Multiplying Fractions
- Multiplying Fractions
- How to Multiply Fractions?
- Representation of Multiplication of Fractions Using Models
- Properties of Multiplying Fractions
- Multiplication of Improper Fractions
- Multiplication of Mixed Fractions
- Multiplying Fractions with Whole Numbers
- Real-Life Applications of Multiplying Fractions
- Solved Examples
- Practice Problem
- Frequently Asked Questions
Multiplying Fractions
Multiplying fractions is a mathematical operation that involves finding the product of two or more fractions. When we multiply two fractions the result is a new fraction representing the product of the two fractions.The process of multiplying fractions is straightforward as shown below:
How to Multiply Fractions?
Multiplying fractions involves multiplying the numerators together to get the new numerator and multiplying the denominators together to get the new denominator. Here is how do you multiply fractions:
Step `1`: Write Down the Fractions to Be Multiplied:
- Write the fractions one next to the other, separated by the multiplication symbol (\(\times\) or \(\cdot\)).
Step `2`: Multiply the Numerators:
- Multiply the numerators (the top numbers) together. The result becomes the new numerator.
Step `3`: Multiply the Denominators:
- Multiply the denominators (the bottom numbers) together. The result becomes the new denominator.
Step `4`: Form the Product Fraction:
- Write the product of the numerators over the product of the denominators to form the new fraction.
Step `5`: Simplify the Fraction (if needed):
- If necessary, simplify the fraction by finding the greatest common factor (GCF) of the numerator and denominator and dividing both by it.
Example `1`: \( \frac{3}{4} \times \frac{4}{5} \)
Solution: To multiply the fractions \( \frac{3}{4} \) and \( \frac{4}{5} \), follow these steps:
`1`. Multiply the Numerators:
\( \text{Numerator} = 3 \times 4 = 12 \)
`2`. Multiply the Denominators:
\( \text{Denominator} = 4 \times 5 = 20 \)
`3`. Form the Product Fraction:
\( \frac{3}{4} \times \frac{4}{5} = \frac{12}{20} \)
`4`. Simplify (if needed):
Simplify the fraction by finding the greatest common factor (GCF) of the numerator and denominator and dividing both by it. In this case, the GCF is `4`.
So, \( \frac{12}{20} = \frac{3}{5} \)
Therefore, \( \frac{3}{4} \times \frac{4}{5} = \frac{3}{5} \).
Example `2`: Multiply \( \frac{2}{3} \) with \( \frac{5}{4} \).
Solution: In this example, the numerators (`2` and `5`) are multiplied together to get the new numerator `(10)`, and the denominators (`3` and `4`) are multiplied together to get the new denominator `(12)`.
\( \frac{2}{3} \times \frac{5}{4} = \frac{2 \times 5}{3 \times 4} = \frac{10}{12} \)
You may need to simplify the result by finding the greatest common factor of the numerator and denominator. In this case, dividing both by `2` gives the simplified fraction \(\frac{5}{6}\).
\( \frac{10}{12} \ = \frac{\frac{10}{2}}{\frac{12}{2}} = \frac{5}{6}\)
So, multiplying fractions involves straightforward multiplication of numerators and denominators to obtain a new fraction, which can then be simplified if necessary.
Representation of Multiplication of Fractions Using Models
We can use visual aids like diagrams or drawings to represent multiplication of fractions . Let's consider an example where we want to represent the multiplication of \( \frac{4}{7} \) and \( \frac{2}{3} \) using a rectangular area model.
Let's represent the multiplication of \( \frac{4}{7} \) and \( \frac{2}{3} \) using a rectangular area model.
Step `1`: Draw a Rectangle:
- Draw a rectangle to represent the whole, which represents \( \frac{4}{7} \times \frac{2}{3} \).
Step `2`: Divide the Rectangle:
- Divide the rectangle into sections based on the denominators of the fractions being multiplied. In this case, divide the rectangle into `7` equal parts horizontally (for \( \frac{4}{7} \)) and into 3 equal parts vertically (for \( \frac{2}{3} \)).
