Fractions

Q1: Classify each fraction. given below, as decimal or vulgar fraction, proper or improper fraction and mixed fraction:

i. \(\frac{3}{5}\)

Step 1: This is a fraction with numerator 3 and denominator 5.
Step 2: The numerator is less than the denominator ⇒ it is a proper fraction.
Step 3: Since it’s a simple fraction, it is a vulgar fraction as well.
Answer: \(\frac{3}{5}\) is a vulgar and proper fraction.

ii. \(\frac{11}{10}\)

Step 1: This is a fraction with numerator 11 and denominator 10.
Step 2: The numerator is greater than the denominator ⇒ it is an improper fraction.
Step 3: Since the denominator is 10, it is a decimal fraction as well.
Answer: \(\frac{11}{10}\) is an decimal and improper fraction.

iii. \(\frac{13}{20}\)

Step 1: This is a fraction with numerator 13 and denominator 20.
Step 2: The numerator is less than the denominator ⇒ it is a proper fraction.
Step 3: Denominator of fraction can be expressed as 100 (\(\frac{13 \times 5}{20 \times 5} = \frac{65}{100}\)), so it is decimal fraction
Answer: \(\frac{13}{20}\) is a decimal and proper fraction.

iv. \(\frac{18}{7}\)

Step 1: This is a fraction with numerator 18 and denominator 7.
Step 2: The numerator is greater than the denominator ⇒ it is an improper fraction.
Step 3: It is also a vulgar fraction as it’s expressed as a simple fraction.
Answer: \(\frac{18}{7}\) is an vulgar and improper fraction.

v. \(3\frac{2}{9}\)

Step 1: This is a mixed fraction, as it contains a whole number (3) and a fraction (\(\frac{2}{9}\)).
Step 2: The whole number part is 3, and the fractional part is \(\frac{2}{9}\).
Step 3: This is a mixed fraction and not just a vulgar or decimal fraction.
Answer: \(3\frac{2}{9}\) is a mixed fraction.

Q2: Express the following improper fractions as mixed fractions:

i. \(\frac{18}{5}\)

Step 1: Divide the numerator by the denominator: \( 18 \div 5 = 3 \) remainder \( 3 \).
Step 2: The quotient is 3, and the remainder is 3. This gives us the mixed fraction: \( 3 \frac{3}{5} \).
Answer: \(\frac{18}{5} = 3 \frac{3}{5}\).

ii. \(\frac{7}{4}\)

Step 1: Divide the numerator by the denominator: \( 7 \div 4 = 1 \) remainder \( 3 \).
Step 2: The quotient is 1, and the remainder is 3. This gives us the mixed fraction: \( 1 \frac{3}{4} \).
Answer: \(\frac{7}{4} = 1 \frac{3}{4}\).

iii. \(\frac{25}{6}\)

Step 1: Divide the numerator by the denominator: \( 25 \div 6 = 4 \) remainder \( 1 \).
Step 2: The quotient is 4, and the remainder is 1. This gives us the mixed fraction: \( 4 \frac{1}{6} \).
Answer: \(\frac{25}{6} = 4 \frac{1}{6}\).

iv. \(\frac{38}{5}\)

Step 1: Divide the numerator by the denominator: \( 38 \div 5 = 7 \) remainder \( 3 \).
Step 2: The quotient is 7, and the remainder is 3. This gives us the mixed fraction: \( 7 \frac{3}{5} \).
Answer: \(\frac{38}{5} = 7 \frac{3}{5}\).

v. \(\frac{22}{5}\)

Step 1: Divide the numerator by the denominator: \( 22 \div 5 = 4 \) remainder \( 2 \).
Step 2: The quotient is 4, and the remainder is 2. This gives us the mixed fraction: \( 4 \frac{2}{5} \).
Answer: \(\frac{22}{5} = 4 \frac{2}{5}\).

