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Division of Fractions

🎓 Class 7 Mathematics CBSE Theory Ch 8 — Working with Fractions ⏱ ~35 min
🌐 Language: [gtranslate]

This MCQ module is based on: Division of Fractions

This mathematics assessment will be based on: Division of Fractions
Targeting Class 7 level in Fractions, with Basic difficulty.

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8.2 Division of Fractions

Division asks: "How many groups?" We know \(12\div 4 = 3\), because we can fit three 4s into 12. Equivalently, division turns into a multiplication problem: find \(?\) such that \(4\times ?=12\). Clearly \(?=3\).

Applying the same idea to fractions:

\(6\div \tfrac12 = ?\ \Leftrightarrow\ \tfrac12\times ?=6\ \Rightarrow\ ?=12.\)

So \(6\div\tfrac12=12\). Notice the answer is larger than 6 — because \(\tfrac12\) fits into 6 twelve times.

The Reciprocal

Definition
The reciprocal? of a fraction \(\tfrac{a}{b}\) (with \(a,b\ne 0\)) is \(\tfrac{b}{a}\). A fraction times its reciprocal equals 1: \(\tfrac{a}{b}\times\tfrac{b}{a}=\tfrac{ab}{ba}=1\).

Examples. Reciprocal of \(\tfrac{3}{5}\) is \(\tfrac{5}{3}\). Reciprocal of 4 is \(\tfrac14\). Reciprocal of \(\tfrac{1}{7}\) is 7.

Brahmagupta's Division Rule

Rule
To divide by a fraction, multiply by its reciprocal: \(\ \dfrac{a}{b}\div\dfrac{c}{d} = \dfrac{a}{b}\times\dfrac{d}{c} = \dfrac{a\,d}{b\,c}.\)

Example 1. \(\tfrac{2}{3}\div\tfrac{4}{5} = \tfrac{2}{3}\times\tfrac{5}{4}=\tfrac{10}{12}=\tfrac{5}{6}\).
Example 2. \(6\div\tfrac{1}{4}=6\times 4=24\).
Example 3. \(\tfrac{7}{8}\div 2 = \tfrac{7}{8}\times\tfrac{1}{2}=\tfrac{7}{16}\).

0 1 2 3 4 5 6 12 half-jumps fit in 6
\(6\div\tfrac12=12\): there are twelve "halves" in 6.

Behaviour of the Quotient

  • If divisor > 1: quotient < dividend (e.g., \(6\div 2 = 3\)).
  • If divisor < 1 (a proper fraction): quotient > dividend (e.g., \(6\div\tfrac14=24\)).
  • If divisor = 1: quotient = dividend.
Historical note
Brahmagupta (628 CE, Brahmasphutasiddhanta) stated: "The division of general fractions is performed by interchanging the numerator and denominator of the divisor, and then multiplying." — essentially our modern rule.

Figure it Out

Q1. Evaluate: (a) \(3\div\tfrac25\), (b) \(\tfrac{14}{9}\div 2\), (c) \(\tfrac{4}{3}\div\tfrac{7}{2}\), (d) \(\tfrac{14}{15}\div\tfrac{7}{3}\).
(a) \(3\times\tfrac52=\tfrac{15}{2}\); (b) \(\tfrac{14}{9}\times\tfrac12=\tfrac{7}{9}\); (c) \(\tfrac{4}{3}\times\tfrac27=\tfrac{8}{21}\); (d) \(\tfrac{14}{15}\times\tfrac37=\tfrac{2}{5}\).
Q2. Maria bought 8 m of lace for 8 bags; she uses \(\tfrac14\) m per bag. How many bags can she decorate?
\(8\div\tfrac14 = 8\times 4 = 32\) bags. (Plenty.)
Q3. \(\tfrac{1}{2}\) m of ribbon is cut into 8 equal pieces. How long is each piece?
\(\tfrac12\div 8 = \tfrac12\times\tfrac18 = \tfrac{1}{16}\) m per piece.
Q4. A baker needs \(\tfrac56\) kg of flour per loaf. If he has 5 kg of flour, how many loaves can he make?
\(5\div\tfrac56 = 5\times\tfrac65 = 6\) loaves.
Q5. If \(4\tfrac12\) kg of flour makes 12 rotis, how much flour is in 6 rotis?
Flour per roti \(= \tfrac{9}{2}\div 12 = \tfrac{9}{24}=\tfrac{3}{8}\) kg. For 6 rotis: \(6\times\tfrac{3}{8}=\tfrac{9}{4}=2\tfrac14\) kg.
Activity: How Many Halves Fit In?
Materials: Paper strips of length 1 unit, scissors, ruler.
  1. Take a strip of 3 units length.
  2. Cut half-unit pieces from it; count how many fit.
  3. Record \(3 \div \tfrac12 = \text{?}\).
  4. Repeat for strips of lengths 2, 4, and \(\tfrac32\).
Length ÷ \(\tfrac12\) = twice the length. So: 3/(1/2) = 6; 2/(1/2) = 4; 4/(1/2) = 8; (3/2)/(1/2)=3. The quotient grows because the divisor is less than 1.

