If you’re like me, you probably learned how to cross multiply fractions in school. But if you’re like me, you also probably forgot how to do it. Don’t worry, though. I’ve got you covered. In this article, I’ll teach you how to cross multiply fractions like a pro. It’s not as hard as you think, I promise.
The first step is to understand what cross multiplication is. Cross multiplication is a method of solving proportions. A proportion is an equation that states that two ratios are equal. For example, the proportion 1/2 = 2/4 is true because both ratios are equal to 1.
To cross multiply fractions, you simply multiply the numerator of the first fraction by the denominator of the second fraction, and then multiply the denominator of the first fraction by the numerator of the second fraction. For example, to solve the proportion 1/2 = 2/4, we would cross multiply as follows: 1 x 4 = 2 x 2. This gives us the equation 4 = 4, which is true. Therefore, the proportion 1/2 = 2/4 is true.
Find the Reciprocal of the Second Fraction
When cross-multiplying fractions, the first step is to find the reciprocal of the second fraction. The reciprocal of a fraction is a new fraction that has the denominator and numerator swapped. In other words, if you have a fraction a/b, its reciprocal is b/a.
To find the reciprocal of a fraction, simply flip the fraction upside down. For example, the reciprocal of 1/2 is 2/1, and the reciprocal of 3/4 is 4/3.
Here’s a table with some examples of fractions and their reciprocals:
| Fraction | Reciprocal |
|---|---|
| 1/2 | 2/1 |
| 3/4 | 4/3 |
| 5/6 | 6/5 |
| 7/8 | 8/7 |
| 9/10 | 10/9 |
Flip the Numerator and Denominator
We flip the numerator and denominator of the fraction we want to divide with, and then change the division sign to a multiplication sign. For instance, let’s say we want to divide 1/2 by 1/4. First, we flip the numerator and denominator of 1/4, which gives us 4/1. Then, we change the division sign to a multiplication sign, which gives us 1/2 multiplied by 4/1.
Why Does Flipping the Numerator and Denominator Work?
Flipping the numerator and denominator of the fraction we want to divide with is valid because of a property of fractions called the reciprocal property. The reciprocal property states that the reciprocal of a fraction is equal to the fraction with its numerator and denominator flipped. For instance, the reciprocal of 1/4 is 4/1, and the reciprocal of 4/1 is 1/4.
When we divide one fraction by another, we are essentially multiplying the first fraction by the reciprocal of the second fraction. By flipping the numerator and denominator of the fraction we want to divide with, we are effectively multiplying by its reciprocal, which is what we want to do in order to divide fractions.
Example
Let’s work through an example to see how flipping the numerator and denominator works in practice. Let’s say we want to divide 1/2 by 1/4. Using the reciprocal property, we know that the reciprocal of 1/4 is 4/1. So, we can rewrite our division problem as 1/2 multiplied by 4/1.
| Original Division Problem | Flipped Numerator and Denominator | Multiplication Problem |
|---|---|---|
| 1/2 ÷ 1/4 | 1/2 × 4/1 | 1 × 4 / 2 × 1 = 4/2 = 2 |
As you can see, flipping the numerator and denominator of the fraction we want to divide with has allowed us to rewrite the division problem as a multiplication problem, which is much easier to solve. By multiplying the numerators and the denominators, we get the answer 2.
Multiply the Numerators and Denominators
To cross multiply fractions, we need to multiply the numerator of the first fraction by the denominator of the second fraction, and vice versa, then divide the product by the other product. In equation form, it looks like this:
(a/b) x (c/d) = (a x c) / (b x d)
For example, to cross multiply 1/2 by 3/4, we would do the following:
| 1 | x | 3 | = | 3 |
| 2 | x | 4 | 8 |
So, 1/2 multiplied by 3/4 is equal to 3/8.
Multiplying Mixed Numbers and Whole Numbers
To multiply a mixed number by a whole number, we first need to convert the mixed number to an improper fraction. For example, to multiply 2 1/2 by 3, we first convert 2 1/2 to an improper fraction:
2 1/2 = (2 x 2) + 1 / 2
2 1/2 = 4/2 + 1/2
2 1/2 = 5/2
Now we can multiply 5/2 by 3:
5/2 x 3 = (5 x 3) / (2 x 1)
5/2 x 3 = 15/2
So, 2 1/2 multiplied by 3 is equal to 15/2, or 7 1/2.
Multiply Whole Numbers and Mixed Numbers
To multiply a whole number and a mixed number, first multiply the whole number by the fraction part of the mixed number. Then, multiply the whole number by the whole number part of the mixed number. Finally, add the two products together.
For example, to multiply 2 by 3 1/2, first multiply 2 by 1/2:
“`
2 x 1/2 = 1
“`
Then, multiply 2 by 3:
“`
2 x 3 = 6
“`
Finally, add 1 and 6 to get:
“`
1 + 6 = 7
“`
Therefore, 2 x 3 1/2 = 7.
Here are some more examples of multiplying whole numbers and mixed numbers:
| Multiplying Whole Numbers and Mixed Numbers | ||
|---|---|---|
| Problem | Solution | Explanation |
| 2 x 3 1/2 | 7 | Multiply 2 by 1/2 to get 1. Multiply 2 by 3 to get 6. Add 1 and 6 to get 7. |
| 3 x 2 1/4 | 8 3/4 | Multiply 3 by 1/4 to get 3/4. Multiply 3 by 2 to get 6. Add 3/4 and 6 to get 8 3/4. |
| 4 x 1 1/3 | 6 | Multiply 4 by 1/3 to get 4/3. Multiply 4 by 1 to get 4. Add 4/3 and 4 to get 6. |
Convert to Improper Fractions
To cross multiply fractions, you must first convert them to improper fractions. An improper fraction is a fraction where the numerator is greater than or equal to the denominator. To convert a proper fraction (where the numerator is less than the denominator) to an improper fraction, multiply the denominator by the whole number and add the numerator. The result is the new numerator, and the denominator remains the same. For example, to convert 1/3 to an improper fraction:
| Multiply the denominator by the whole number: | 3 x 1 = 3 |
|---|---|
| Add the numerator: | 3 + 1 = 4 |
| The result is the new numerator: | Numerator = 4 |
| The denominator remains the same: | Denominator = 3 |
| Therefore, the improper fraction is: | 4/3 |
Now that you have converted the fractions to improper fractions, you can cross multiply to solve the equation.