Q3: Express the following mixed fractions as improper fractions:

i. \(2\frac{4}{9}\)

Step 1: Multiply the whole number (2) by the denominator (9): \( 2 \times 9 = 18 \).
Step 2: Add the result to the numerator (4): \( 18 + 4 = 22 \).
Step 3: Write the result as a fraction with the same denominator (9): \( \frac{22}{9} \).
Answer: \( 2\frac{4}{9} = \frac{22}{9} \).

ii. \(7\frac{5}{13}\)

Step 1: Multiply the whole number (7) by the denominator (13): \( 7 \times 13 = 91 \).
Step 2: Add the result to the numerator (5): \( 91 + 5 = 96 \).
Step 3: Write the result as a fraction with the same denominator (13): \( \frac{96}{13} \).
Answer: \( 7\frac{5}{13} = \frac{96}{13} \).

iii. \(3\frac{1}{4}\)

Step 1: Multiply the whole number (3) by the denominator (4): \( 3 \times 4 = 12 \).
Step 2: Add the result to the numerator (1): \( 12 + 1 = 13 \).
Step 3: Write the result as a fraction with the same denominator (4): \( \frac{13}{4} \).
Answer: \( 3\frac{1}{4} = \frac{13}{4} \).

iv. \(2\frac{5}{48}\)

Step 1: Multiply the whole number (2) by the denominator (48): \( 2 \times 48 = 96 \).
Step 2: Add the result to the numerator (5): \( 96 + 5 = 101 \).
Step 3: Write the result as a fraction with the same denominator (48): \( \frac{101}{48} \).
Answer: \( 2\frac{5}{48} = \frac{101}{48} \).

v. \(12\frac{7}{11}\)

Step 1: Multiply the whole number (12) by the denominator (11): \( 12 \times 11 = 132 \).
Step 2: Add the result to the numerator (7): \( 132 + 7 = 139 \).
Step 3: Write the result as a fraction with the same denominator (11): \( \frac{139}{11} \).
Answer: \( 12\frac{7}{11} = \frac{139}{11} \).

Q4: Reduce the given fractions to lowest terms:

i. \(\frac{8}{18}\)

Step 1: Find the greatest common divisor (GCD) of 8 and 18. The GCD of 8 and 18 is 2.
Step 2: Divide both the numerator and denominator by 2: \( \frac{8 \div 2}{18 \div 2} = \frac{4}{9} \).
Answer: \(\frac{8}{18} = \frac{4}{9}\).

ii. \(\frac{27}{36}\)

Step 1: Find the greatest common divisor (GCD) of 27 and 36. The GCD of 27 and 36 is 9.
Step 2: Divide both the numerator and denominator by 9: \( \frac{27 \div 9}{36 \div 9} = \frac{3}{4} \).
Answer: \(\frac{27}{36} = \frac{3}{4}\).

iii. \(\frac{18}{42}\)

Step 1: Find the greatest common divisor (GCD) of 18 and 42. The GCD of 18 and 42 is 6.
Step 2: Divide both the numerator and denominator by 6: \( \frac{18 \div 6}{42 \div 6} = \frac{3}{7} \).
Answer: \(\frac{18}{42} = \frac{3}{7}\).

iv. \(\frac{35}{75}\)

Step 1: Find the greatest common divisor (GCD) of 35 and 75. The GCD of 35 and 75 is 5.
Step 2: Divide both the numerator and denominator by 5: \( \frac{35 \div 5}{75 \div 5} = \frac{7}{15} \).
Answer: \(\frac{35}{75} = \frac{7}{15}\).

v. \(\frac{18}{45}\)

Step 1: Find the greatest common divisor (GCD) of 18 and 45. The GCD of 18 and 45 is 9.
Step 2: Divide both the numerator and denominator by 9: \( \frac{18 \div 9}{45 \div 9} = \frac{2}{5} \).
Answer: \(\frac{18}{45} = \frac{2}{5}\).

Q5: State true or false:

i. \(\frac{30}{40}\) and \(\frac{12}{16}\) are equivalent fractions.

