CBSE Class 7 Mathematics Working with Fractions MCQs Set 11

Answer: (b) \( \frac{7}{3} \)

Explanation:
1. To get the reciprocal, swap the numerator and the denominator.
2. So the reciprocal of \( \frac{3}{7} \) is \( \frac{7}{3} \).
3. Check: \( \frac{3}{7}\times\frac{7}{3}=\frac{21}{21}=1 \).

Teacher's Note:
1. A number times its reciprocal is always 1.
2. Use this product as a quick check.

Answer: (c) 1

Explanation:
1. The reciprocal of \( \frac{a}{b} \) is \( \frac{b}{a} \).
2. \( \frac{a}{b}\times\frac{b}{a}=\frac{a\times b}{b\times a} \), and the numerator and denominator are equal.
3. So the product is always 1.

Teacher's Note:
1. This is exactly how the reciprocal is defined.
2. Try it with 2/5 × 5/2 to see.

Answer: (d) Brahmagupta in the Brahmasphutasiddhanta

Explanation:
1. The rule 'product of the numerators over the product of the denominators' was first stated by Brahmagupta.
2. He gave it in his book Brahmasphutasiddhanta in 628 CE.
3. So the answer is Brahmagupta in the Brahmasphutasiddhanta.

Teacher's Note:
1. Brahmagupta also gave the rule for dividing fractions.
2. Umasvati is linked with reducing fractions to lowest terms.

Answer: (a) \( \frac{9}{4} \)

Explanation:
1. \( 2\frac{1}{4} \) means \( 2+\frac{1}{4} \).
2. 2 wholes are \( \frac{8}{4} \), so \( \frac{8}{4}+\frac{1}{4}=\frac{9}{4} \).
3. So the answer is \( \frac{9}{4} \).

Teacher's Note:
1. Shortcut: (whole × denominator + numerator) over the denominator.
2. Here (2 × 4 + 1)/4 = 9/4.

Answer: (c) \( \frac{2}{3} \) unit and \( \frac{5}{7} \) unit

Explanation:
1. The area of a rectangle = length × breadth, even when the sides are fractions.
2. So \( \frac{2}{3}\times\frac{5}{7} \) is the area of a rectangle with sides \( \frac{2}{3} \) unit and \( \frac{5}{7} \) unit.

Teacher's Note:
1. The two fractions being multiplied are the two sides.
2. This is the area model of multiplication.

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

Explanation:
1. R is true: \( \frac{1}{b}\times\frac{1}{d}=\frac{1}{b\times d} \).
2. Using R, \( \frac{1}{5}\times\frac{1}{4}=\frac{1}{5\times4}=\frac{1}{20} \), so A is true.
3. A is just R applied to 1/5 and 1/4, so R correctly explains A.

Teacher's Note:
1. A fractional unit has numerator 1.
2. The unit square is cut into 5 × 4 = 20 equal parts.

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

Explanation:
1. R is true: any number × 1 = the same number.
2. So a proper fraction × 1 is equal to the fraction, not smaller. A is false.
3. So A is false but R is true.

Teacher's Note:
1. 1 is the number that changes nothing in multiplication.
2. Only a number between 0 and 1 makes the product smaller.

Answer: (a) Only Statement I is true.

Explanation:
1. Both sides of Statement I equal \( \frac{6}{20}=\frac{3}{10} \), so Statement I is true.
2. The order of multiplication never changes the product, so Statement II is false.
3. So only Statement I is true.

Teacher's Note:
1. A rectangle has the same area whichever side is called the length.
2. Statement I itself disproves Statement II.

Answer: (c) Both statements are true.

Explanation:
1. Multiplication by a whole number is repeated addition, so Statement I is true.
2. \( \frac{2}{3}+\frac{2}{3}+\frac{2}{3}+\frac{2}{3}+\frac{2}{3}=\frac{10}{3} \), so Statement II is true.
3. So both statements are true.

Teacher's Note:
1. Multiply only the numerator by the whole number.
2. The denominator stays 3.

Answer: (b) \( \frac{3}{2} \)

Explanation:
1. \( \frac{3}{8}\times4=\frac{3\times4}{8}=\frac{12}{8} \).
2. Cancel the common factor 4: \( \frac{12}{8}=\frac{3}{2} \).
3. So the answer is \( \frac{3}{2} \).

Teacher's Note:
1. The whole number multiplies the numerator, not the denominator.
2. 3/32 comes from wrongly multiplying the denominator.

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

Explanation:
1. (i) \( \frac{1}{2}\times\frac{1}{3}=\frac{1}{6} \) (r). (ii) \( \frac{1}{2}\div\frac{1}{3}=\frac{1}{2}\times3=\frac{3}{2} \) (s).
2. (iii) \( \frac{1}{3}\div\frac{1}{2}=\frac{1}{3}\times2=\frac{2}{3} \) (p). (iv) \( \frac{1}{3}\times6=2 \) (q).
3. So the match is (i)-(r), (ii)-(s), (iii)-(p), (iv)-(q).

