CBSE Class 6 Mathematics Fractions MCQs Set 12

Answer: (b) a fractional unit

Explanation:
1. Splitting one whole into equal parts gives pieces such as \( \frac{1}{2} \), \( \frac{1}{3} \) or \( \frac{1}{4} \).
2. Each such equal piece is called a fractional unit (also called a unit fraction).

Teacher's Note:
1. A fractional unit always has 1 on top.
2. The parts must be equal for it to be a fractional unit.

Answer: (c) \( \frac{1}{6} \) chapati

Explanation:
1. One whole chapati is cut into 6 equal shares.
2. Each child gets one of these 6 equal shares, that is \( \frac{1}{6} \) chapati.

Teacher's Note:
1. The number of sharers goes in the denominator.
2. \( \frac{6}{1} \) means 6 whole chapatis, which is far too much.

Answer: (a) denominator

Explanation:
1. The top number of a fraction is the numerator and the bottom number is the denominator.
2. 12 is the bottom number, so it is the denominator. It names the fractional unit \( \frac{1}{12} \).

Teacher's Note:
1. Denominator = 'down' number.
2. The denominator tells how many equal parts make one whole.

Answer: (d) \( \frac{3}{8} \)

Explanation:
1. The bar has 8 equal pieces, so each piece is \( \frac{1}{8} \) of the bar.
2. 3 pieces are shaded, so the shaded portion is 3 times \( \frac{1}{8} \), that is \( \frac{3}{8} \).

Teacher's Note:
1. Count all the pieces for the denominator and the shaded pieces for the numerator.
2. \( \frac{5}{8} \) is the unshaded part, a common slip.

Answer: (b) \( \frac{1}{4} \)

Explanation:
1. The more equal parts a whole is cut into, the smaller each part is.
2. 4 is the smallest denominator here, so \( \frac{1}{4} \) is the largest fractional unit.

Teacher's Note:
1. For fractional units, smaller denominator means bigger piece.
2. Think: 1 roti among 4 children or among 25 children?

Answer: (a) Both A and R are true, and R is the correct explanation of A.

Explanation:
1. \( \frac{9}{5} \) = \( \frac{5}{5} \) + \( \frac{4}{5} \) = \( 1\frac{4}{5} \), which is more than 1. So A is true.
2. When the numerator is bigger than the denominator, there are more fractional units than one whole needs, so R is true.
3. 9 > 5 is exactly why \( \frac{9}{5} \) is more than 1, so R explains A.

Teacher's Note:
1. Five fifths make one whole; anything beyond that is more than 1.
2. Compare top and bottom numbers to judge quickly.

Answer: (c) \( \frac{3}{4} \)

Explanation:
1. The gap from 0 to 1 is split into 4 equal parts, so each small part is \( \frac{1}{4} \).
2. P is at the third mark after 0, so P = 3 times \( \frac{1}{4} \) = \( \frac{3}{4} \).

Teacher's Note:
1. Count the parts in one unit (from 0 to 1), not from 0 to 2.
2. Then count the parts from 0 up to the point.

Answer: (d) \( \frac{4}{8} \)

Explanation:
1. Multiply the numerator and the denominator of \( \frac{1}{2} \) by 4: \( \frac{1\times 4}{2\times 4} \) = \( \frac{4}{8} \).
2. So \( \frac{4}{8} \) is equivalent to \( \frac{1}{2} \).

Teacher's Note:
1. Equivalent fractions: multiply top and bottom by the same number.
2. A quick check: the numerator should be half the denominator.

Answer: (b) \( \frac{5}{8} \)

Explanation:
1. The highest common factor of 30 and 48 is 6.
2. \( \frac{30\div 6}{48\div 6} \) = \( \frac{5}{8} \), and 5 and 8 have no common factor except 1.
3. \( \frac{15}{24} \) and \( \frac{10}{16} \) are equal to it but are not in lowest terms.

Teacher's Note:
1. Divide by the HCF to reach lowest terms in one step.
2. Check that no common factor other than 1 is left.