Step 1: To check if the fractions are equivalent, reduce both to their lowest terms.
For \(\frac{30}{40}\), the GCD of 30 and 40 is 10. Dividing both by 10: \( \frac{30 \div 10}{40 \div 10} = \frac{3}{4} \).
For \(\frac{12}{16}\), the GCD of 12 and 16 is 4. Dividing both by 4: \( \frac{12 \div 4}{16 \div 4} = \frac{3}{4} \).
Since both fractions reduce to \(\frac{3}{4}\), they are equivalent.
Answer: True.

ii. \(\frac{10}{25}\) and \(\frac{25}{10}\) are equivalent fractions.

Step 1: Check the fractions: For \(\frac{10}{25}\), the GCD of 10 and 25 is 5. Dividing both by 5: \( \frac{10 \div 5}{25 \div 5} = \frac{2}{5} \).
For \(\frac{25}{10}\), the GCD of 25 and 10 is 5. Dividing both by 5: \( \frac{25 \div 5}{10 \div 5} = \frac{5}{2} \).
Since \(\frac{2}{5} \neq \frac{5}{2}\), these fractions are not equivalent.
Answer: False.

iii. \(\frac{35}{49}, \frac{20}{28}, \frac{45}{63}\) and \(\frac{100}{140}\) are equivalent fractions.

Step 1: Check if all fractions reduce to the same value.
For \(\frac{35}{49}\), the GCD of 35 and 49 is 7. Dividing both by 7: \( \frac{35 \div 7}{49 \div 7} = \frac{5}{7} \).
For \(\frac{20}{28}\), the GCD of 20 and 28 is 4. Dividing both by 4: \( \frac{20 \div 4}{28 \div 4} = \frac{5}{7} \).
For \(\frac{45}{63}\), the GCD of 45 and 63 is 9. Dividing both by 9: \( \frac{45 \div 9}{63 \div 9} = \frac{5}{7} \).
For \(\frac{100}{140}\), the GCD of 100 and 140 is 20. Dividing both by 20: \( \frac{100 \div 20}{140 \div 20} = \frac{5}{7} \).
Since all fractions reduce to \(\frac{5}{7}\), they are equivalent.
Answer: True.

Q6: Distinguish each of the fractions, given below, as a simple fraction or a complex fraction:

i. \(\frac{0}{8}\)

Step 1: This is a fraction where both the numerator and denominator are integers, and the denominator is non-zero. It is a simple fraction.
Answer: Simple Fraction.

ii. \(\frac{3}{8}\)

Step 1: This is a fraction with an integer numerator and an integer denominator, with no fractions in the numerator or denominator. It is a simple fraction.
Answer: Simple Fraction.

iii. \(\frac{5}{7}\)

Step 1: This is a fraction with integer values in the numerator and denominator. It is a simple fraction.
Answer: Simple Fraction.

iv. \(\frac{3\frac{3}{5}}{18}\)

Step 1: The numerator is a mixed fraction, which contains both a whole number and a fraction. Since the numerator itself is a fraction, this is a complex fraction.
Answer: Complex Fraction.

v. \(\frac{6}{2\frac{2}{5}}\)

Step 1: The denominator is a mixed fraction. A fraction with a mixed number in the denominator is classified as a complex fraction.
Answer: Complex Fraction.

vi. \(\frac{3\frac{1}{3}}{7\frac{2}{7}}\)

Step 1: Both the numerator and denominator are mixed fractions. This makes the fraction a complex fraction.
Answer: Complex Fraction.

vii. \(\frac{5\frac{2}{9}}{5}\)

Step 1: The numerator is a mixed fraction, and the denominator is an integer. This is a complex fraction.
Answer: Complex Fraction.

viii. \(\frac{8}{0}\)

Step 1: This is an invalid fraction as division by zero is undefined. It cannot be classified as either a simple or complex fraction.
Answer: Undefined (Not a Simple or Complex Fraction).