Step `3`: Shade the Appropriate Areas:
- Shade the area representing \( \frac{4}{7} \) of the whole. This means shading `4` out of the `7` equal parts horizontally. Likewise shade `2` out of the `3` equal parts horizontally.
Step `4`: Shade the Intersection Areas:
- Imagine overlapping the first rectangle over the second rectangle and shade the areas where the shaded regions from \( \frac{4}{7} \) and \( \frac{2}{3} \) intersect. Here the orange cells represent the intersection / overlapping area.
Step `5`: Find the Total Number of Overlapping Parts:
- Count the total number of equal parts in the intersection area. This determines the numerator of the product.
Step `6`: Find the Total Number of Parts:
- Determine the total number of equal parts in the rectangle. This determines the denominator of the product.
Here's a visual representation all put together:
In this example, the orange shaded area represents the product \( \frac{4}{7} \times \frac{2}{3} \). By counting the cells in the orange shaded part, you can determine the numerator of the product, and the total number of equal parts gives you the denominator.
Properties of Multiplying Fractions
Multiplying fractions follows several properties that are essential to understand. Here are the key properties of multiplying fractions:
`1`. Closure Property: The product of two fractions is always a fraction.
\( \frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd} \)
`2`. Commutative Property: The order of multiplication does not affect the result.
\( \frac{a}{b} \times \frac{c}{d} = \frac{c}{d} \times \frac{a}{b} \)
`3`. Associative Property: The grouping of fractions does not affect the result.
\( \left(\frac{a}{b} \times \frac{c}{d}\right) \times \frac{e}{f} = \frac{a}{b} \times \left(\frac{c}{d} \times \frac{e}{f}\right) \)
`4`. Identity Property: Multiplying any fraction by `1` results in the original fraction.
\( \frac{a}{b} \times 1 = \frac{a}{b} \)
`5`. Zero Property: Multiplying any fraction by `0` results in `0`.
\( \frac{a}{b} \times 0 = 0 \)
`6`. Multiplicative Inverse Property: The product of a fraction and its reciprocal (multiplicative inverse) is always `1`.
\( \frac{a}{b} \times \frac{b}{a} = 1 \)
`7`. Multiplying by a Whole Number: To multiply a fraction by a whole number, simply multiply the numerator by the whole number. Keep the denominator same.
\( \frac{a}{b} \times c = \frac{ac}{b} \)
`8`. Distributive Property: Multiplying a fraction by a sum or difference distributes the multiplication to each term.
\( \frac{a}{b} \times (c + d) = \frac{a}{b} \times c + \frac{a}{b} \times d \)
Multiplication of Improper Fractions
To multiply improper fractions, you can follow the same steps as multiplying regular fractions. Here's the general procedure:
`1`. Multiply the numerators together.
`2`. Multiply the denominators together.
`3`. Write the product of the numerators over the product of the denominators to form the new fraction.
`4`. If needed, simplify the fraction by finding the greatest common factor (GCF) of the numerator and denominator and dividing both by it.
Let's go through an example with two improper fractions:
Example: \( \frac{5}{4} \times \frac{7}{3} \)
Solution:
`1`. Multiply the numerators: \(5 \times 7 = 35\).
`2`. Multiply the denominators: \(4 \times 3 = 12\).
`3`. Form the product fraction: \( \frac{5}{4} \times \frac{7}{3} = \frac{35}{12} \).
The GCF of `35` and `12` is `1` , so the fraction is already in its simplest form:
\( \frac{35}{12} \)
Therefore, \( \frac{5}{4} \times \frac{7}{3} = \frac{35}{12} \).
Multiplication of Mixed Fractions
To multiply mixed fractions, you can follow these steps:
`1`. Convert mixed fractions to improper fractions.
`2`. Multiply the numerators of the two fractions together.
`3`. Multiply the denominators of the two fractions together.
`4`. Write the product of the numerators over the product of the denominators to form the new fraction.
`5`. If the result is an improper fraction, convert it back to a mixed fraction.
