Multiplication of Fractions and Whole Numbers

Theory: To multiply a fraction by a whole number, multiply the numerator by the whole number and keep the denominator unchanged.
Example:

Question: 3 \(\times \frac{2}{5}\) = ?

Solution: \(\frac{3 \times 2}{5} = \frac{6}{5}\)

Theory: Multiplying a fraction by a whole number is the same as repeated addition of the fraction.

Example:

Question: \(4 \times \frac{1}{3}\) = ?

Solution: \(\frac{1}{3} + \frac{1}{3} + \frac{1}{3} + \frac{1}{3} = \frac{4}{3}\)

Theory: Visual grids and area diagrams can be used to show multiplication of fractions by modeling rows and columns.

Example:

Question: Show 2 \(\times \frac{3}{4}\) using an area model.

Solution:  Draw 2 rectangles, divide each into 4 parts, shade 3 in each\(\Rightarrow\) Total shaded = 6 out of 4 = \(\frac{6}{4} = \frac{3}{2}\)

Theory: Multiply numerator of proper fraction by whole number. Denominator remains the same. Simplify if possible.

Example:

Question: \(5 \times \frac{2}{3}\) = ?

Solution: \(\frac{5 \times 2}{3} = \frac{10}{3}\)

Theory: Improper fractions can be multiplied like proper fractions. Convert to mixed form if needed.

Example:

Question: \(2 \times \frac{7}{4}\) = ?

Solution: \(\frac{2 \times 7}{4} = \frac{14}{4} = \frac{7}{2}\)

Theory: Simplifying common factors before multiplying reduces calculation effort and gives simplified results.

Example:

Question: \(6 \times \frac{4}{12}\) = ?

Solution: \(\frac{4}{12} = \frac{1}{3}, \ 6 \times \frac{1}{3} = \frac{6}{3}\) = 2

Theory: Used in calculating portions like food distribution, repeated tasks, and scaling recipes.

Example:

Question: If 1 bag has \(\frac{2}{5}\)  kg flour, how much  in 4 bags?

Solution: \(4 \times \frac{2}{5} = \frac{8}{5} = 1 \frac{3}{5}\) kg

Theory: Word problems require interpreting scenarios correctly and applying the multiplication process.

Example:

Question: A pipe fills \(frac{3}{4}\) of a  tank in 1 hour. How  much in 3 hours?

Solution: \(3 \times \frac{3}{4} = \frac{9}{4} = 2 \frac{1}{4}\)

Theory: Finding fraction of a number means multiplying the number by the fraction.

Example:

Question: What is \(frac{2}{5}\) of 40?

Solution: \(\frac{2}{5} \times 40 = \frac{80}{5} = 16\)

Theory: Convert mixed number to improper fraction, multiply by whole number, and convert back if needed.

Example:

Question: \(2 \times 1\frac{1}{2}\) = ?

Solution: Convert: \(\ 1\frac{1}{2} = \frac{3}{2}, \ 2 \times \frac{3}{2} = \frac{6}{2} = 3\)

Theory: Estimating helps check if the product is reasonable and identify errors.

Example:

Question: Estimate 3 \(times \frac{2}{3}\)

Solution: \(\frac{2}{3} \approx 0.67, \ 3 \times 0.67 \approx 2, \ Actual = \frac{6}{3} = 2\)

Theory: Students may wrongly multiply numerator with denominator or add instead of multiply. Clarity on steps is essential.

Example:

Question: A student finds 4 \(times \frac{3}{7} = \frac{4}{21}\).  Is it correct?

Solution: No. Correct: 4 \(\times \frac{3}{7} = \frac{12}{7}\)

Theory: Multiplication of fractions and whole numbers is commutative: \(a \times \frac{b}{c} = \frac{b}{c} \times a\)

Example:

Question: \(Show\ \frac{2}{5} \times 3 = 3 \times \frac{2}{5}\)

Solution: \(\frac{6}{5} = \frac{6}{5}\)       

Theory: Multiplication of fractions follows associative property: \(a × (b × c) = (a × b) × c\)

Example:

Question: Verify 2 \(\times (\frac{1}{2} \times 3) = (2 \times \frac{1}{2}) \times 3\)

Solution: LHS = 2 \(\times \frac{3}{2} = 3\)
\(\ RHS = \frac{2}{2} \times 3 = 1 \times 3 = 3\)

Theory: Used to find areas, lengths, and scale models using fractional quantities.

Example:

Question: A rectangle has length 6 cm and width \(\frac{2}{5}\ cm\).Find area.

Solution: Area = \(6 \times \frac{2}{5} = \frac{12}{5} = 2.4\ cm^2\)