When you’re like me, you in all probability discovered the right way to cross multiply fractions at school. However should you’re like me, you additionally in all probability forgot the right way to do it. Don’t be concerned, although. I’ve bought you coated. On this article, I am going to educate you the right way to cross multiply fractions like a professional. It isn’t as laborious as you assume, I promise.
Step one is to know what cross multiplication is. Cross multiplication is a technique of fixing proportions. A proportion is an equation that states that two ratios are equal. For instance, the proportion 1/2 = 2/4 is true as a result of each ratios are equal to 1.
To cross multiply fractions, you merely multiply the numerator of the primary fraction by the denominator of the second fraction, after which multiply the denominator of the primary fraction by the numerator of the second fraction. For instance, to resolve the proportion 1/2 = 2/4, we might cross multiply as follows: 1 x 4 = 2 x 2. This offers us the equation 4 = 4, which is true. Subsequently, the proportion 1/2 = 2/4 is true.
Discover the Reciprocal of the Second Fraction
When cross-multiplying fractions, step one is to search out the reciprocal of the second fraction. The reciprocal of a fraction is a brand new fraction that has the denominator and numerator swapped. In different phrases, if in case you have a fraction a/b, its reciprocal is b/a.
To search out the reciprocal of a fraction, merely flip the fraction the other way up. For instance, the reciprocal of 1/2 is 2/1, and the reciprocal of three/4 is 4/3.
This is a desk 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 wish to divide with, after which change the division signal to a multiplication signal. For example, as an instance we wish to divide 1/2 by 1/4. First, we flip the numerator and denominator of 1/4, which provides us 4/1. Then, we modify the division signal to a multiplication signal, which provides us 1/2 multiplied by 4/1.
Why Does Flipping the Numerator and Denominator Work?
Flipping the numerator and denominator of the fraction we wish to divide with is legitimate due to a property of fractions known as the reciprocal property. The reciprocal property states that the reciprocal of a fraction is the same as the fraction with its numerator and denominator flipped. For example, the reciprocal of 1/4 is 4/1, and the reciprocal of 4/1 is 1/4.
After we divide one fraction by one other, we’re basically multiplying the primary fraction by the reciprocal of the second fraction. By flipping the numerator and denominator of the fraction we wish to divide with, we’re successfully multiplying by its reciprocal, which is what we wish to do with a view to divide fractions.
Instance
Let’s work via an instance to see how flipping the numerator and denominator works in observe. For example we wish to divide 1/2 by 1/4. Utilizing the reciprocal property, we all know that the reciprocal of 1/4 is 4/1. So, we are able to rewrite our division drawback as 1/2 multiplied by 4/1.
| Authentic Division Drawback | Flipped Numerator and Denominator | Multiplication Drawback |
|---|---|---|
| 1/2 ÷ 1/4 | 1/2 × 4/1 | 1 × 4 / 2 × 1 = 4/2 = 2 |
As you’ll be able to see, flipping the numerator and denominator of the fraction we wish to divide with has allowed us to rewrite the division drawback as a multiplication drawback, which is way simpler to resolve. By multiplying the numerators and the denominators, we get the reply 2.
Multiply the Numerators and Denominators
To cross multiply fractions, we have to multiply the numerator of the primary fraction by the denominator of the second fraction, and vice versa, then divide the product by the opposite product. In equation type, it seems like this:
(a/b) x (c/d) = (a x c) / (b x d)
For instance, to cross multiply 1/2 by 3/4, we might do the next:
| 1 | x | 3 | = | 3 |
| 2 | x | 4 | 8 |
So, 1/2 multiplied by 3/4 is the same as 3/8.
Multiplying Blended Numbers and Entire Numbers
To multiply a blended quantity by a complete quantity, we first must convert the blended quantity to an improper fraction. For instance, 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 are able to 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 the same as 15/2, or 7 1/2.
Multiply Entire Numbers and Blended Numbers
To multiply a complete quantity and a blended quantity, first multiply the entire quantity by the fraction a part of the blended quantity. Then, multiply the entire quantity by the entire quantity a part of the blended quantity. Lastly, add the 2 merchandise collectively.
For instance, to multiply 2 by 3 1/2, first multiply 2 by 1/2:
“`
2 x 1/2 = 1
“`
Then, multiply 2 by 3:
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2 x 3 = 6
“`
Lastly, add 1 and 6 to get:
“`
1 + 6 = 7
“`
Subsequently, 2 x 3 1/2 = 7.
Listed here are some extra examples of multiplying complete numbers and blended numbers:
| Multiplying Entire Numbers and Blended Numbers | ||
|---|---|---|
| Drawback | Resolution | Rationalization |
| 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 could first convert them to improper fractions. An improper fraction is a fraction the place the numerator is bigger than or equal to the denominator. To transform a correct fraction (the place the numerator is lower than the denominator) to an improper fraction, multiply the denominator by the entire quantity and add the numerator. The result’s the brand new numerator, and the denominator stays the identical. For instance, to transform 1/3 to an improper fraction:
| Multiply the denominator by the entire quantity: | 3 x 1 = 3 |
|---|---|
| Add the numerator: | 3 + 1 = 4 |
| The result’s the brand new numerator: | Numerator = 4 |
| The denominator stays the identical: | Denominator = 3 |
| Subsequently, the improper fraction is: | 4/3 |
Now that you’ve got transformed the fractions to improper fractions, you’ll be able to cross multiply to resolve the equation.
