This MCQ module is based on: Multiplying Two Fractions
TOPIC 29 OF 31
Multiplying Two Fractions
🎓 Class 7
Mathematics
CBSE
Theory
Ch 8 — Working with Fractions
⏱ ~35 min
🌐 Language: [gtranslate]
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This mathematics assessment will be based on: Multiplying Two Fractions
Targeting Class 7 level in Fractions, with Basic difficulty.
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Multiplying Two Fractions
The tortoise walks \(\tfrac14\) km in 1 hour. How far does it walk in \(\tfrac12\) hour? Half of \(\tfrac14\) km is \(\tfrac12\times\tfrac14=\tfrac{1}{8}\) km.
Why the Area Model Works
Take a unit square (side 1). Shade the left \(\tfrac14\) of it as a vertical strip — that area is \(\tfrac14\) of the whole. Now shade the bottom \(\tfrac12\) horizontally. The overlap is a rectangle of breadth \(\tfrac14\) and height \(\tfrac12\). Its area, counted as small-cells out of the total 8 cells, is \(\tfrac18\).
Fig 8.1 — Area model: \(\tfrac12\times\tfrac14 = \tfrac18\) of the unit square.
Rule (Brahmagupta's formula)
For any two fractions \(\dfrac{a}{b}\) and \(\dfrac{c}{d}\), \(\ \dfrac{a}{b}\times\dfrac{c}{d} = \dfrac{a\times c}{b\times d}\). That is — multiply numerators and multiply denominators.Check: \(\tfrac12\times\tfrac14 = \tfrac{1\times 1}{2\times 4} = \tfrac18\). ✓
More examples: \(\tfrac34\times\tfrac25 = \tfrac{6}{20} = \tfrac{3}{10}\); \(\tfrac{5}{7}\times\tfrac{2}{9}=\tfrac{10}{63}\).
Connection with Area of a Rectangle
If a rectangle has length \(\tfrac12\) unit and breadth \(\tfrac14\) unit, its area equals product of its sides, \(\tfrac12\times\tfrac14 = \tfrac18\) sq units. The area-of-a-rectangle formula still holds when the sides are fractions.
Order of Multiplication
Multiplication of fractions is commutative: \(\tfrac{a}{b}\times\tfrac{c}{d} = \tfrac{c}{d}\times\tfrac{a}{b}\). In the area model, swapping length and breadth doesn't change the shaded area.
Cancelling Common Factors
Before multiplying, cancel common factors to keep numbers small.
\(\displaystyle\frac{14}{15}\times\frac{25}{42}=\frac{\cancel{14}^{\,1}\times \cancel{25}^{\,5}}{\cancel{15}^{\,3}\times \cancel{42}^{\,3}}=\frac{1\times5}{3\times3}=\frac{5}{9}\)
A Pinch of History
Indian mathematicians from the 5th century onwards wrote clear rules for fraction multiplication. Brahmasphutasiddhanta (628 CE) and later Bhaskara I give precisely the formula \(\tfrac{a}{b}\times\tfrac{c}{d}=\tfrac{a\,c}{b\,d}\). Jaina scholar Umasvati (c. 150 CE) used the same rule in a philosophical text.Figure it Out
Q1. Using a unit-square grid, represent: (a) \(\tfrac13\times\tfrac15\), (b) \(\tfrac14\times\tfrac13\), (c) \(\tfrac15\times\tfrac12\), (d) \(\tfrac16\times\tfrac15\). State each product.
(a) \(\tfrac{1}{15}\); (b) \(\tfrac{1}{12}\); (c) \(\tfrac{1}{10}\); (d) \(\tfrac{1}{30}\). In each case the product equals 1 cell out of the total (numerators × denominators).
Q2. Compute after cancelling: (a) \(\tfrac{3}{7}\times\tfrac{14}{9}\), (b) \(\tfrac{5}{12}\times\tfrac{8}{15}\), (c) \(\tfrac{9}{10}\times\tfrac{25}{27}\).
(a) \(\tfrac{3\times 14}{7\times 9}=\tfrac{2}{3}\); (b) \(\tfrac{5\times 8}{12\times 15}=\tfrac{2}{9}\); (c) \(\tfrac{9\times 25}{10\times 27}=\tfrac{5}{6}\).
