $$\frac{(1 \times 3)}{(2 \times 1)} = \frac{3}{2}$$.

=> Correct Answer: A ($$\frac{3}{2}$$)

Answer: $$\frac{5}{6}$$

Step 1: Apply Keep-Change-Flip:

• $$\frac{5}{8} \div \frac{3}{4} = \frac{5}{8} \times \frac{4}{3}$$

Step 2: Cross-cancel before multiplying:

• Cancel 4 and 8 (divide by 4): 4 becomes 1, 8 becomes 2
• Calculation becomes: $$(\frac{5}{2}) \times (\frac{1}{3})$$

Step 3: Multiply results:

• $$\frac{(5 \times 1)}{(2 \times 3)} = \frac{5}{6}$$.

=> Correct Answer: A ($$\frac{5}{6}$$)

Answer: 1

Step 1: Calculate inside parentheses first:

• $$\frac{2}{3} \div \frac{4}{5} = \frac{2}{3} \times \frac{5}{4} = \frac{10}{12}$$
• Simplify $$\frac{10}{12}$$ by dividing by 2: $$\frac{5}{6}$$.

Step 2: Calculate outer expression:

• Divide result $$(\frac{5}{6})$$ by $$\frac{5}{6}$$:
• $$\frac{5}{6} \div \frac{5}{6} = \frac{5}{6} \times \frac{6}{5} = \frac{30}{30} = 1$$.

=> Correct Answer: B (1).

Answer: 0.5

Perform long division $$1 \div 2$$:

• $$1 \div 2 = 0$$ with a decimal point (0.)
• Add zero to 1 to make 10
• $$10 \div 2 = 5 \; (\mathrm{remainder} \; 0)$$

=> Result: 0.5.

=> Correct Answer: A (0.5).

Answer: 0.875

Step-by-step long division of 7 by 8:

• $$7 \div 8 = 0$$., add zero to make 70.
• $$70 \div 8 = 8 (8 \times 8 = 64)$$, remainder = 6.
• Add zero -> 60. $$60 \div 8 = 7 (8 \times 7 = 56)$$, remainder = 4.
• Add zero -> 40. $$40 \div 8 = 5 (8 \times 5 = 40)$$, remainder = 0.

=> Combine digits: 0.875.

=> Correct Answer: A (0.875).

Answer: 0.8125

Step-by-step long division of 13 by 16:

• $$13 \div 16 = 0$$., add zero -> 130.
• $$130 \div 16 = 8 (16 \times 8 = 128)$$, remainder = 2.
• Add zero -> 20. $$20 \div 16 = 1 (16 \times 1 = 16)$$, remainder = 4.
• Add zero -> 40. $$40 \div 16 = 2 (16 \times 2 = 32)$$, remainder = 8.
• Add zero -> 80. $$80 \div 16 = 5 (16 \times 5 = 80)$$, remainder = 0.

=> Exact decimal: 0.8125.

=> Correct Answer: B (0.8125).