Teacher's Note:
1. To divide, multiply by the reciprocal of the divisor.
2. Swapping dividend and divisor gives the reciprocal answer: 3/2 and 2/3.

Answer: (c) \( \frac{1}{12} \) square unit

Explanation:
1. The unit square has 3 rows and 4 columns, so it is cut into 3 × 4 = 12 equal parts.
2. One part is shaded, and its sides are \( \frac{1}{3} \) unit and \( \frac{1}{4} \) unit.
3. Area = \( \frac{1}{3}\times\frac{1}{4}=\frac{1}{12} \), that is \( \frac{1}{12} \) square unit.

Teacher's Note:
1. Multiply the denominators; do not add them.
2. 1/7 comes from adding 3 and 4 by mistake.

Answer: (a) \( \frac{a\times d}{b\times c} \)

Explanation:
1. To divide, multiply the dividend by the reciprocal of the divisor.
2. The reciprocal of \( \frac{c}{d} \) is \( \frac{d}{c} \).
3. \( \frac{a}{b}\times\frac{d}{c}= \) \( \frac{a\times d}{b\times c} \).

Teacher's Note:
1. Only the divisor is turned upside down.
2. Option (b) is the rule for multiplication.

Answer: (b) less than the first number

Explanation:
1. Multiplying by a number between 0 and 1 means taking only a part of the first number.
2. Example: \( \frac{3}{4}\times5=\frac{15}{4}=3\frac{3}{4} \), which is less than 5.
3. So the product is less than the first number.

Teacher's Note:
1. 'Of' a part is always smaller than the whole.
2. Multiplying by a number greater than 1 does the opposite.

Answer: (a) \( \frac{2}{3} \)

Explanation:
1. The divisor is the number we divide by.
2. Here \( \frac{7}{8} \) is the dividend and we divide it by \( \frac{2}{3} \).
3. So the divisor is \( \frac{2}{3} \) (and the quotient is \( \frac{21}{16} \)).

Teacher's Note:
1. Dividend ÷ divisor = quotient.
2. 3/2 is the reciprocal of the divisor, not the divisor.

Answer: (b) Both A and R are true, but R is not the correct explanation of A.

Explanation:
1. \( \frac{1}{8}\div\frac{1}{4}=\frac{1}{8}\times4=\frac{4}{8}=\frac{1}{2} \), so A is true.
2. R is also true, e.g. \( \frac{1}{2}\div2=\frac{1}{4} \).
3. But in A the divisor is the fraction 1/4, not a whole number, so R does not explain A.

Teacher's Note:
1. Check whether R talks about the same kind of division as A.
2. Here the divisor is a fraction.

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

Explanation:
1. \( \frac{14}{15}\times\frac{25}{42}=\frac{14\times25}{15\times42} \); cancel 14 with 42 (by 14) and 25 with 15 (by 5) to get \( \frac{1\times5}{3\times3}=\frac{5}{9} \). A is true.
2. Cancelling is always done between a numerator and a denominator, never between two numerators. R is false.
3. So A is true but R is false.

Teacher's Note:
1. Cancel only top with bottom.
2. Cancelling two numerators changes the value.

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

Explanation:
1. \( \frac{2}{3}=\frac{10}{15} \) and \( \frac{4}{5}=\frac{12}{15} \); \( \frac{8}{15} \) is less than both. A is true.
2. R is the true rule for a multiplier between 0 and 1.
3. Here each number lies between 0 and 1, so by R the product is less than each of them. R explains A.

Teacher's Note:
1. Apply R twice, once for each factor.
2. Use a common denominator to compare.

Answer: (b) Only Statement I is true.

Explanation:
1. \( \frac{7}{4}\div\frac{1}{7}=\frac{7}{4}\times7=\frac{49}{4} \), so Statement I is true.
2. The divisor 1/7 lies between 0 and 1, so the quotient is greater than the dividend. Statement II is false.
3. So only Statement I is true.

Teacher's Note:
1. Note the options here are in a different order.
2. Dividing by a proper fraction makes the result bigger.

Answer: (d) Neither statement is true.

Explanation:
1. \( \frac{1}{5}\div\frac{1}{9}=\frac{1}{5}\times9=\frac{9}{5} \), not 1/45. Statement I is false.
2. \( \frac{1}{6}\div\frac{11}{12}=\frac{1}{6}\times\frac{12}{11}=\frac{2}{11} \), not 11/72. Statement II is false.
3. So neither statement is true.

Teacher's Note:
1. Both wrong values come from multiplying instead of dividing.
2. Always flip the divisor first.

Answer: (c) 45 km

Explanation:
1. \( 2\frac{1}{2}=\frac{5}{2} \) litres.
2. Distance = \( 18\times\frac{5}{2}=\frac{90}{2}=45 \) km.
3. So the car goes 45 km.

Teacher's Note:
1. Change the mixed number to an improper fraction first.
2. Quick check: 18 × 2 = 36 and half of 18 is 9; 36 + 9 = 45.