Answer: (a) Both statements are true.

Explanation:
1. \( \frac{5}{8} \) is 5 fractional units of \( \frac{1}{8} \) put together, so Statement I is true.
2. 8 pieces of \( \frac{1}{8} \) make the whole, so \( \frac{8}{8} \) = 1 and Statement II is true.
3. So both statements are true.

Teacher's Note:
1. Read a fraction as 'so many times the fractional unit'.
2. Same numerator and denominator always means one whole.

Answer: (c) A is true, but R is false.

Explanation:
1. One roti among 100 children gives a bigger piece than one roti among 200 children, so A is true.
2. A bigger denominator means more equal parts, so each part becomes smaller. R is false.
3. So A is true but R is false.

Teacher's Note:
1. Bigger denominator, smaller fractional unit.
2. R reverses the rule, so it is false.

Answer: (d) broken

Explanation:
1. The Sanskrit word bhinna means “broken”, which suits a fraction as a broken part of a whole.
2. Fractions were also called bhaga or ansha, meaning “part” or “piece”.

Teacher's Note:
1. Link bhinna with a broken piece of a whole.
2. Remember bhaga and ansha too.

Answer: (a) Only Statement I is true.

Explanation:
1. With the same fractional unit we only need to compare numerators, so Statement I is true.
2. With the same numerator, a bigger denominator means smaller units, so that fraction is smaller, e.g. \( \frac{4}{9} \) < \( \frac{4}{7} \). Statement II is false.
3. So only Statement I is true.

Teacher's Note:
1. Same top number: the bigger bottom number gives the smaller fraction.
2. Test a statement with a simple example.

Answer: (c) 10

Explanation:
1. Each friend gets \( \frac{5}{4} \) glass.
2. Double the friends from 4 to 8, so the glasses must also double: \( \frac{5\times 2}{4\times 2} \) = \( \frac{10}{8} \).
3. So 10 glasses among 8 friends give the same share.

Teacher's Note:
1. Equal shares stay equal when both numbers are multiplied by the same number.
2. Here the multiplier is 2.

Answer: (b) The shaded part of A is longer than the shaded part of B.

Explanation:
1. Strip A has 4 equal parts with 3 shaded, so it shows \( \frac{3}{4} \). Strip B has 3 equal parts with 2 shaded, so it shows \( \frac{2}{3} \).
2. With the fractional unit \( \frac{1}{12} \): \( \frac{3}{4} \) = \( \frac{9}{12} \) and \( \frac{2}{3} \) = \( \frac{8}{12} \).
3. 9 > 8, so the shaded part of A is longer.

Teacher's Note:
1. Read each strip separately: parts in all, then parts shaded.
2. Use a common fractional unit to compare.

Answer: (d) \( 5\frac{2}{9} \)

Explanation:
1. 9 ninths make one whole. 47 = 5 × 9 + 2.
2. So \( \frac{47}{9} \) = 5 + \( \frac{2}{9} \) = \( 5\frac{2}{9} \).

Teacher's Note:
1. Divide the numerator by the denominator: quotient is the whole part, remainder is the new numerator.
2. The denominator stays the same.

Answer: (c) (i)-(r), (ii)-(s), (iii)-(q), (iv)-(p)

Explanation:
1. \( \frac{16}{20} \) = \( \frac{4}{5} \) (divide by 4); \( \frac{36}{60} \) = \( \frac{3}{5} \) (divide by 12).
2. \( \frac{42}{56} \) = \( \frac{3}{4} \) (divide by 14); \( \frac{17}{51} \) = \( \frac{1}{3} \) (divide by 17).
3. So the matching is (i)-(r), (ii)-(s), (iii)-(q), (iv)-(p).

Teacher's Note:
1. Divide top and bottom by their HCF.
2. 17 × 3 = 51 is the hidden one.

Answer: (a) Both A and R are true, and R is the correct explanation of A.

