What Is A Reciprocal In Math? How To Find One, With Examples
The short version
- The reciprocal of a number is what you multiply it by to get 1. For a fraction, flip it: the reciprocal of 3/4 is 4/3.
- For a whole number, put it under 1: the reciprocal of 6 is 1/6. For a mixed number or decimal, turn it into a fraction first, then flip.
- A negative number’s reciprocal stays negative. The reciprocal of -5/8 is -8/5.
- Zero has no reciprocal, because nothing times 0 equals 1.
- Reciprocals are the reason “keep, change, flip” works when you divide fractions.
A reciprocal in math is the number you multiply by to get 1. For any nonzero number n, the reciprocal is 1/n. For a fraction, you just swap the top and bottom: the reciprocal of 2/5 is 5/2, because 2/5 × 5/2 = 10/10 = 1. Another name for it is the multiplicative inverse.
Here’s how to find the reciprocal of every kind of number your kid will see, why it matters for dividing fractions and solving equations, and practice problems with an answer key.
What is a reciprocal?
Two numbers are reciprocals of each other when their product is exactly 1. That’s the whole definition. A few pairs:
4 × 1/4 = 1
2/3 × 3/2 = 6/6 = 1
0.2 × 5 = 1
So 4 and 1/4 are reciprocals, 2/3 and 3/2 are reciprocals, and 0.2 and 5 are reciprocals. The relationship goes both ways: if 5 is the reciprocal of 0.2, then 0.2 is the reciprocal of 5.
The simplest way to think about it: the reciprocal “undoes” multiplication by a number. Multiply by 3/2, then multiply by 2/3, and you’re back where you started.
How to find the reciprocal of any number
| Type of number | What to do | Example | Reciprocal |
|---|---|---|---|
| Whole number | Write it over 1, then flip | 6 = 6/1 | 1/6 |
| Fraction | Swap numerator and denominator | 3/4 | 4/3 |
| Unit fraction | Flip; you get a whole number | 1/8 | 8 |
| Mixed number | Convert to an improper fraction, then flip | 2 1/2 = 5/2 | 2/5 |
| Decimal | Convert to a fraction, simplify, then flip | 0.25 = 1/4 | 4 |
| Negative | Flip and keep the negative sign | -5/8 | -8/5 |
Worked example: a mixed number
Find the reciprocal of 3 2/5.
Step 1: Convert. 3 × 5 = 15, and 15 + 2 = 17, so 3 2/5 = 17/5.
Step 2: Flip. The reciprocal is 5/17.
Check: 17/5 × 5/17 = 85/85 = 1.
Worked example: a decimal
Find the reciprocal of 0.6.
Step 1: Convert. 0.6 = 6/10 = 3/5.
Step 2: Flip. The reciprocal is 5/3, which is 1 2/3 or about 1.667.
Check: 0.6 × 5/3 = 3/5 × 5/3 = 15/15 = 1.
You can also get a decimal’s reciprocal with a calculator: 1 ÷ 0.6 ≈ 1.667. For homework, though, teachers usually want the fraction.
Special cases
1 and -1 are their own reciprocals, because 1 × 1 = 1 and -1 × -1 = 1.
0 has no reciprocal. Any number times 0 is 0, so there’s no way to get 1. That’s the same reason you can’t divide by zero.
Reciprocal vs opposite
Kids mix these up constantly, so it’s worth putting them side by side. The opposite (additive inverse) is what you add to get 0. The reciprocal (multiplicative inverse) is what you multiply by to get 1.
| Number | Opposite (adds to 0) | Reciprocal (multiplies to 1) |
|---|---|---|
| 4 | -4 | 1/4 |
| 2/3 | -2/3 | 3/2 |
| -7 | 7 | -1/7 |
| 0 | 0 | none |
The opposite changes the sign. The reciprocal flips the fraction and leaves the sign alone.
Using the reciprocal to divide fractions
This is where most kids first meet the word. To divide by a fraction, you multiply by its reciprocal. Many classes call it “keep, change, flip”: keep the first fraction, change ÷ to ×, flip the second fraction.
Worked example: fraction ÷ fraction
3/4 ÷ 2/5
Keep 3/4, change to ×, flip 2/5 to 5/2.
3/4 × 5/2 = 15/8
15/8 = 1 7/8
Worked example: whole number ÷ fraction
6 ÷ 1/3
6 × 3/1 = 18
That makes sense if you picture it. How many thirds fit in 6 whole pizzas? Each pizza has 3 thirds, so 6 × 3 = 18.