Example: \( 2\frac{1}{3} \times 3\frac{2}{5} \)
Solution:
`1`. Convert mixed fractions to improper fractions:
\(2\frac{1}{3} = \frac{7}{3}\)
\(3\frac{2}{5} = \frac{17}{5}\)
`2`. Multiply the numerators: \(7 \times 17 = 119\).
`3`. Multiply the denominators: \(3 \times 5 = 15\).
`4`. Form the product fraction: \( \frac{119}{15} \).
`5`. Simplify (if needed): The GCF of `119` and `15` is `1`. Hence, \( \frac{119}{15} \) is in its simplest form.
`6`. Convert the result to a mixed fraction:\( \frac{119}{15} = 7\frac{14}{15} \).
Therefore, \(2\frac{1}{3} \times 3\frac{2}{5} = 7\frac{14}{15}\).
Multiplying Fractions with Whole Numbers
When multiplying a fraction by a whole number, the process is relatively straightforward. Here are the steps:
`1`. Write the Whole Number as a Fraction. For example, if you have the whole number `4`, you can write it as \( \frac{4}{1} \).
`2`. Multiply the numerator (the top number) of the fraction and the whole number.
`3`. Multiply the denominator (the bottom number) of the fraction and `1`.
`4`. Write the product of the numerators over the product of the denominators to form the new fraction.
`5`. If needed, simplify the fraction by finding the greatest common factor (GCF) of the numerator and denominator and dividing both by it.
Example: \( \frac{3}{5} \times 4 \)
Solution:
`1`. Write `4` as a fraction: \( \frac{4}{1} \).
`2`. Multiply the numerators: \(3 \times 4 = 12\).
`3`. Multiply the denominators: \(5 \times 1 = 5\).
`4`. Form the product fraction: \( \frac{12}{5} \).
`5`. Simplify (if needed): The GCF of `12` and `5` is `1`. Hence, \( \frac{12}{5} \) is already in its simplest form.
Therefore, \( \frac{3}{5} \times 4 = \frac{12}{5} \).
Real-Life Applications of Multiplying Fractions
Multiplying fractions have various real-life applications, especially in situations involving measurements, ratios, and proportions. Here are a few examples of how multiplying fractions is used in everyday scenarios:
`1`. Cooking and Baking: Recipes often involve multiplying fractions when adjusting the quantities of ingredients. For instance, if a recipe requires doubling or halving, you might need to multiply fractions to determine the new amounts of each ingredient.
`2`. Construction and Carpentry: In construction, measurements may be in fractional units. Multiplying fractions is crucial when calculating the area of a room or determining the amount of material needed for a project.
`3`. Resizing Images or Objects: Graphic designers or individuals working with images may need to resize pictures or objects. When scaling proportionally, multiplying fractions helps maintain the correct aspect ratio.
`4`. Medical Dosages: In healthcare, medication dosages are often prescribed based on a patient's weight or age. Calculating the correct dosage may involve multiplying fractions to adjust the concentration of a solution.
`5`. Fuel Efficiency: Calculating fuel efficiency in miles per gallon (mpg) or liters per 100 kilometers often involves multiplying fractions. For instance, determining the average fuel consumption over a specific distance.
`6`. Time and Work Calculations: In work scenarios, calculating rates or determining how long a task will take can involve multiplying fractions. For example, if two workers can complete a task at different rates, multiplying their rates by time helps estimate the total work done.
`7`. Scaling Maps: When working with maps, especially for navigation or urban planning, multiplying fractions helps scale distances accurately.
Solved Examples
Here are a few examples of multiplying fractions:
Example `1`: \( \frac{1}{3} \times \frac{5}{4} \)
Solution:
`1`. Multiply the numerators: \(1 \times 5 = 5\).
`2`. Multiply the denominators: \(3 \times 4 = 12\).
`3`. Form the product fraction: \( \frac{1}{3} \times \frac{5}{4} = \frac{5}{12} \).
`4`. Simplify the fraction: \( \frac{5}{12} \) is in its simplest form.
So, \( \frac{1}{3} \times \frac{5}{4} = \frac{5}{12} \).