Multiply Same-Denominator Fractions
When multiplying fractions with the same denominator, we can simply multiply the numerators and keep the denominator. For instance, to multiply 2/5 by 3/5:
“`
(2/5) x (3/5) = (2 x 3) / (5 x 5) = 6/25
“`
To help visualize this, we can create a table to show the cross-multiplication process:
| Numerator | Denominator | |
|---|---|---|
| Fraction 1 | 2 | 5 |
| Fraction 2 | 3 | 5 |
| Product | 6 | 25 |
Multiplying Fractions with Different Denominators
When multiplying fractions with different denominators, we need to find a common denominator. The common denominator is the least common multiple (LCM) of the denominators of the two fractions. For instance, to multiply 1/2 by 3/4:
“`
1/2 x 3/4 = (1 x 3) / (2 x 4) = 3/8
“`
Multiply Mixed Number Fractions
To multiply mixed number fractions, first convert them to improper fractions. To do this, multiply the whole number by the denominator of the fraction and add the numerator. The result is the new numerator. The denominator remains the same.
Example:
Convert the mixed number fraction 2 1/2 to an improper fraction.
2 x 2 + 1 = 5/2
Now multiply the improper fractions as you would with any other fraction. Multiply the numerators and multiply the denominators.
Example:
Multiply the improper fractions 5/2 and 3/4.
(5/2) x (3/4) = 15/8
Converting the Improper Fraction Back to Mixed Number
If the result of multiplying improper fractions is an improper fraction, you can convert it back to a mixed number.
To do this, divide the numerator by the denominator. The quotient is the whole number. The remainder is the numerator of the fraction. The denominator remains the same.
Example:
Convert the improper fraction 15/8 to a mixed number.
15 ÷ 8 = 1 remainder 7
So 15/8 is equal to the mixed number 1 7/8.
| Fraction | Improper Fraction | Improper Fraction Product | Mixed Number |
|---|---|---|---|
| 2 1/2 | 5/2 | 15/8 | 1 7/8 |
| 1 3/4 | 7/4 | 35/8 | 4 3/8 |
Use Parentheses for Clarity
In some cases, using parentheses can help to improve clarity and avoid confusion. For example, consider the following fraction:
“`
$\frac{(2/3) \times (3/4)}{(5/6) \times (1/2)}$
“`
Without parentheses, this fraction could be interpreted in two different ways:
“`
$\frac{2/3 \times 3/4}{5/6 \times 1/2}$
or
$\frac{2/3 \times (3/4 \times 5/6 \times 1/2)}{1}$
“`
By using parentheses, we can specify the order of operations and ensure that the fraction is interpreted correctly:
“`
$\frac{(2/3) \times (3/4)}{(5/6) \times (1/2)}$
“`
In this case, the parentheses indicate that the numerators and denominators should be multiplied first, before the fractions are simplified.
Here is a table summarizing the two interpretations of the fraction without parentheses:
| Interpretation | Result |
|---|---|
| $\frac{2/3 \times 3/4}{5/6 \times 1/2}$ | $\frac{1}{2}$ |
| $\frac{(2/3 \times 3/4) \times 5/6 \times 1/2}{1}$ | $\frac{5}{12}$ |
As you can see, the use of parentheses can have a significant impact on the result of the fraction.
Review and Check Your Answer
Step 10: Check Your Answer
Once you have cross-multiplied and simplified the fractions, you should check your answer to ensure its accuracy. Here’s how you can do this:
- Multiply the numerators and denominators of the original fractions: Calculate the products of the numerators and denominators of the two fractions you started with.
- Compare the results: If the products are the same, your cross-multiplication is correct. If they are different, you have made an error and should review your calculations.
Example:
Let’s check the answer we obtained earlier: 2/3 = 8/12.
| Original fractions: | Cross-multiplication: |
|---|---|
| 2/3 | 2 x 12 = 24 |
| 8/12 | 8 x 3 = 24 |
As the products are the same (24), our cross-multiplication is correct.
How to Cross Multiply Fractions
Cross multiplication is a method for solving proportions that involves multiplying the numerators (top numbers) of the fractions on opposite sides of the equal sign and doing the same with the denominators (bottom numbers). To cross multiply fractions:
- Multiply the numerator of the first fraction by the denominator of the second fraction.
- Multiply the numerator of the second fraction by the denominator of the first fraction.
- Set the results of the multiplications equal to each other.
- Solve the resulting equation to find the value of the variable.
For example, to solve the proportion 1/x = 2/3, we would cross multiply as follows:
1 · 3 = x · 2
3 = 2x
x = 3/2
People Also Ask
How do you cross multiply percentages?
To cross multiply percentages, convert each percentage to a fraction and then cross multiply as usual.
How do you cross multiply fractions with variables?
When cross multiplying fractions with variables, treat the variables as if they were numbers.
What is the shortcut for cross multiplying fractions?
There is no shortcut for cross multiplying fractions. The method outlined above is the most efficient way to do so.
