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It’s movie night! You have 3 friends over and 1 family-sized pizza.
How do you split the pizzas, so everyone gets a fair share?
In this case, the pizza represents the whole number, while you and your friends represent the smallest number of slices, i.e. fractions, you should divide them by.
Let’s see the math behind sharing!
In this easy-to-follow guide, we'll show you how to divide whole numbers by fractions with simple explanations, solved examples, and practice questions.
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Dividing a whole number by a fraction is like figuring out how many pieces fit into a larger whole.
When we divide a whole number by a fraction, we're asking how many times that fraction can "fit" into that whole number.
For example, how many slices can fit into 1 whole pizza.
You can divide the whole pizza into 4 slices so everyone gets 1 large slice, or you can decide to cut the pizza into 8 smaller slices and give everyone 2 servings.
So, think of dividing by a fraction as a way of expanding or increasing the number of pieces you have.
Dividing a whole number by a fraction is the same as asking how many parts of that fraction fit into the whole number.
Imagine you have 5 whole chocolate bars, and you want to know how many pieces you’d get if you broke each chocolate bar into 3 pieces.
Let’s phrase it like a math problem: How many third-sized \( \Large \frac{1}{3}\) pieces are in 5 whole chocolate bars?
You have 5 chocolate bars, 3 pieces each, all you have to do is multiply 5 by 3 and the answer is: 15!
Let’s see how we arrived at this answer mathematically:
5 \( \Large \frac{1}{3}\)
To find out how many thirds we have, we multiplied the whole number 5 with the reciprocal of our fraction, 3.
The reciprocal of a fraction is what you get when you flip the fraction upside down, swapping its top (numerator) and bottom (denominator) numbers.
The reciprocal of \( \Large \frac{1}{3}\) is, therefore, \( \Large \frac{3}{1}\), or simply, 3.
Going back to our division, to divide our whole number, we simply multiply it by the reciprocal of the fraction:
5 ÷ \( \Large \frac{3}{1}\)
=\( \Large \frac{15}{1}\)
=15
The result is a larger number, as we're determining how many of those fractional parts fit into the whole number.
Let’s rehearse these steps using another example:
13 ÷ \( \Large \frac{2}{4}\)=?
To divide the whole number by a fraction, we multiply it by the reciprocal of that fraction.
Our first step is to find the reciprocal by flipping the fraction we’re dividing by.
Our ‘original’ fraction is \( \Large \frac{2}{4}\).
When we flip the nominator (2) with the denominator (4), we get \( \Large \frac{4}{2}\).
So, the reciprocal of \( \Large \frac{2}{4}\) is \( \Large \frac{4}{2}\).
Now we multiply the whole number by the reciprocal of the fraction.
13×\( \Large \frac{4}{2}\)=?
Finally, solve the multiplication problem.
In this case:
13×\( \Large \frac{4}{2}\)=26
13×\( \Large \frac{4}{2}\)
=\( \Large \frac{52}{3}\)
=26
That’s it! The answer to our problem is 26.
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Let’s look at a few more examples:
A) 6÷\( \Large \frac{1}{3}\)
Reciprocal of \( \Large \frac{1}{3}\) is \( \Large \frac{3}{1}\), or simply 3.
Multiply: 6×\( \Large \frac{3}{1}\)=18
Answer: 18.
B) 10÷\( \Large \frac{2}{5}\)
Reciprocal of \( \Large \frac{2}{5}\) is \( \Large \frac{5}{2}\)
Multiply: 10×\( \Large \frac{5}{2}\)=\( \Large \frac{50}{2}\)=25
Answer: 25
C) 12÷\( \Large \frac{3}{4}\)
Reciprocal of \( \Large \frac{3}{4}\) is \( \Large \frac{4}{3}\)
Multiply: 12×\( \Large \frac{4}{3}\)=\( \Large \frac{48}{3}\)=16.
Answer: 16
16
D) 14÷\( \Large \frac{7}{8}\)
Reciprocal of \( \Large \frac{7}{8}\) is \( \Large \frac{8}{7}\).
Multiply: 14×\( \Large \frac{8}{7}\)=\( \Large \frac{112}{7}\)=16
Answer: 16
E) 30÷\( \Large \frac{1}{5}\)
Reciprocal of \( \Large \frac{1}{5}\) is \( \Large \frac{5}{1}\), or simply 5.
Multiply: 30×5=150
Answer: 150
Now, your turn!
Try these exercises and see if you can divide whole numbers by fractions like a pro. For more comprehensive exercises, schedule a free assessment at the Mathnasium of Allen.
A) 12÷\( \Large \frac{2}{3}\)=?
B) 13÷\( \Large \frac{1}{9}\)=?
C) 7÷\( \Large \frac{1}{7}\)=?
D) 10÷\( \Large \frac{1}{4}\)=?
E) 4÷\( \Large \frac{1}{8}\)=?
F) 17÷\( \Large \frac{2}{8}\)=?
Emma is making friendship bracelets. She has 8 yards of string, and each bracelet needs \( \Large \frac{2}{3}\) of a yard of string. How many full bracelets can she make with the string she has?
A bakery has 5 pounds of dough, and each small loaf requires \( \Large \frac{1}{3}\)of a pound.
How many small loaves can the bakery make?
Jasper has a 7-pound bag of birdseed, and each bird feeder needs \( \Large \frac{1}{2}\) pound of seed.
How many bird feeders can he fill?
Remember to first find the reciprocal, multiply the whole number with reciprocal, and solve the problem.
We’ve put together answers to some common questions students have while learning about dividing whole numbers by fractions.
Dividing a whole number by a fraction feels like multiplying because the goal is to understand how many pieces fit into this whole number.
To get to this number of pieces, we flip the fraction and get the answer directly, so it seems like you’re multiplying rather than dividing.
It’s similar but not exactly the same process. Dividing by a decimal often means converting it to a fraction first.
For example, if you’re dividing by 0.5, you could turn it to \( \Large \frac{1}{2}\) and then divide as you would with a fraction.
Explore our latest guide to converting decimals to fractions
When you divide by a fraction larger than 1, it’s similar to dividing by a whole number, but it gives a smaller result than dividing by 1.
For example, dividing 8 by \( \Large \frac{5}{4}\) asks how many \( \Large \frac{5}{4}\) fit into 8, which will be less than 8.
You can check your answer by doing the reverse operation. After dividing by a fraction, multiply the answer by that fraction to see if it gives you the original number.
For example, if you think 8÷\( \Large \frac{2}{3}\)=12 then check by doing 12×\( \Large \frac{2}{3}\), which should be 8.
Mathnasium of Allen’s specially trained math tutors work with elementary students of all skill levels to help them learn and master any K-12 math topic, including fractions.
We empower students to unlock their full math potential using assessment-based, personalized learning plans and a proven teaching approach.
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At Mathnasium of Allen, we follow personalized learning plans designed to put each student on their best path towards math excellence.
Whether your student is looking to catch up, keep up, or get ahead in your math class, schedule a free assessment, and enroll them today!
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Here are the answers to the exercise problems, let’s see how you did!
Exercise 1:
A) 18
B) 117
C) 49
D) 40
E) 32
F) 68
Exercise 2:
Emma will make 12 friendship bracelets from 8 yards of string.
Exercise 3:
The bakery can make 15 small loaves.
Exercise 4:
Jasper can fill 14 bird feeders.
