When you’re like me, you in all probability discovered the way to cross multiply fractions at school. However in case you’re like me, you additionally in all probability forgot the way to do it. Don’t fret, although. I’ve acquired you coated. On this article, I will educate you the 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 provides 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 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, 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.
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 example, for example 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 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. As an example, the reciprocal of 1/4 is 4/1, and the reciprocal of 4/1 is 1/4.
Once we divide one fraction by one other, we’re primarily 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 so as to divide fractions.
Instance
Let’s work by means of an instance to see how flipping the numerator and denominator works in apply. To illustrate 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 will rewrite our division downside as 1/2 multiplied by 4/1.
| Unique 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 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 Complete 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 will 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 Complete 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:
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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 Complete 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 need to 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 |
| Due to this fact, the improper fraction is: | 4/3 |
Now that you’ve 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 will merely multiply the numerators and maintain the denominator. As an 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 will create a desk to point out 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 totally 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 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 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 number one 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 may help to enhance readability and keep away from confusion. For instance, take into account the next fraction:
“`
$frac{(2/3) occasions (3/4)}{(5/6) occasions (1/2)}$
“`
With out parentheses, this fraction may very well be interpreted in two other ways:
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$frac{2/3 occasions 3/4}{5/6 occasions 1/2}$
or
$frac{2/3 occasions (3/4 occasions 5/6 occasions 1/2)}{1}$
“`
By utilizing parentheses, we will specify the order of operations and be sure that the fraction is interpreted accurately:
“`
$frac{(2/3) occasions (3/4)}{(5/6) occasions (1/2)}$
“`
On this case, the parentheses point out that the numerators and denominators ought 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 | End result |
|---|---|
| $frac{2/3 occasions 3/4}{5/6 occasions 1/2}$ | $frac{1}{2}$ |
| $frac{(2/3 occasions 3/4) occasions 5/6 occasions 1/2}{1}$ | $frac{5}{12}$ |
As you may see, the usage of parentheses can have a major influence on the results of the fraction.
Overview and Examine Your Reply
Step 10: Examine Your Reply
After getting cross-multiplied and simplified the fractions, you must verify 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 totally different, you’ve got made an error and may assessment your calculations.
Instance:
Let’s verify the reply we obtained earlier: 2/3 = 8/12.
| Unique 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.
Easy methods to Cross Multiply Fractions
Cross multiplication is a technique for fixing proportions that includes 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
Individuals Additionally Ask
How do you cross multiply percentages?
To cross multiply percentages, convert every proportion to a fraction after which cross multiply as common.
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 isn’t a shortcut for cross multiplying fractions. The strategy outlined above is essentially the most environment friendly approach to take action.