Worked example: mixed numbers
2 1/2 ÷ 1 1/4
Convert: 5/2 ÷ 5/4
Flip the divisor: 5/2 × 4/5 = 20/10 = 2
Check with decimals: 2.5 ÷ 1.25 = 2.
Why flipping works
Dividing by 2 is the same as multiplying by 1/2: half of 10 is 5, and 10 ÷ 2 is 5. The same idea holds for every nonzero number. Dividing by a number gives the same result as multiplying by its reciprocal. “Keep, change, flip” is that fact turned into a slogan.
Reciprocals in algebra
When a variable is multiplied by a fraction, multiply both sides by the reciprocal to get the variable alone.
(2/3)x = 8
Multiply both sides by 3/2.
(3/2)(2/3)x = 8 × 3/2
x = 24/2 = 12
Check: 2/3 × 12 = 24/3 = 8.
Later on, reciprocals show up as negative exponents (5⁻¹ = 1/5), as the slopes of perpendicular lines (a line with slope 2 is perpendicular to one with slope -1/2), and in trig functions like cosecant, which is 1 ÷ sine.
Common mistakes with reciprocals
- Flipping the wrong fraction. In 3/4 ÷ 2/5, only the second fraction flips. A kid who flips the first gets 4/3 × 2/5 = 8/15 instead of 15/8.
- Changing the sign. The reciprocal of -4 is -1/4. Writing 1/4 confuses the reciprocal with the opposite.
- Flipping a mixed number directly. The reciprocal of 2 1/2 isn’t 2 2/1. Convert to 5/2 first, then flip to 2/5.
- Moving the decimal point. The reciprocal of 0.5 is 2, not 5.
- Giving 0 a reciprocal. There isn’t one. Writing 1/0 isn’t a number.
Practice problems
- What’s the reciprocal of 9?
- What’s the reciprocal of 5/7?
- What’s the reciprocal of -3?
- What’s the reciprocal of 3 1/3?
- What’s the reciprocal of 0.8?
- Divide: 5/6 ÷ 2/3
- Divide: 4 ÷ 2/7
- Solve for x: (3/5)x = 12
Answers
- 1/9
- 7/5
- -1/3
- 3 1/3 = 10/3, so the reciprocal is 3/10.
- 0.8 = 4/5, so the reciprocal is 5/4 (or 1.25).
- 5/6 × 3/2 = 15/12 = 5/4 = 1 1/4
- 4 × 7/2 = 28/2 = 14
- x = 12 × 5/3 = 60/3 = 20. Check: 3/5 × 20 = 60/5 = 12.
If your kid is stuck on reciprocals
Finding a reciprocal is a one-step move. When a student can’t do it reliably, the gap is almost always fraction sense from grades 4 and 5: converting mixed numbers to improper fractions, writing decimals as fractions, and knowing what a fraction actually represents. You can see where those skills are supposed to land in our guide to math skills by grade level.
This matters beyond one worksheet. Reciprocals come back in every algebra unit that has a fraction in it, which is most of them. We explain that chain in struggling with algebra. If you’re not sure whether your child’s trouble is a single topic or something wider, a proper math assessment is the fastest way to find out.
Frequently asked questions
What does reciprocal mean in math?
It’s the number you multiply by to get 1. For a fraction, flip it. For a whole number n, it’s 1/n.
Is a reciprocal the same as a multiplicative inverse?
Yes. They’re two names for the same thing. Textbooks in middle and high school use “multiplicative inverse” more often.
What’s the reciprocal of a whole number?
One over that number. The reciprocal of 12 is 1/12.
What’s the reciprocal of a negative number?
It stays negative. The reciprocal of -2/9 is -9/2, because -2/9 × -9/2 = 18/18 = 1.
Does 0 have a reciprocal?
No. Nothing multiplied by 0 gives 1, so 0 is the one number with no reciprocal.
Find the actual gap
A kid who struggles to flip a fraction usually has a fraction gap from a few grades back, and it’ll keep showing up until someone closes it. Math Anon starts with a short adaptive assessment that finds exactly which skills are missing, then gives you a written report that spells them out.
Your child then practices those specific skills for about 20 minutes a day, and you get a progress report every week. The assessment is $49, credited back in full on the first month. Start the assessment or look over pricing.
