In case you’re like me, you in all probability realized methods to cross multiply fractions in class. However in case you’re like me, you additionally in all probability forgot methods to do it. Don’t fret, although. I’ve acquired you lined. On this article, I will train you methods to cross multiply fractions like a professional. It isn’t as onerous as you suppose, 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 unravel 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, when you’ve got 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. For example, 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. For 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 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 the intention to divide fractions.
Instance
Let’s work by way 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 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’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 downside as a multiplication downside, which is way simpler to unravel. 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 appears 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 Combined Numbers and Complete 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 Complete Numbers and Combined 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:
“`
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 blended numbers:
| Multiplying Complete Numbers and Combined Numbers | ||
|---|---|---|
| Drawback | Answer | 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, 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 |
| Due to this fact, 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 unravel the equation.
Multiply Identical-Denominator Fractions
When multiplying fractions with the identical denominator, we are able to merely multiply the numerators and preserve 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 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. For example, to multiply 1/2 by 3/4:
“`
1/2 x 3/4 = (1 x 3) / (2 x 4) = 3/8
“`
Multiply Combined 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 some other 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 Combined 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 #1 7/8.
| Fraction | Improper Fraction | Improper Fraction Product | Combined 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, 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 alternative ways:
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$frac{2/3 instances 3/4}{5/6 instances 1/2}$
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 be certain 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 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 |
|---|---|
| $frac{2/3 instances 3/4}{5/6 instances 1/2}$ | $frac{1}{2}$ |
| $frac{(2/3 instances 3/4) instances 5/6 instances 1/2}{1}$ | $frac{5}{12}$ |
As you’ll be able to see, the usage of parentheses can have a big affect on the results of the fraction.
Evaluate and Examine Your Reply
Step 10: Examine Your Reply
After you have 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 appropriate. If they’re totally different, you could have made an error and will assessment 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 appropriate.
How one can 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.
- Resolve the ensuing equation to seek out the worth of the variable.
For instance, to unravel 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 traditional.
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 approach to take action.