Competency-Based Questions

Scenario: A cook has 4 kg of rice. Each portion of rice biryani needs \(\tfrac25\) kg.
Q1. How many portions can be prepared?
L3 Apply
\(4\div\tfrac25 = 4\times\tfrac52 = 10\) portions.
Q2. If the cook decides to make smaller portions of \(\tfrac15\) kg each, analyse how the count changes.
L4 Analyse
\(4\div\tfrac15 = 20\) portions. Since the divisor halved, the count doubled from 10 to 20.
Q3. Evaluate: "Dividing a quantity by \(\tfrac12\) gives the same answer as multiplying it by 2." True or false?
L5 Evaluate
True — the reciprocal of \(\tfrac12\) is 2, and dividing by a fraction is multiplying by its reciprocal.
Q4. Create a 2-step problem involving both multiplication and division of fractions with a cooking theme.
L6 Create
Sample: "A recipe needs \(\tfrac34\) cup of milk for 2 servings. (a) How much for 5 servings? (b) If only \(\tfrac52\) cups of milk are available, how many servings can be made?" (a) \(\tfrac34\div 2\times 5 = \tfrac{15}{8}\) cups; (b) \(\tfrac52 \div \tfrac{3}{8}=\tfrac52\times\tfrac{8}{3}=\tfrac{40}{6}=6\tfrac23\) servings.

Assertion–Reason Questions

A: The reciprocal of \(\tfrac{3}{7}\) is \(\tfrac{7}{3}\).
R: A fraction times its reciprocal equals 1.
(a) Both true, R explains A.
(b) Both true, R doesn't explain A.
(c) A true, R false.
(d) A false, R true.
(a) — \(\tfrac37\times\tfrac73=1\). R defines the reciprocal and explains A.
A: \(8\div\tfrac14 = 32\).
R: Dividing by a fraction < 1 produces a quotient larger than the dividend.
(a) Both true, R explains A.
(b) Both true, R doesn't explain A.
(c) A true, R false.
(d) A false, R true.
(a) — R gives the general reason A holds.
A: Zero has a reciprocal equal to 0.
R: Every real number has a reciprocal.
(a) Both true, R explains A.
(b) Both true, R doesn't explain A.
(c) A true, R false.
(d) A false, R true.
Both false — zero has no reciprocal (1/0 is undefined), and hence R is false. "Both false" is the correct response.

Frequently Asked Questions

What is the reciprocal of a fraction?
The reciprocal of a/b is b/a, obtained by swapping numerator and denominator. The reciprocal of 3/7 is 7/3.
How do you divide 4/5 by 2/3?
Multiply 4/5 by the reciprocal of 2/3, which is 3/2. So 4/5 times 3/2 equals 12/10, which simplifies to 6/5 or 1 1/5.
Why does 'invert and multiply' work?
Dividing by a number is the same as multiplying by its reciprocal. So dividing by 2/3 equals multiplying by 3/2, because multiplying by the reciprocal cancels the divisor.
How do you divide a whole number by a fraction?
Write the whole number as a fraction over 1, multiply by the reciprocal of the divisor fraction, and simplify.
What is the result when you divide a fraction by 1?
Any fraction divided by 1 equals itself, because the reciprocal of 1 is 1 and multiplying by 1 leaves a number unchanged.
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