Multiply Similar-Denominator Fractions
When multiplying fractions with the identical denominator, we are able to merely multiply the numerators and hold the denominator. For example, to multiply 2/5 by 3/5:
“`
(2/5) x (3/5) = (2 x 3) / (5 x 5) = 6/25
“`
To assist visualize this, we are able to create a desk to indicate the cross-multiplication course of:
| Numerator | Denominator | |
|---|---|---|
| Fraction 1 | 2 | 5 |
| Fraction 2 | 3 | 5 |
| Product | 6 | 25 |
Multiplying Fractions with Totally different Denominators
When multiplying fractions with completely different denominators, we have to discover a widespread denominator. The widespread denominator is the least widespread a number of (LCM) of the denominators of the 2 fractions. For example, to multiply 1/2 by 3/4:
“`
1/2 x 3/4 = (1 x 3) / (2 x 4) = 3/8
“`
Multiply Blended Quantity Fractions
To multiply blended quantity fractions, first convert them to improper fractions. To do that, multiply the entire quantity by the denominator of the fraction and add the numerator. The result’s the brand new numerator. The denominator stays the identical.
Instance:
Convert the blended quantity fraction 2 1/2 to an improper fraction.
2 x 2 + 1 = 5/2
Now multiply the improper fractions as you’ll with another fraction. Multiply the numerators and multiply the denominators.
Instance:
Multiply the improper fractions 5/2 and three/4.
(5/2) x (3/4) = 15/8
Changing the Improper Fraction Again to Blended Quantity
If the results of multiplying improper fractions is an improper fraction, you’ll be able to convert it again to a blended quantity.
To do that, divide the numerator by the denominator. The quotient is the entire quantity. The rest is the numerator of the fraction. The denominator stays the identical.
Instance:
Convert the improper fraction 15/8 to a blended quantity.
15 ÷ 8 = 1 the rest 7
So 15/8 is the same as the blended no 1 7/8.
| Fraction | Improper Fraction | Improper Fraction Product | Blended Quantity |
|---|---|---|---|
| 2 1/2 | 5/2 | 15/8 | 1 7/8 |
| 1 3/4 | 7/4 | 35/8 | 4 3/8 |
Use Parentheses for Readability
In some instances, utilizing parentheses can assist to enhance readability and keep away from confusion. For instance, think about the next fraction:
“`
$frac{(2/3) instances (3/4)}{(5/6) instances (1/2)}$
“`
With out parentheses, this fraction might be interpreted in two other ways:
“`
$frac{2/3 instances 3/4}{5/6 instances 1/2}$
or
$frac{2/3 instances (3/4 instances 5/6 instances 1/2)}{1}$
“`
By utilizing parentheses, we are able to specify the order of operations and make sure that the fraction is interpreted appropriately:
“`
$frac{(2/3) instances (3/4)}{(5/6) instances (1/2)}$
“`
On this case, the parentheses point out that the numerators and denominators needs to be multiplied first, earlier than the fractions are simplified.
Here’s a desk summarizing the 2 interpretations of the fraction with out parentheses:
| Interpretation | Outcome |
|---|---|
| $frac{2/3 instances 3/4}{5/6 instances 1/2}$ | $frac{1}{2}$ |
| $frac{(2/3 instances 3/4) instances 5/6 instances 1/2}{1}$ | $frac{5}{12}$ |
As you’ll be able to see, using parentheses can have a major influence on the results of the fraction.
Assessment and Test Your Reply
Step 10: Test Your Reply
After getting cross-multiplied and simplified the fractions, it’s best to verify your reply to make sure its accuracy. This is how you are able to do this:
- Multiply the numerators and denominators of the unique fractions: Calculate the merchandise of the numerators and denominators of the 2 fractions you began with.
- Evaluate the outcomes: If the merchandise are the identical, your cross-multiplication is appropriate. If they’re completely different, you have got made an error and will overview your calculations.
Instance:
Let’s verify the reply we obtained earlier: 2/3 = 8/12.
| Authentic fractions: | Cross-multiplication: |
|---|---|
| 2/3 | 2 x 12 = 24 |
| 8/12 | 8 x 3 = 24 |
Because the merchandise are the identical (24), our cross-multiplication is appropriate.
Tips on how to Cross Multiply Fractions
Cross multiplication is a technique for fixing proportions that entails multiplying the numerators (prime numbers) of the fractions on reverse sides of the equal signal and doing the identical with the denominators (backside numbers). To cross multiply fractions:
- Multiply the numerator of the primary fraction by the denominator of the second fraction.
- Multiply the numerator of the second fraction by the denominator of the primary fraction.
- Set the outcomes of the multiplications equal to one another.
- Resolve the ensuing equation to search out the worth of the variable.
For instance, to resolve the proportion 1/x = 2/3, we might cross multiply as follows:
1 · 3 = x · 2
3 = 2x
x = 3/2
Folks Additionally Ask
How do you cross multiply percentages?
To cross multiply percentages, convert every share to a fraction after which cross multiply as ordinary.
How do you cross multiply fractions with variables?
When cross multiplying fractions with variables, deal with the variables as in the event that they had been numbers.
What’s the shortcut for cross multiplying fractions?
There is no such thing as a shortcut for cross multiplying fractions. The strategy outlined above is probably the most environment friendly method to take action.