Example `2`: \( \frac{3}{5} \times \frac{4}{7} \)
Solution:
`1`. Multiply the numerators: \(3 \times 4 = 12\).
`2`. Multiply the denominators: \(5 \times 7 = 35\).
`3`. Form the product fraction: \( \frac{3}{5} \times \frac{4}{7} = \frac{12}{35} \).
`4`. Simplify the fraction: \( \frac{12}{35} \) is in its simplest form.
So, \( \frac{3}{5} \times \frac{4}{7} = \frac{12}{35} \).
Example `3`: \( \frac{2}{3} \times 10 \)
Solution:
`1`. Write `10` as a fraction: \( \frac{10}{1} \).
`2`. Multiply the numerators: \(2 \times 10 = 20\).
`3`. Multiply the denominators: \(3 \times 1 = 3\).
`4`. Form the product fraction: \( \frac{2}{3} \times 10 = \frac{20}{3} \).
So, \( \frac{2}{3} \times 10 = \frac{20}{3} \), which can be expressed as \(6\frac{2}{3}\) in mixed fraction form.
Example `4`: \( 3\frac{2}{5} \times 4\frac{1}{2} \)
Solution:
`1`. Convert the mixed fractions to improper fractions:
\(3\frac{2}{5} = \frac{17}{5}\)
\(4\frac{1}{2} = \frac{9}{2}\)
`2`. Multiply the numerators: \(17 \times 9 = 153\).
`3`. Multiply the denominators: \(5 \times 2 = 10\).
`4`. Form the product fraction: \( \frac{17}{5} \times \frac{9}{2} = \frac{153}{10} \).
`5`. Simplify (if needed):
\( \frac{153}{10} \) can be written as mixed fraction \(15\frac{3}{10} \).
Practice Problems
Q`1`. Solve the \( \frac{2}{3} \times \frac{5}{6} \).
- \(\frac{5}{9} \)
- \(\frac{4}{3} \)
- \(\frac{5}{11} \)
- \(\frac{9}{5} \)
Answer: a
Q`2`. What is the product of \(2 \frac{3}{4}\) and \(1 \frac{1}{2}\)?
- \(3 \frac{1}{8}\)
- \(4 \frac{1}{8}\)
- \(4 \frac{9}{4}\)
- `13`
Answer: b
Q`3`. Multiplying \( \frac{3}{4} \) by what fraction will result in `2`?
- \( \frac{3}{8} \)
- \( \frac{3}{4} \)
- \( \frac{8}{3} \)
- \( \frac{4}{3} \)
Answer: c
Q`4`. What is the product of \( \frac{3}{5} \) and \( \frac{2}{20} \)?
- \( \frac{50}{3} \)
- \( \frac{5}{3} \)
- \( \frac{8}{22} \)
- \( \frac{3}{50} \)
Answer: d
Q`5`. Solve \( 4\frac{2}{3} \times 6\frac{1}{2} \) .
- \( 30\frac{1}{2} \)
- \( 30\frac{2}{3} \)
- \( 30\frac{1}{3} \)
- \( 24\frac{1}{3} \)
Answer: c
Frequently Asked Questions
Q`1`. What is the rule for multiplying fractions?
Answer: The rule for multiplying fractions is to multiply the numerators together to get the new numerator and multiply the denominators together to get the new denominator.
Q`2`. Do you need a common denominator to multiply fractions?
Answer: No, you don't need a common denominator to multiply fractions. You can multiply fractions directly, and the result may need to be simplified.
Q`3`. How do you multiply mixed numbers?
Answer: To multiply mixed numbers, convert them to improper fractions, then multiply the numerators and denominators as you would with regular fractions.
Q`4`. How do you simplify the product of fractions?
Answer: To simplify the product of fractions, find the greatest common factor (GCF) of the numerator and denominator and divide both by it.
Q`5`. Can the product of two fractions be greater than both fractions?
Answer: No, the product of two proper fractions will always be smaller than either of the original fractions.
Q`6`. Can you multiply a fraction by a fraction to get a whole number?
Answer: Yes, multiplying certain fractions can result in a whole number, especially when the numerator and denominator have a common factor that cancels out.