Multiplying FractionsNow that we’ve learned how to add and subtract fractions, we will learn how to multiply fractions. Multiplying fractions is a lot simpler than adding or subtracting fractions because we don’t need to find a common denominator, instead we just multiply across numerators and denominators. The following video will explain why this works and show a few examples. Show
Video Source (05:48 mins) | Transcript When multiplying fractions we simply multiply the numerators together and the denominators together. Remember, any whole number can be represented as a fraction by putting it over 1. Example: \(3=\frac{3}{1}\) Reduce when needed. Example when reducing is not needed: \(\frac{2}{5}\cdot\frac{2}{3}=\frac{2\cdot2}{5\cdot3}=\frac{4}{15}\) Example when reducing is needed: \(\frac{2}{5}\cdot\frac{3}{4}=\frac{2\cdot3}{5\cdot2\cdot2}=\frac{2}{2}\cdot\frac{3}{5\cdot2}=1\cdot\frac{3}{10}=\frac{3}{10}\) Additional Resources
Practice Problems Multiply the following fractions:
Multiplying fractions: When a fraction is multiplied by another fraction the resultant is a fraction or a whole number. We know, a fraction has two parts: numerator and denominator. Thus, when we multiply any two fractions, then numerators and denominators are multiplied, respectively. Example of multiplying fractions is ⅔ x ¼ = (2 x 1)/(3 x 4) = 2/12 = ⅙. Multiplying fractions is not like the addition or subtraction of fractions, where the denominators of both the fractions should be the same. Two fractions can be multiplied easily, even if the denominators are different. While multiplying fractions, it should be kept in mind that the fractions can be in a proper fraction or improper fraction, but they cannot be a mixed fraction. Table of Contents:
Fractions and TypesIf a fraction is written in the form of a/b, then a and b are the parts of the fraction, where a is called a numerator and b is called the denominator. For example, Suppose ⅖ is a fraction, then 2 is the numerator and 5 is the denominator. There are three main types of fractions which are proper fractions, improper fractions, and mixed fractions. Below is a brief explanation on each of the types. Proper fractionsWhen the numerator of a fraction is less than the denominator. Example: ½, ⅗, 7/9 Improper fractionsWhen the numerator is greater than the denominator Example: 3/2, 5/4, 8/3 Mixed FractionThe combination of a whole number and a fraction. It is also called a mixed number. Example: 13/4 = 3 ¼ 21/5 = 4 ⅕ How to Multiply Fractions?Multiplying fractions is defined as the product of a fraction with a fraction or with an integer or with the variables. The procedure to multiply the fractions are:
For example, 3/2 and ⅓ are the two fractions The multiplication of two fractions is given by: (3/2)× (⅓) = [3×1]/[2×3] (3/2)× (⅓) = 3/6 Now, simplify the fraction, we get ½ Therefore, the multiplication of two fractions 3/2 and ⅓ is ½. Simply, we can write the formula for multiplication of fraction as; If “a/b” and “p/q are the multiplicand and multiplier, then the product of (a/b) and (p/q) is given by “ap/bq”
Thus, The product of Fraction = Product of Numerator/Product of Denominator Try This: Multiplying Fractions Calculator Dividing FractionsWhen we divide the fraction by another fraction, we convert the latter into reciprocal and then multiply with the former fraction. Learn dividing fractions in detail at BYJU’S. Example: ⅔ ÷ ¾ Solution: Convert ¾ into its reciprocal, to get 4/3. Now multiply ⅔ by 4/3 ⇒ ⅔ x 4/3 ⇒ (2×4)/(3×3) ⇒ 8/9. Simplification of FractionsIn multiplying fractions, we generally multiply the top numbers (numerators) with each other, and the bottom numbers (denominators) with each other. To make the fractional multiplication simpler, we can reduce the fraction by cancelling off the common factors. It means that you can cancel out the common factors from one side of the fraction, which is duplicated on the other side of the fractional part. For example, (4/9) and (3/16) are the two fractions. (4/9) can be written as (2×2)/(3×3) (3/16) can be written as (1×3)/(2×2×2×2) Therefore, \(\begin{array}{l}\frac{4}{9}\times \frac{3}{16} = \frac{2\times 2}{3\times 3} \times \frac{1\times 3}{2\times 2\times 2\times 2}\end{array} \) Now, cancel out the common factors, we get \(\begin{array}{l}\frac{4}{9}\times \frac{3}{16} = \frac{1}{3}\times \frac{1}{4}\end{array} \) Now, we can multiply numerator with numerator and denominator with denominator. (4/9) × (3/16)= 1/12 In case, if the fraction has no common factors, then we should directly multiply the numerators and denominators to get the product of the fractions. Multiplication of Fractions with FractionsMultiplying Proper FractionsMultiplication of proper fractions is simple, as we can directly multiply the numerator of one fraction with the other fraction and the denominator of one fraction with the other fraction. If