Question 1

What is the rule for multiplying two fractions $\frac{a}{b}$ and $\frac{c}{d}$?

Accepted Answer

Multiply numerators together and denominators together to get $\frac{a 	imes c}{b 	imes d}$ — The multiplication of two fractions $\frac{a}{b}$ and $\frac{c}{d}$ is done by multiplying the numerators to get the new numerator and the denominators to get the new denominator, resulting in $\frac{a 	imes c}{b 	imes d}$. This method reflects finding a part of a part and is the standard rule for fraction multiplication.

Question 2

When multiplying two fractions both less than 1, how does the product compare to the original fractions?

Accepted Answer

The product is smaller than either fraction — When two fractions less than 1 are multiplied, the product represents a part of a part, which is always smaller than either of the original fractions. For example, $\frac{1}{2} 	imes \frac{1}{3} = \frac{1}{6}$, which is smaller than both $\frac{1}{2}$ and $\frac{1}{3}$.

Question 3

Multiply $3$ by $\frac{2}{5}$ using repeated addition.

Accepted Answer

$\frac{6}{5}$ — Given: 3 (whole number), $\frac{2}{5}$ (fraction)
Find: Product of 3 and $\frac{2}{5}$ using repeated addition
Formula: Multiplying a fraction by a whole number is repeated addition of the fraction that many times.
Solution: Step 1: Add $\frac{2}{5}$ three times: $\frac{2}{5} + \frac{2}{5} + \frac{2}{5}$
Step 2: Add numerators: $\frac{2+2+2}{5} = \frac{6}{5}$
Answer: $\frac{6}{5}$
Note: Students often forget to add numerators correctly or keep denominator same.

Question 4

Convert the whole number 4 into a fraction and multiply it by $\frac{3}{7}$ using the multiplication rule.

Accepted Answer

$\frac{12}{7}$ — Given: Whole number 4, fraction $\frac{3}{7}$
Find: Product using multiplication rule
Formula: Convert whole number to fraction by placing over 1, then multiply numerators and denominators.
Solution: Step 1: Convert 4 to $\frac{4}{1}$
Step 2: Multiply: $\frac{4}{1} 	imes \frac{3}{7} = \frac{4 	imes 3}{1 	imes 7} = \frac{12}{7}$
Answer: $\frac{12}{7}$
Note: Students sometimes forget to convert whole number to fraction before multiplying.

Question 5

Explain how multiplying a fraction by a whole number can be represented visually using a rectangle divided into equal parts.

Accepted Answer

Multiplying a fraction by a whole number can be shown by shading parts of a rectangle divided into equal parts. For example, if a rectangle is divided into 5 equal parts and 2 parts are shaded to represent $\frac{2}{5}$, then multiplying by 3 means shading this $\frac{2}{5}$ three times side by side. This visualizes repeated addition and shows the total shaded parts as the product. — This visual method helps students understand multiplication as repeated addition of fractions by showing repeated shaded parts in the rectangle. It makes the abstract concept concrete and easier to grasp.

Question 6

Multiply $\frac{2}{3}$ by $\frac{3}{4}$ and simplify the result.

Accepted Answer

$\frac{1}{2}$ — Given: $\frac{2}{3}$ and $\frac{3}{4}$
Find: Product and simplify
Formula: Multiply numerators and denominators, then simplify.
Solution: Step 1: Multiply numerators: $2 	imes 3 = 6$
Step 2: Multiply denominators: $3 	imes 4 = 12$
Step 3: Product is $\frac{6}{12}$
Step 4: Simplify $\frac{6}{12} = \frac{1}{2}$
Answer: $\frac{1}{2}$
Note: Students often forget to simplify the fraction.

Question 7

Describe the area model used to explain multiplication of fractions $\frac{2}{3}$ and $\frac{3}{4}$.

Accepted Answer

The area model uses a rectangle divided into 3 equal parts horizontally and 4 equal parts vertically. Two parts are shaded horizontally to represent $\frac{2}{3}$, and three parts are shaded vertically to represent $\frac{3}{4}$. The overlapping shaded area, which is 6 parts out of 12 total, represents the product $\frac{6}{12}$, which simplifies to $\frac{1}{2}$. This model shows multiplication as finding a part of a part. — This visual representation helps students understand why multiplying numerators and denominators gives the product fraction. It concretely shows the overlapping area as the product fraction.

Question 8

Multiply the mixed numbers $2 \frac{1}{3}$ and $1 \frac{1}{2}$ and express the answer as a mixed number.

