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Lesson 3 of 8

Working with Fractions · Lesson 3 of 8

The Multiplication Rule and Simplification

“Derive the fraction multiplication rule and use common factors to simplify calculations.”

Learning Objectives

• Explain why numerators and denominators multiply in a fraction product. • Use the multiplication rule for proper, improper, and mixed fractions. • Cancel common factors without changing the value of a product. • Solve problems involving fractional time, area, sharing, and measurement.

Multiplying Numerators and Denominators

Drawing hundreds of small cells is not an efficient way to calculate every product. However, the grid tells us exactly what to count. Use that reasoning to build a general rule, so that the calculation keeps its meaning even when a diagram becomes too large to draw.

For 1/12 of 1/18, divide the unit square into 18 rows and 12 columns. There are 18 × 12 = 216 equal cells, and the selected amount is one of them. Thus the product is 1/216. For 5/12 of 7/18, select five columns within seven rows: 5 × 7 = 35 cells out of the same 216.

Multiplication of unit fractionsLaTeX
The whole is divided into b columns and d rows, giving b × d equal pieces. Here b and d are positive denominators.
Example — A larger grid without drawing every cell

Problem
Calculate 5/12 × 7/18.

  1. 1.The denominators tell us that one whole contains 12 × 18 = 216 equal cells.
  2. 2.The selected region contains 5 × 7 = 35 of those cells.
  3. 3.The product is 35/216. Its numerator and denominator have no common factor greater than 1, so this is its lowest form.

The same counting works with letters. If a of b equal columns and c of d equal rows are selected, the region contains a × c cells out of b × d cells. This explains both parts of the multiplication rule; we are not choosing a denominator by adding or finding a common denominator.

Multiplication of fractionsLaTeX
a and c are the numerators; b and d are the nonzero denominators. In the positive fraction models here, they count selected pieces and equal divisions.

A whole number fits this rule when written over 1. For instance, 3 × 3/4 becomes 3/1 × 3/4 = 9/4. Likewise, 3/5 × 4 becomes 3/5 × 4/1 = 12/5. The earlier whole-number method is therefore part of this general rule.

Multiplication of Fractions — Simplifying to Lowest Form

A fraction is in lowest form when numerator and denominator share no common factor greater than 1. We can multiply first and then simplify, but sometimes that makes unnecessarily large numbers. Because the fraction value does not change when numerator and denominator are divided by the same nonzero number, we can simplify the factors before multiplying.

Example — Cancel before multiplying

Problem
Find 12/7 × 5/24 in lowest form.

  1. 1.Write the product as (12 × 5)/(7 × 24). The factor 12 occurs in the numerator, and 24 in the denominator.
  2. 2.Divide both by 12: 12 becomes 1 and 24 becomes 2. The product becomes (1 × 5)/(7 × 2).
  3. 3.Multiply the remaining factors to get 5/14. Multiplying first would give 60/168, which simplifies to the same 5/14.
Example — Two cancellations

Problem
Find 14/15 × 25/42 in lowest form.

  1. 1.Write (14 × 25)/(15 × 42). Divide 14 and 42 by their common factor 14, leaving 1 and 3.
  2. 2.Divide 25 and 15 by 5, leaving 5 and 3. The product is now (1 × 5)/(3 × 3).
  3. 3.Multiply to obtain 5/9. Each cancellation divided the complete numerator and denominator by the same factor.
Common mistake

Cancellation removes common factors, not equal-looking digits. For example, cancelling the digit 2 from 12 and 24 would be meaningless. Also, a cancellation must involve one numerator factor and one denominator factor. Dividing two numerator factors alone changes the product.

A Pinch of History

The chapter describes apavartana, the reduction of fractions to lowest terms, as a familiar idea in ancient Indian writing. It also connects the general fraction multiplication rule with Brahmagupta’s Brāhmasphuṭasiddhānta, written in 628 CE. These notes show that convenient arithmetic rules grew from ideas people studied and used over many centuries.

Using the Rule in a Situation

Before applying the rule, decide what the fraction refers to. Half of the land that remains after a road is built is different from half of the original land. In measurement problems, convert quantities to the same unit before comparing them. In mixed-number products, convert each mixed number to an improper fraction first.