Explanation:
1. Both fractions have 4 units, but ninths are smaller than sevenths. So \( \frac{4}{7} \) > \( \frac{4}{9} \) and A is true.
2. R is the equal-sharing idea, and it is true.
3. 4 rotis among 9 children gives less than 4 rotis among 7 children, so R explains A.

Teacher's Note:
1. Same numerator: fewer sharers means bigger share.
2. R is the reason behind A here.

Answer: (b) \( \frac{2}{15} \) litre

Explanation:
1. Use the common fractional unit \( \frac{1}{15} \): \( \frac{4}{5} \) = \( \frac{12}{15} \) and \( \frac{2}{3} \) = \( \frac{10}{15} \).
2. Juice left = \( \frac{12}{15} \) − \( \frac{10}{15} \) = \( \frac{2}{15} \) litre.

Teacher's Note:
1. Change both fractions to the same unit before subtracting.
2. Never subtract the denominators.

Answer: (a) Only Statement I is true.

Explanation:
1. \( \frac{2}{5} \) and \( \frac{1}{5} \) have the same unit \( \frac{1}{5} \), so \( \frac{2}{5} \) + \( \frac{1}{5} \) = \( \frac{3}{5} \). Statement I is true.
2. Denominators are never added: \( \frac{1}{4} \) + \( \frac{1}{3} \) = \( \frac{3}{12} \) + \( \frac{4}{12} \) = \( \frac{7}{12} \), not \( \frac{2}{7} \). Statement II is false.
3. So only Statement I is true.

Teacher's Note:
1. Same unit: add the numerators only.
2. Adding denominators is the most common fraction mistake.

Answer: (c) \( \frac{2}{3} \) = \( \frac{8}{12} \)

Explanation:
1. Strip P has 3 equal parts with 2 shaded, so it shows \( \frac{2}{3} \).
2. Strip Q has 12 equal parts with 8 shaded, so it shows \( \frac{8}{12} \).
3. The shaded lengths are equal, so \( \frac{2}{3} \) = \( \frac{8}{12} \).

Teacher's Note:
1. Equal shaded lengths mean equivalent fractions.
2. Check: \( \frac{2\times 4}{3\times 4} \) = \( \frac{8}{12} \).

Answer: (c) A is true, but R is false.

Explanation:
1. Each of the 6 equal parts is the fractional unit \( \frac{1}{6} \), so A is true.
2. The parts only need to be equal in size; they can be cut in different shapes. So R is false.

Teacher's Note:
1. Equal size matters, not equal shape.
2. A chikki can be cut into sixths in many ways.

Answer: (d) (i)-(r), (ii)-(p), (iii)-(s), (iv)-(q)

Explanation:
1. \( 3\frac{1}{4} \) = \( \frac{3\times 4+1}{4} \) = \( \frac{13}{4} \) and \( 7\frac{2}{3} \) = \( \frac{7\times 3+2}{3} \) = \( \frac{23}{3} \).
2. \( 2\frac{3}{11} \) = \( \frac{2\times 11+3}{11} \) = \( \frac{25}{11} \) and \( 3\frac{9}{10} \) = \( \frac{3\times 10+9}{10} \) = \( \frac{39}{10} \).
3. So the matching is (i)-(r), (ii)-(p), (iii)-(s), (iv)-(q).

Teacher's Note:
1. Whole × denominator + numerator gives the new numerator.
2. The denominator does not change.

Answer: (b) Al-Hassar

Explanation:
1. Indian texts wrote the numerator above the denominator without any line.
2. The dividing line was added by the Moroccan mathematician Al-Hassar in the 12th century, and the notation then spread to Europe.

Teacher's Note:
1. The Indian way: number above number, no line.
2. The bar came later, from Al-Hassar.

Answer: (a) Both A and R are true, and R is the correct explanation of A.

