How To Find Reciprocal Of A Fraction

Learning Outcomes

Find the reciprocal of a fraction

The fractions [latex]\Large\frac{two}{3}[/latex] and [latex]\Large\frac{three}{two}[/latex] are related to each other in a special mode. So are [latex]\Large-\frac{10}{vii}[/latex] and [latex]\Big-\frac{vii}{10}[/latex]. Practise you meet how? Besides looking like upside-down versions of ane another, if we were to multiply these pairs of fractions, the product would be [latex]1[/latex].

[latex]\Large\frac{2}{3}\cdot \frac{3}{2}\normalsize=1\text{ and }-\Large\frac{10}{7}\left(-\frac{vii}{10}\right)\normalsize=ane[/latex]

Such pairs of numbers are chosen reciprocals.

Reciprocal

The reciprocal of the fraction [latex]\Large\frac{a}{b}[/latex] is [latex]\Large\frac{b}{a}[/latex], where [latex]a\ne 0[/latex] and [latex]b\ne 0[/latex],
A number and its reciprocal have a product of [latex]one[/latex].

[latex]\Large\frac{a}{b}\cdot \frac{b}{a}\normalsize=1[/latex]

Hither are some examples of reciprocals:

Original number	Reciprocal	Product
[latex]\dfrac{3}{4}[/latex]	[latex]\dfrac{iv}{3}[/latex]	[latex]\dfrac{3}{4}\cdot\dfrac{4}{three}=\dfrac{3\cdot 4}{4\cdot iii}=\dfrac{12}{12}=i[/latex]
[latex]\dfrac{1}{2}[/latex]	[latex]\dfrac{two}{ane}[/latex]	[latex]\dfrac{one}{2}\cdot\dfrac{2}{ane}=\dfrac{one\cdot2}{2\cdot1}=\dfrac{ii}{ii}=1[/latex]
[latex] iii=\dfrac{3}{1}[/latex]	[latex]\dfrac{1}{3}[/latex]	[latex]\dfrac{three}{one}\cdot\dfrac{one}{three}=\dfrac{iii\cdot 1}{1\cdot 3}=\dfrac{iii}{3}=one[/latex]
[latex]2\dfrac{1}{3}=\dfrac{seven}{3}[/latex]	[latex]\dfrac{iii}{7}[/latex]	[latex]\dfrac{seven}{3}\cdot\dfrac{3}{vii}=\dfrac{7\cdot3}{three\cdot7}=\dfrac{21}{21}=\normalsize 1[/latex]

To find the reciprocal of a fraction, we invert the fraction. This ways that we place the numerator in the denominator and the denominator in the numerator. You tin recollect of it as switching the numerator and denominator: bandy the [latex]2[/latex] with the [latex]5[/latex] in [latex]\dfrac{2}{5}[/latex] to go the reciprocal [latex]\dfrac{5}{2}[/latex].

Brand sure that if it's a negative fraction, the reciprocal is also negative. This is because the product of ii negative numbers will give you the positive i that you are looking for. To get a positive result when multiplying two numbers, the numbers must accept the same sign. And then reciprocals must accept the aforementioned sign.

To discover the reciprocal, keep the same sign and capsize the fraction.

Instance

Find the reciprocal of each number. Then check that the production of each number and its reciprocal is [latex]1[/latex].

[latex]\Large\frac{4}{9}[/latex]
[latex]\Large-\frac{1}{vi}[/latex]
[latex]\Large-\frac{14}{5}[/latex]
[latex]7[/latex]

Solution:
To notice the reciprocals, we go on the sign and invert the fractions.

1.
Find the reciprocal of [latex]\Large\frac{4}{9}[/latex]	The reciprocal of [latex]\Large\frac{four}{nine}[/latex] is [latex]\Large\frac{9}{4}[/latex]
Bank check:
Multiply the number and its reciprocal.	[latex]\Large\frac{4}{9}\cdot \frac{nine}{4}[/latex]
Multiply numerators and denominators.	[latex]\Large\frac{36}{36}[/latex]
Simplify.	[latex]ane\quad\checkmark [/latex]

2.
Notice the reciprocal of [latex]\Large-\frac{1}{half dozen}[/latex]	[latex]\Big-\frac{half dozen}{1}[/latex]
Simplify.	[latex]-half-dozen[/latex]
Check:	[latex]\Large-\frac{1}{6}\normalsize\cdot \left(-6\right)[/latex]
	[latex]ane\quad\checkmark [/latex]

three.
Find the reciprocal of [latex]\Big-\frac{14}{5}[/latex]	[latex]\Large-\frac{v}{14}[/latex]
Bank check:	[latex]\Large-\frac{xiv}{5}\cdot \left(-\frac{5}{14}\right)[/latex]
	[latex]\Large\frac{70}{seventy}[/latex]
	[latex]i\quad\checkmark [/latex]

iv.
Find the reciprocal of [latex]7[/latex]
Write [latex]7[/latex] equally a fraction.	[latex]\Large\frac{7}{1}[/latex]
Write the reciprocal of [latex]\Large\frac{7}{1}[/latex]	[latex]\Big\frac{ane}{vii}[/latex]
Check:	[latex]7\cdot\Big\left(\frac{1}{7}\right)[/latex]
	[latex]1\quad\checkmark [/latex]

Endeavour It

In the following video nosotros will show more examples of how to find the reciprocal of integers, fractions and mixed numbers.

Caution Caution! Division past nil is undefined so is the reciprocal of any fraction that has a zero in the numerator. For whatsoever existent number a, [latex]\dfrac{a}{0}[/latex] is undefined. Additionally, the reciprocal of [latex]\dfrac{0}{a}[/latex] will e'er be undefined.

Sectionalization by Zero

You know what it ways to split up by [latex]two[/latex] or divide past [latex]10[/latex], but what does it hateful to split a quantity by [latex]0[/latex]? Is this even possible? On the flip side, can you divide [latex]0[/latex] by a number? Consider the fraction

[latex]\dfrac{0}{8}[/latex]

We can read it every bit, "zero divided by eight." Since multiplication is the inverse of division, we could rewrite this as a multiplication problem. What number times [latex]eight[/latex] equals [latex]0[/latex]?

[latex]\text{?}\cdot{viii}=0[/latex]

We can infer that the unknown must exist [latex]0[/latex] since that is the only number that will give a effect of [latex]0[/latex] when information technology is multiplied by [latex]8[/latex].

Now let'southward consider the reciprocal of [latex]\dfrac{0}{8}[/latex] which would be [latex]\dfrac{eight}{0}[/latex]. If we rewrite this every bit a multiplication problem, we will accept "what times [latex]0[/latex] equals [latex]8[/latex]?"

[latex]\text{?}\cdot{0}=eight[/latex]

This doesn't make whatsoever sense. There are no numbers that you can multiply by cypher to get a result of 8. In fact, any number divided by [latex]0[/latex] is incommunicable, or meliorate divers, all division by zero is undefined.