Multiplying a Fraction by a Whole Number
When Aaron walks 3 km in 1 hour, in 5 hours he walks 5 × 3 = 15 km (repeated addition: 3 + 3 + 3 + 3 + 3). What if his pet tortoise walks only 1/4 km in 1 hour? In 3 hours, the tortoise walks 3 × (1/4) = 1/4 + 1/4 + 1/4 = 3/4 km!
Interactive Fraction Strip Multiplier
Choose a fraction and a multiplier to combine unit strips into a single product.
Fractional Time: When the Multiplier is a Fraction
Aaron walks 3 km in 1 hour. How far does he walk in 1/5 hour? Dividing the 1-hour distance of 3 km into 5 equal parts gives 3/5 km. Then, for 2/5 hour, it is twice that: 2 × (3/5) = 6/5 km = 1 1/5 km!
Convert Before Multiplying: Mixed Numbers Tool
Textbook Example 2: 1 1/4 hours of internet at ₹8/hour requires converting 1 1/4 to 5/4 first.
Formula: (Whole × Den + Num) / Den = (1 × 4 + 1) / 4 = 5/4
7 ÷ 4 = 1 remainder 3
Figure It Out Problems
Practice multiplication of fractions by whole numbers in daily real-world rate contexts.
40 Chapter Practice Questions
20-Question Comprehensive Quiz
10 Common Mistakes & How to Avoid Them
Chapter Key Terms & Math Toolbox
The operation where numerators are multiplied together to form the new numerator, and denominators are multiplied together to form the new denominator.
The number which, when multiplied by a given non-zero fraction, results in the product 1.
Division of a fraction by another is carried out by multiplying the dividend by the reciprocal of the divisor.
A geometric square of side length 1 partitioned into horizontal rows and vertical columns to visualize fraction products as overlapping areas.
The ancient Indian technique (codified by Brahmagupta & Umasvati) of dividing numerator and denominator by common factors before multiplying.
A fraction whose numerator is greater than or equal to its denominator, representing a value greater than or equal to 1.
A number composed of a whole number and a proper fraction, such as 2 1/4 = 9/4.
A product where intermediate numerators and denominators cancel in pairs, leaving only the boundary factors.
The property stating that the order in which two fractions are multiplied does not change their product, corresponding to rotating a rectangle.
Multiplication by k > 1 enlarges a positive quantity, while multiplication by 0 < k < 1 shrinks it.
Frequently Asked Questions
What is the rule for multiplying two fractions?
To multiply two fractions, multiply their numerators together to get the new numerator, and multiply their denominators together to get the new denominator: (a/b) × (c/d) = (a × c) / (b × d). For example, (2/3) × (4/5) = (2 × 4) / (3 × 5) = 8/15.
Why do we multiply numerators and denominators?
Visually, when a unit square is divided into b rows and d columns, the total number of small parts created is b × d (the product of denominators). The region of overlap spans a rows and c columns, giving a × c parts (the product of numerators). Thus, the overlapping fraction is (a × c) / (b × d).
What is a reciprocal of a fraction?
The reciprocal (or multiplicative inverse) of a non-zero fraction a/b is b/a. When a fraction is multiplied by its reciprocal, the product is always 1: (a/b) × (b/a) = 1. For example, the reciprocal of 3/5 is 5/3, and the reciprocal of 8 (which is 8/1) is 1/8.
How do you divide two fractions?
To divide by a fraction, multiply the dividend by the reciprocal of the divisor: (a/b) ÷ (c/d) = (a/b) × (d/c) = (a × d) / (b × c). For example, (2/3) ÷ (3/5) = (2/3) × (5/3) = 10/9.
Why does dividing by a fraction sometimes give an answer larger than the dividend?
Division answers: 'How many groups of the divisor fit inside the dividend?' When the divisor is smaller than 1 (like 1/4), each group is smaller than a whole, so MORE groups fit inside. For example, 6 ÷ (1/4) = 24 because twenty-four quarter-portions fit inside 6 whole units.
Can we cancel numbers before multiplying?
Yes! If a numerator and a denominator share a common factor, you can divide both by that factor before multiplying. This technique, called apavartana in Indian mathematics, prevents having to deal with very large numbers. For example, in (12/7) × (5/24), 12 and 24 can be cancelled by 12, leaving (1 × 5) / (7 × 2) = 5/14.
Why doesn't multiplication always make a number larger?
When multiplying positive numbers, if the multiplier is between 0 and 1, you are taking a fraction OF the quantity, which reduces it. For example, (1/2) × 10 = 5, which is less than 10. The product is only greater than both numbers when BOTH numbers are greater than 1.
How is fraction multiplication connected to the area of a rectangle?
A rectangle with fractional side lengths length = a/b and breadth = c/d has an area equal to (a/b) × (c/d) square units. Just as area of whole-number rectangles is length × breadth, the same geometric formula holds true for fractional dimensions.
How do you multiply mixed numbers?
First convert each mixed number into an improper fraction. For example, 2 1/3 = 7/3 and 1 1/2 = 3/2. Then multiply the improper fractions: (7/3) × (3/2) = 21/6 = 7/2 = 3 1/2. Never multiply the whole parts and fraction parts separately.
Who was Brahmagupta and what was his contribution to fractions?
Brahmagupta was an illustrious Indian mathematician and astronomer who, in 628 CE in his treatise Brāhmasphuṭasiddhānta, gave the first explicit general mathematical rules for both multiplication ((ac)/(bd)) and division ((a/b) × (d/c)) of fractions in the modern form used worldwide today.
Why does zero have no reciprocal?
By definition, multiplying a number by its reciprocal must yield 1. Since multiplying 0 by any real or rational number always yields 0, there is no number x such that 0 × x = 1. Therefore, zero has no reciprocal, and division by zero is undefined.
What is a telescoping product of fractions?
A telescoping product is a series of fraction multiplications where the numerator of each fraction cancels with the denominator of the previous fraction. For example, (1 - 1/2)(1 - 1/3)(1 - 1/4)...(1 - 1/n) = (1/2)(2/3)(3/4)...((n-1)/n) = 1/n, because all intermediate terms cancel.
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