When rewriting the expression (4)/(9) ÷ (2)/(5) as a multiplication problem, which fraction is inverted?

Only the divisor, (2)/(5), which becomes (5)/(2). To divide by a fraction, we multiply by its reciprocal. The dividend ((4)/(9)) remains unchanged, while the divisor ((2)/(5)) is inverted to its reciprocal ((5)/(2)). Therefore, the expression becomes (4)/(9) × (5)/(2).

Grade 10 · Core Mathematics · Lesson 8 of 11

Reciprocals in Fraction Computation

A Grade 10 Core Mathematics CBE lesson from the sub-strand Real Numbers. In it, learners rewrite division by a nonzero fraction as multiplication by its reciprocal.

Strand Numbers and AlgebraSub-strand Real NumbersLesson code 10CM-11-L840 minutes · 2 slides

KICD learning outcome (e)

Apply reciprocal of real numbers in mathematical computations.

By the end of this lesson, learners should be able to

Rewrite division by a nonzero fraction as multiplication by its reciprocal.
Simplify products before calculating when appropriate.
Check units and signs in a fraction computation.

Key vocabulary

Quotient: The result obtained when one quantity is divided by another.; The quotient of three-fourths and two is three-eighths.
Cancellation: Removing equal nonzero factors from numerator and denominator.; Cancellation reduces six times five over five to six.

Lesson preview

Slide 1 · Division Becomes MultiplicationOpening · 18 min

Figure 1: A diagram showing a reciprocal turnstile that flips a divisor fraction upside down and changes the division sign into a multiplication sign.

To divide by a nonzero fraction, multiply by that fraction's reciprocal. The operation changes because multiplying by an inverse undoes the divisor's effect while preserving the original dividend. Use Figure 1 to visualise how the operation and divisor transform.

Keep the dividend unchanged during the division rewrite.
Invert only the divisor fraction, never the dividend.
Use parentheses to show the complete divisor clearly.
Check by reversing the operation rewrite if uncertain.

Worked example. Identify two-thirds as dividend and five-sevenths as divisor. Find the reciprocal of the divisor: seven-fifths. Replace the division symbol with multiplication. Answer: Two-thirds divided by five-sevenths equals two-thirds multiplied by seven-fifths..

Check question: When rewriting the expression $\frac{4}{9} \div \frac{2}{5}$ as a multiplication problem, which fraction is inverted?

Only the divisor, $\frac{2}{5}$ , which becomes $\frac{5}{2}$ .
Only the dividend, $\frac{4}{9}$ , which becomes $\frac{9}{4}$ .
Both fractions, which become $\frac{9}{4}$ and $\frac{5}{2}$ .
Neither fraction; the expression becomes $\frac{4}{9} \times \frac{2}{5}$ .

Show answer

Only the divisor, $\frac{2}{5}$ , which becomes $\frac{5}{2}$ . To divide by a fraction, we multiply by its reciprocal. The dividend ( $\frac{4}{9}$ ) remains unchanged, while the divisor ( $\frac{2}{5}$ ) is inverted to its reciprocal ( $\frac{5}{2}$ ). Therefore, the expression becomes $\frac{4}{9} \times \frac{5}{2}$ .

Slide 2 · Simplify Factors StrategicallyPractice · 22 min

Figure 2: A chalkboard showing how to correctly cancel common factors across numerators and denominators in multiplication, alongside an invalid cancellation in addition.

Teach the full lesson on the projector

This lesson, and every other Grade 1–10 CBE lesson, comes ready to project with the EAP ClassKit. It runs without electricity or internet.

Both slides with diagrams and class activities
Teacher notes, SNE adaptations and assessment checks
Homework and the other 10 lessons in this sub-strand

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