In case you’re like me, you most likely realized the best way to cross multiply fractions in class. However if you happen to’re like me, you additionally most likely forgot the best way to do it. Don’t be concerned, although. I’ve acquired you coated. On this article, I am going to train you the best way to cross multiply fractions like a professional. It isn’t as onerous as you suppose, I promise.
Step one is to grasp 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. Due to this fact, the proportion 1/2 = 2/4 is true.
Discover the Reciprocal of the Second Fraction
When cross-multiplying fractions, step one is to seek 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, in case you have a fraction a/b, its reciprocal is b/a.
To seek 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.
Here 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. As an illustration, for instance we wish to divide 1/2 by 1/4. First, we flip the numerator and denominator of 1/4, which supplies us 4/1. Then, we alter the division signal to a multiplication signal, which supplies 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 referred to 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. As an illustration, 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 as a way to divide fractions.
Instance
Let’s work by an instance to see how flipping the numerator and denominator works in apply. 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 downside 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 may see, flipping the numerator and denominator of the fraction we wish to divide with has allowed us to rewrite the division downside as a multiplication downside, which is far 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 kind, it seems to be 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 combined quantity by a complete quantity, we first have to convert the combined 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 combined quantity, first multiply the entire quantity by the fraction a part of the combined quantity. Then, multiply the entire quantity by the entire quantity a part of the combined 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:
“`
2 x 3 = 6
“`
Lastly, add 1 and 6 to get:
“`
1 + 6 = 7
“`
Due to this fact, 2 x 3 1/2 = 7.
Listed below are some extra examples of multiplying entire numbers and combined numbers:
| Multiplying Entire Numbers and Blended Numbers | ||
|---|---|---|
| Drawback | Resolution | Clarification |
| 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, it’s essential to first convert them to improper fractions. An improper fraction is a fraction the place the numerator is larger 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 |
| Due to this fact, the improper fraction is: | 4/3 |
Now that you’ve got transformed the fractions to improper fractions, you may 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. As an illustration, 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 Completely 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. As an illustration, 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 combined 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 combined 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 may convert it again to a combined 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 combined quantity.
15 ÷ 8 = 1 the rest 7
So 15/8 is the same as the combined #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 circumstances, utilizing parentheses might help to enhance readability and keep away from confusion. For instance, contemplate the next fraction:
“`
$frac{(2/3) instances (3/4)}{(5/6) instances (1/2)}$
“`
With out parentheses, this fraction could possibly be interpreted in two other ways:
“`
$frac{2/3 instances 3/4}{5/6 instances 1/2}$
or
$frac{2/3 instances (3/4 instances 5/6 instances 1/2)}{1}$
“`
Through the use of parentheses, we are able to specify the order of operations and make sure that the fraction is interpreted accurately:
“`
$frac{(2/3) instances (3/4)}{(5/6) instances (1/2)}$
“`
On this case, the parentheses point out that the numerators and denominators must be multiplied first, earlier than the fractions are simplified.
Here’s a desk summarizing the 2 interpretations of the fraction with out parentheses:
| Interpretation | End result |
|---|---|
| $frac{2/3 instances 3/4}{5/6 instances 1/2}$ | $frac{1}{2}$ |
| $frac{(2/3 instances 3/4) instances 5/6 instances 1/2}{1}$ | $frac{5}{12}$ |
As you may see, the usage of parentheses can have a big influence on the results of the fraction.
Evaluation and Examine Your Reply
Step 10: Examine Your Reply
Upon getting cross-multiplied and simplified the fractions, it is best to test your reply to make sure its accuracy. Here 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.
- Examine the outcomes: If the merchandise are the identical, your cross-multiplication is right. If they’re completely different, you could have made an error and may evaluate your calculations.
Instance:
Let’s test 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 right.
The best way to Cross Multiply Fractions
Cross multiplication is a technique for fixing proportions that entails multiplying the numerators (high 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.
- Remedy the ensuing equation to seek 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
Individuals Additionally Ask
How do you cross multiply percentages?
To cross multiply percentages, convert every proportion to a fraction after which cross multiply as normal.
How do you cross multiply fractions with variables?
When cross multiplying fractions with variables, deal with the variables as in the event that they have been numbers.
What’s the shortcut for cross multiplying fractions?
There is no such thing as a shortcut for cross multiplying fractions. The tactic outlined above is essentially the most environment friendly means to take action.