Example — The land that remains

Problem
A road uses 1/6 of Somu’s land. She gives half of the remaining land to Krishna and one third of that same remaining land to Bora. What fraction of the original land does each person receive or keep?

  1. 1.The road leaves 1 − 1/6 = 5/6 of the original land.
  2. 2.Krishna receives 1/2 × 5/6 = 5/12. Bora receives 1/3 × 5/6 = 5/18. Both shares refer to the 5/6 remainder before either gift.
  3. 3.Krishna and Bora together receive 5/12 + 5/18 = 15/36 + 10/36 = 25/36. The pre-gift remainder is 5/6 = 30/36.
  4. 4.Somu keeps 30/36 − 25/36 = 5/36. Check: road 6/36 + Krishna 15/36 + Bora 10/36 + Somu 5/36 = 36/36.
Example — Rectangle with mixed-number sides

Problem
Find the area of a rectangle measuring 3 3/4 ft by 9 3/5 ft.

  1. 1.Convert the sides: 3 3/4 = 15/4 and 9 3/5 = 48/5.
  2. 2.Area = 15/4 × 48/5 square feet. Cancel 15 with 5 to leave 3 and 1; cancel 48 with 4 to leave 12 and 1.
  3. 3.The area is 3 × 12 = 36 square feet. The unit is square feet because two lengths have been multiplied.
Example — Which amount is heavier?

Problem
Compare 12/15 of 500 g with 3/20 of 4 kg.

  1. 1.Simplify 12/15 to 4/5. Then 4/5 × 500 g = 400 g.
  2. 2.Convert 4 kg to 4000 g. Then 3/20 × 4000 g = 600 g.
  3. 3.Since 600 g > 400 g, 3/20 of 4 kg is heavier. Comparing 12/15 and 3/20 alone would ignore that their wholes differ.

Quiz

Quick check

Which expression correctly gives 5/8 × 3/7?

Quick check

Why can 12 and 24 be simplified before multiplying in 12/7 × 5/24?

Quick check

What is 14/15 × 25/42 in lowest form?

Quick check

What is 2 1/2 × 3/5?

Quick check

Four saplings are equally spaced, with 3/4 m between neighbours. How far apart are the first and last?

Practice Problems

Practice Problems
  1. A tap fills 7/10 of a tank in one hour at a steady rate. Find the fraction filled in 1/3, 2/3, 3/4, and 7/10 hour.
  2. For the same tank, use multiplication to find the time needed to fill the tank. What number multiplied by 7/10 gives 1?
  3. Simplify before multiplying: 18/25 × 15/24 and 21/16 × 8/35. Explain each common factor used.
  4. Draw four equally spaced saplings, with 3/4 m between neighbours. Find the distance from the first to the last and explain why it uses three gaps.
  5. A rectangle is 2 1/4 m by 1 2/3 m. Find its area.
  6. After 1/5 of a field is used for a path, a gardener plants vegetables on 3/4 of the remaining field. What fraction of the original field has vegetables?
  7. Explain why you can cancel factors in (6 × 5)/(7 × 15), but cannot remove 5 from the sum (6 + 5)/(7 + 15).

Tank fractions: 7/30, 7/15, 21/40, 49/100. Full tank: 10/7 hours, since 10/7 × 7/10 = 1. Products: 9/20 and 3/10. Saplings: 3 × 3/4 = 9/4 = 2 1/4 m. Rectangle: 9/4 × 5/3 = 15/4 = 3 3/4 m². Vegetables: 3/4 × 4/5 = 3/5 of the original field. Cancellation applies to common multiplicative factors of the complete numerator and denominator; an individual term in a sum is not such a factor.

Key Takeaways

Key Takeaways

• Multiply numerators together and denominators together. • The multiplication rule comes from counting selected cells within equal rows and columns. • Write a whole number over 1 and convert mixed numbers before multiplying. • Cancelling common numerator-denominator factors preserves the value and keeps calculations manageable. • Identify the correct whole, count intervals carefully, and keep measurement units consistent.

Previous · Lesson 2

Multiplying Two Fractions

Next · Lesson 4

Understanding the Size of a Product