Explanation:
1. 45 is a common multiple of 5 and 9: \( \frac{4}{5} \) = \( \frac{36}{45} \) and \( \frac{7}{9} \) = \( \frac{35}{45} \). So R is true.
2. 36 > 35, so \( \frac{4}{5} \) > \( \frac{7}{9} \) and A is true.
3. R gives exactly the equivalent fractions that prove A, so R explains A.

Teacher's Note:
1. Use the product of the denominators as a common denominator.
2. Then compare only the numerators.

Answer: (d) \( \frac{2}{5} \), \( \frac{7}{10} \), \( \frac{11}{15} \)

Explanation:
1. 30 is a common multiple of 10, 15 and 5: \( \frac{7}{10} \) = \( \frac{21}{30} \), \( \frac{11}{15} \) = \( \frac{22}{30} \), \( \frac{2}{5} \) = \( \frac{12}{30} \).
2. 12 < 21 < 22, so the order is \( \frac{2}{5} \), \( \frac{7}{10} \), \( \frac{11}{15} \).

Teacher's Note:
1. Least to most means ascending order.
2. Pick the smallest common multiple to keep numbers small.

Answer: (b) Only Statement II is true.

Explanation:
1. The common fractional unit for sixths and fourths is \( \frac{1}{12} \).
2. \( \frac{1}{6} \) + \( \frac{1}{4} \) = \( \frac{2}{12} \) + \( \frac{3}{12} \) = \( \frac{5}{12} \), so Statement II is true.
3. Statement I adds the denominators, which is wrong. So only Statement II is true.

Teacher's Note:
1. \( \frac{2}{10} \) is smaller than \( \frac{1}{4} \) alone, so it cannot be the sum.
2. Use the smallest common multiple, 12.

Answer: (c) (i)-(r), (ii)-(s), (iii)-(p), (iv)-(q)

Explanation:
1. \( \frac{1}{6} \) + \( \frac{1}{3} \) = \( \frac{1}{6} \) + \( \frac{2}{6} \) = \( \frac{3}{6} \) = \( \frac{1}{2} \); \( \frac{5}{6} \) − \( \frac{4}{9} \) = \( \frac{15}{18} \) − \( \frac{8}{18} \) = \( \frac{7}{18} \).
2. \( \frac{2}{3} \) + \( \frac{3}{4} \) = \( \frac{8}{12} \) + \( \frac{9}{12} \) = \( \frac{17}{12} \); \( \frac{3}{5} \) − \( \frac{1}{2} \) = \( \frac{6}{10} \) − \( \frac{5}{10} \) = \( \frac{1}{10} \).
3. So the matching is (i)-(r), (ii)-(s), (iii)-(p), (iv)-(q).

Teacher's Note:
1. Work each one out, then look for that option.
2. Reduce \( \frac{3}{6} \) to lowest terms.

Answer: (a) \( \frac{7}{12} \) km

Explanation:
1. Use the fractional unit \( \frac{1}{12} \): \( \frac{5}{6} \) = \( \frac{10}{12} \) and \( \frac{1}{4} \) = \( \frac{3}{12} \).
2. Walking distance = \( \frac{10}{12} \) − \( \frac{3}{12} \) = \( \frac{7}{12} \) km.

Teacher's Note:
1. 'The rest' means subtract.
2. \( \frac{13}{12} \) is what you get by adding by mistake.

Answer: (d) A is false, but R is true.

Explanation:
1. The largest fractional unit is \( \frac{1}{2} \), and \( \frac{1}{2} \) + \( \frac{1}{2} \) = 1. Two different units must use something smaller than \( \frac{1}{2} \), so their sum is less than 1. A is false.
2. \( \frac{1}{2} \) + \( \frac{1}{3} \) + \( \frac{1}{6} \) = \( \frac{3}{6} \) + \( \frac{2}{6} \) + \( \frac{1}{6} \) = \( \frac{6}{6} \) = 1, so R is true.

Teacher's Note:
1. Two different units always fall short of 1.
2. Three different units can make 1: \( \frac{1}{2} \), \( \frac{1}{3} \) and \( \frac{1}{6} \).