required, we can simplify the resultant fractions into their lowest term. For example, the multiplication of 5/9 and 2/3. (5/9)×(2/3)= (5×2)/(9×3) = 10/27. Multiplying Improper FractionsWe know that in an improper fraction, the numerator is greater than the denominator. While multiplying two improper fractions, it will also result in the improper fraction. For example, multiplying two improper fractions, such as 9/2 and 6/5, results in: (9/2)×(6/5) = (9/1)×(3/5)= 27/5. If required, we can convert the improper fraction into a mixed fraction. Example 1: Solve ⅔×½ Solution: ⅔×½ = 2×1/3×2 = 2/6 = ⅓ Therefore, from the above example, we can observe, by multiplying two fractions we get a fraction number. This is a proper fraction. Example 2: Multiply ⅘×⅞ Solution: ⅘×⅞ = 4×7 / 5×8 = 28/40 We can further simplify it as; 28/40 = 7/10 If we have to multiply three fractions, then the above formula remains the same. Example 3: Multiply ¼×⅖×⅛ Solution: Multiplying the given fraction ¼×⅖×⅛, we get Product = 1×2×1 / 4×5×8 = 2 / 160 = 1 / 80 Multiplying Fractions with Whole numbersIf a whole number or real number is multiplied with a fraction, then it is equal to the real number times the fraction is added. Example 4: Multiply 5×½ Solution: 5×½ means 5 times of ½ This means, if we add ½ five times, we get the answer. Therefore, ½ + ½ + ½ + ½ +½ = (1+1+1+1+1)/2 = 5/2 = 2.5 Example 5: Multiply 8/7×10 Solution: Given, 8/7×10 We can write it as 8×10/7 Therefore, 80/7 is the answer. In decimal, it is 11.42.
Example 6: Multiply \(\begin{array}{l}3\frac{1}{5}\end{array} \) ×12Solution: Simplifying the value \(\begin{array}{l}3\frac{1}{5}\end{array} \) we get,16/5×12 = 16×12 / 5 = 192 / 5 = 38.4 Multiplying Fractions with VariablesNow, consider the fraction is multiplied with a variable, then the results or outcome will be as per the below example. Example 7: Multiply 5x/2y × 2x/3z Solution: Given, 5x/2y × 2x/3z Therefore, we can solve the above-given expression as; \(\begin{array}{l}\frac{5x\times 2x}{2y\times 3z} = \frac{10x^{2}}{6yz}\end{array} \) Properties of Fractional MultiplicationThe following are the properties of multiplication of fractions:
For example, (⅔) × (4/6) = 8/18 = 4/9 Similarly, (4/6)×(⅔) = 8/18 = 4/9
For example, (⅘)× (1/1) = (⅘)
For example, (½)× 0 = 0 Multiplication of simple fractions is easy, we just need to multiply numerators and denominators respectively. But to multiply mixed numbers or fractions we need to add one more step.
Multiplying Fractions Tricks:
Video Lesson on FractionsSolved Examples
We can write, 21/3 = 7/3 Now multiply 7/3 and 3 7/3 x 3 = 7
We can write, 11/2 = 3/2 21/5 = 11/5 Now multiply both the fractions. 3/2 x 11/5 (3 x 11)/(2 x 5) 33/10 Now convert 33/10 into a mixed fraction 33/10 = 33/10 Problems and SolutionsQ.1: Multiply ⅖ and 6/7. Solution: ⅖ x 6/7 ⇒ (2×6)/(5×7) ⇒ 12/35 Q.2: Multiply ⅓ and 1/10. Solution: ⅓ x 1/10 ⇒ 1/(3 x 10) ⇒ 1/30 Q.3: Find the product of ⅝ and 4/10. Solution: ⅝ x 4/10 ⇒ (5 x 4)/(8 x 10) ⇒ 20/80 ⇒ 1/4 Practice Problems
Related Topics on FractionsFrequently Asked Questions on Multiplying FractionsThe
fraction is defined as the ratio of two numbers. It generally represents the parts of the whole. The fraction can be written in the form “a/b”. Where, the top number “a” is called the numerator and the bottom number “b” is called the denominator. To multiply fractions, first simply the fraction to its lowest term. In the case of mixed fractions, simplify it. After simplifying the fraction, multiply the numerator with the numerator and the denominator with the denominator. Then, the product of fractions is obtained in p/q form. To multiply a fraction with a whole number, represent the whole number as a fraction by putting 1 in the denominator. Then, multiply the numerator with the numerator and the denominator with the denominator to get the product. No, there is no need for a common denominator to multiply fractions. Any two fractions can be multiplied in which numerators are multiplied with each other and the denominators are multiplied with each other. If a fraction has to be multiplied with a mixed number (fraction), simplify the fraction first. Once the mixed fraction is in the form of p/q, multiply the numerators with numerators and denominators with denominators. Can you multiply a fraction by the denominator?Yes, you can multiply two fractions with different denominators. First, multiply the numerators. Second, multiply the denominators. Third, simplify the product if needed.
How do you multiply mixed numbers with the same denominator?How to Multiply Mixed Fractions with the Same Denominators? Multiplying mixed fractions with the same denominators are done by first converting the mixed fractions to improper followed by multiplying the numerators and denominators separately and simplifying it to get the result.
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