Answer: (a) \( \frac{3}{5} \)

Explanation:
1. In \( \frac{3}{4} \) hour it fills \( \frac{3}{4} \) of \( \frac{4}{5} \) of the tank.
2. Part filled = \( \frac{3}{4}\times\frac{4}{5}=\frac{12}{20}=\frac{3}{5} \) of the tank.
3. So the answer is \( \frac{3}{5} \).

Teacher's Note:
1. The answer must be less than 4/5, since less than an hour passes.
2. That rules out 16/15 and 15/16 at once.

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

Explanation:
1. (i) \( \frac{5}{6}\times\frac{3}{10}=\frac{15}{60}=\frac{1}{4} \) (r). (ii) \( \frac{5}{6}\times\frac{10}{3}=\frac{50}{18}=\frac{25}{9} \) (p).
2. (iii) \( \frac{3}{10}\times\frac{6}{5}=\frac{18}{50}=\frac{9}{25} \) (s). (iv) \( \frac{25}{30}+\frac{9}{30}=\frac{34}{30}=\frac{17}{15} \) (q).
3. So the match is (i)-(r), (ii)-(p), (iii)-(s), (iv)-(q).

Teacher's Note:
1. Addition needs a common denominator; multiplication does not.
2. (ii) and (iii) give reciprocal answers.

Answer: (b) \( \frac{3}{10} \) square unit

Explanation:
1. The unit square has 4 rows and 5 columns, that is 20 equal parts.
2. The shaded rectangle covers 2 × 3 = 6 of these parts.
3. Area = \( \frac{2}{5}\times\frac{3}{4}=\frac{6}{20}=\frac{3}{10} \) square unit.

Teacher's Note:
1. Count the shaded cells and the total cells.
2. 17/20 comes from adding the fractions by mistake.

Answer: (b) \( \frac{5}{3} \) and \( \frac{7}{4} \)

Explanation:
1. The product is greater than both numbers only when both numbers are greater than 1.
2. \( \frac{5}{3} \) and \( \frac{7}{4} \) are both greater than 1; their product \( \frac{35}{12} \) is bigger than both.
3. In (a) the product \( \frac{3}{8} \) is smaller than both; in (c) \( \frac{12}{5} \) lies between them; in (d) the product equals \( \frac{2}{7} \). So the answer is (b).

Teacher's Note:
1. Look for numbers greater than 1 (numerator bigger than denominator).
2. Multiplying by 1 changes nothing.

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

Explanation:
1. \( \frac{3}{4}\div\frac{5}{2}=\frac{3}{4}\times\frac{2}{5}=\frac{6}{20}=\frac{3}{10} \), which is smaller than 3/4. A is false.
2. R is the correct rule: a divisor greater than 1 gives a quotient less than the dividend.
3. So A is false but R is true.

Teacher's Note:
1. 5/2 is greater than 1, so expect a smaller answer.
2. R actually shows why A is wrong.

Answer: (c) Only Statement II is true.

Explanation:
1. Counter-example: \( \frac{3}{2}\times\frac{4}{5}=\frac{12}{10}=\frac{6}{5} \), which is greater than 1. Statement I is false.
2. Dividing by a fraction between 0 and 1 means multiplying by its reciprocal, which is greater than 1. Statement II is true.
3. So only Statement II is true.

Teacher's Note:
1. One counter-example is enough to show a statement is false.
2. The product is less than the bigger number, not less than 1.

Answer: (d) 190

Explanation:
1. Yesterday: \( \frac{1}{4}\times360=90 \) pages. Today: \( \frac{2}{9}\times360=80 \) pages.
2. Pages read = 90 + 80 = 170.
3. Pages left = 360 − 170 = 190.

Teacher's Note:
1. Both fractions are of the whole book, 360 pages.
2. 170 is the pages read, not the pages left.

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

Explanation:
1. (i) \( \frac{1}{2}\times\frac{2}{3}=\frac{1}{3} \) (q). (ii) \( \frac{1}{3}\times\frac{3}{4}=\frac{1}{4} \) (s).
2. (iii) \( \frac{1}{4}\times\frac{4}{5}=\frac{1}{5} \) (p). (iv) \( \frac{1}{5}\times\frac{5}{6}=\frac{1}{6} \) (r).
3. So the match is (i)-(q), (ii)-(s), (iii)-(p), (iv)-(r).

Teacher's Note:
1. Each new numerator cancels the previous denominator.
2. The answer is 1 over the last denominator.

Answer: (d) \( \frac{1}{16} \)

Explanation:
1. Each small square is \( \frac{1}{4} \) of the big square.
2. The diagonal cuts it into two equal triangles, each \( \frac{1}{2}\times\frac{1}{4}=\frac{1}{8} \) of the big square.
3. The shaded part is half of that triangle: \( \frac{1}{2}\times\frac{1}{8}=\frac{1}{16} \). So the answer is \( \frac{1}{16} \).

Teacher's Note:
1. Take the big square as 1 and work step by step.
2. Each halving multiplies by 1/2.