Accepted Answer

3 \frac{1}{2} — Given: $2 \frac{1}{3}$ and $1 \frac{1}{2}$
Find: Product as a mixed number
Formula: Convert mixed numbers to improper fractions, multiply, then simplify and convert back.
Solution: Step 1: Convert $2 \frac{1}{3} = \frac{7}{3}$
Step 2: Convert $1 \frac{1}{2} = \frac{3}{2}$
Step 3: Multiply: $\frac{7}{3} 	imes \frac{3}{2} = \frac{21}{6}$
Step 4: Simplify $\frac{21}{6} = \frac{7}{2}$
Step 5: Convert $\frac{7}{2}$ to mixed number: $3 \frac{1}{2}$
Answer: $3 \frac{1}{2}$
Note: Students often forget to convert back to mixed numbers or simplify.

Question 9

Explain the process of converting a mixed number to an improper fraction with an example.

Accepted Answer

To convert a mixed number to an improper fraction, multiply the whole number by the denominator of the fraction, add the numerator, and write the result over the original denominator. For example, $2 \frac{1}{3}$ is converted as (2 × 3 + 1)/3 = $\frac{7}{3}$. This conversion is necessary for performing multiplication or division of mixed numbers. — This method helps in simplifying operations with mixed numbers by converting them into improper fractions which are easier to multiply or divide.

Question 10

Which property of multiplication states that changing the order of factors does not change the product for fractions?

Accepted Answer

Commutative property — The commutative property of multiplication states that for any two numbers, including fractions, changing the order of factors does not affect the product. That is, $\frac{a}{b} 	imes \frac{c}{d} = \frac{c}{d} 	imes \frac{a}{b}$.

Question 11

Explain the associative property of multiplication with fractions using an example.

Accepted Answer

The associative property states that when multiplying three or more fractions, the way in which they are grouped does not affect the product. For example, $\left(\frac{1}{2} 	imes \frac{3}{4}ight) 	imes \frac{2}{3} = \frac{1}{2} 	imes \left(\frac{3}{4} 	imes \frac{2}{3}ight)$. Both groupings give the same result, showing that grouping does not change the product. — This property helps simplify calculations by allowing flexibility in grouping factors without changing the product.

Question 12

Use the distributive property to simplify $\frac{2}{3} 	imes \left(\frac{3}{4} + \frac{1}{4}ight)$.

Accepted Answer

$\frac{2}{3}$ — Given: Expression $\frac{2}{3} 	imes \left(\frac{3}{4} + \frac{1}{4}ight)$
Find: Simplify using distributive property
Formula: $a 	imes (b + c) = a 	imes b + a 	imes c$
Solution: Step 1: Apply distributive property:
$\frac{2}{3} 	imes \frac{3}{4} + \frac{2}{3} 	imes \frac{1}{4}$
Step 2: Multiply fractions:
$\frac{2 	imes 3}{3 	imes 4} + \frac{2 	imes 1}{3 	imes 4} = \frac{6}{12} + \frac{2}{12}$
Step 3: Add fractions:
$\frac{6 + 2}{12} = \frac{8}{12}$
Step 4: Simplify:
$\frac{8}{12} = \frac{2}{3}$
Answer: $\frac{2}{3}$
Note: Students may forget to apply distributive property correctly or simplify final answer.

Question 13

What is the identity property of multiplication for fractions?

Accepted Answer

Multiplying any fraction by 1 leaves it unchanged / Multiplying by 1 does not change the fraction — The identity property of multiplication states that multiplying any number, including fractions, by 1 results in the same number. For example, $\frac{3}{5} 	imes 1 = \frac{3}{5}$. This property is fundamental and helps in simplifying expressions.

Question 14

True or False: When multiplying fractions, $\frac{2}{5} 	imes \frac{3}{7} = \frac{5}{12}$.

Accepted Answer

False. The correct product is $\frac{6}{35}$ obtained by multiplying numerators and denominators. — Multiplying $\frac{2}{5}$ and $\frac{3}{7}$ involves multiplying numerators (2 × 3 = 6) and denominators (5 × 7 = 35), so the correct product is $\frac{6}{35}$, not $\frac{5}{12}$.

Question 15

Match the following properties of multiplication of fractions with their correct descriptions.

Accepted Answer

— Matching properties with descriptions helps students recall and differentiate these fundamental properties.

Working With

Working With — Study Notes

Introduction to Multiplication of Fractions

Multiplying a Fraction by a Whole Number

Multiplying a Fraction by a Fraction

Practice Questions — Working With

All 8 Chapters in Ganita Prakash