Q3. A water tank is filled by a tap in 1 hour. What fraction is filled in \(\tfrac{7}{10}\) h?
\(1\times\tfrac{7}{10}=\tfrac{7}{10}\) of the tank.
Q4. The government has taken \(\tfrac15\) of Somu's land. How much remains?
Remaining fraction \(=1-\tfrac15=\tfrac45\) of the land.
Q5. What is \(\tfrac34\times\tfrac12\)? Verify with the area model.
\(\tfrac{3}{8}\). In a \(4\times 2\) grid, shading 3 columns and 1 row gives 3 overlapping cells out of 8.
Activity: Draw the Area Product
Materials: 1 cm grid paper, crayons.
- Draw a rectangle 6 cm wide and 4 cm tall (it contains 24 small squares).
- Shade the first \(\tfrac13\) of the width (2 cm) with one colour.
- Shade the bottom \(\tfrac14\) of the height (1 cm) with a second colour.
- Count how many cells receive both colours.
- Compute \(\tfrac13\times\tfrac14\) and check your count divided by 24 equals the fraction.
Overlap = 2 cells. Fraction = \(\tfrac{2}{24}=\tfrac{1}{12}=\tfrac13\times\tfrac14\). ✓
Competency-Based Questions
Scenario: A farmer has a rectangular field measuring \(\tfrac34\) km long and \(\tfrac23\) km wide. He plans to plant paddy on \(\tfrac14\) of the total area.
Q1. Find the area of the field (sq km).
L3 Apply\(\tfrac34\times\tfrac23 = \tfrac{6}{12}=\tfrac12\) sq km.
Q2. Find the paddy area (sq km).
L4 Analyse\(\tfrac14\times\tfrac12=\tfrac18\) sq km.
Q3. Evaluate whether commuting (swapping) the side lengths changes the field's area.
L5 EvaluateNo — multiplication is commutative: \(\tfrac34\times\tfrac23 = \tfrac23\times\tfrac34\). Swapping length and breadth does not change the area.
Q4. Create an original problem whose answer is exactly \(\tfrac{5}{12}\), using multiplication of two fractions.
L6 CreateSample: "A juice bottle is \(\tfrac56\) full. Of that juice, \(\tfrac12\) is mango juice. What fraction of the bottle is mango juice?" \(\tfrac56\times\tfrac12=\tfrac{5}{12}\).
Assertion–Reason Questions
A: \(\tfrac23\times\tfrac34 = \tfrac12\).
R: Multiplying two fractions means multiplying numerators and multiplying denominators, then simplifying.
R: Multiplying two fractions means multiplying numerators and multiplying denominators, then simplifying.
(a) — \(\tfrac{2\times3}{3\times4}=\tfrac{6}{12}=\tfrac12\). R explains A.
A: The product of two proper fractions is always smaller than either factor.
R: A proper fraction lies between 0 and 1, and multiplying by a number < 1 decreases the value.
R: A proper fraction lies between 0 and 1, and multiplying by a number < 1 decreases the value.
(a) — e.g. \(\tfrac34\times\tfrac25=\tfrac{3}{10}\), smaller than both \(\tfrac34\) and \(\tfrac25\). R explains A.
A: Fraction multiplication is not commutative.
R: The area of a rectangle depends on which side is called length.
R: The area of a rectangle depends on which side is called length.
Both A and R are false. A: it is commutative. R: area is independent of which side we label length. So none of (a)-(d) if a "both false" option is absent, the closest is — both false.
Frequently Asked Questions
How do you multiply 2/3 by 4/5?
Multiply the numerators: 2 times 4 is 8. Multiply the denominators: 3 times 5 is 15. The product is 8/15, already in simplest form.
Can you simplify fractions before multiplying?
Yes. Cancel any common factor between a numerator and a denominator before multiplying. This keeps numbers small and avoids heavy simplification afterwards.
Why is the product of two proper fractions smaller than each?
Because each fraction is less than 1. Multiplying by a value less than 1 always reduces the other number, so the product is smaller than both original fractions.
How do you multiply mixed fractions?
Convert each mixed fraction to an improper fraction, multiply numerators and denominators, simplify and convert back to mixed form.
What is the area example of fraction multiplication?
The area of a rectangle with length 2/3 m and breadth 1/2 m is 2/3 times 1/2, which equals 2/6 or 1/3 square metres.
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