6.4.1 Dividing Fractions with the Calculator

In this exploration, you will learn how to use your calculator to divide fractions, and also discover a useful method for dividing fractions by hand.

Calculator symbol The calculator can be accessed on the left-hand side bar under the Toolkit.

Activity symbol Calculator Exploration: Dividing Fractions

For this activity, suppose that you have three pizzas, and you divide each of them in half.

How many halves do you have? You can see that there are six halves in three pizzas. When you calculate three division one divided by two, you can think of this as “How many halves are there in three?”

Try the calculation on the calculator. Remember to put one divided by two in parentheses, because you are dividing three by the entire fraction one divided by two. Do you get the answer six?

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Discussion

The key sequence is

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Answer

Remember that a division sign is equivalent to a fraction bar, and this is how the calculator displays the calculation. The fraction bar under the three is larger than the one in the middle of the one divided by two, indicating that you are dividing 3 by the fraction one divided by two.

To show that the parentheses are not optional, enter the same calculation without parentheses.

The calculator follows the order of operations. There are two divisions, so the calculator works from the left and divides three by one, then the result by two.

Can you see how the length of the fraction bars shows this? Compare this result to your previous expression, where you put the one divided by two into parentheses. Whenever you divide fractions, take great care to use parentheses around each fraction so that the calculator does as you intend.

Activity symbol Calculator Exploration: Practice with Dividing Fractions

Now try the following divisions. First, think about what the answer should be, and then try the calculation on the calculator.

(a) three division one divided by three

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Discussion

Remember to clear the last calculation before you start, and don’t forget to put the fraction in parentheses.

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Answer

(a) three division one divided by three is asking how many thirds there are in three. Think about the pizzas; you should be able to see that there are nine thirds in three pizzas.

The calculator agrees.

(b) three division one divided by four

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Answer

(b) How many fourths (or quarters) are there in three? The answer is 12.

Can you see a pattern in the calculations? Look at them again and see if you can describe what’s happening.

Look at the last three calculations. They asked you to divide 3 by one divided by two, by one divided by three, and 3 by one divided by four.

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Discussion

How do you create the result from the numbers involved?

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Answer

When you divide 3 by one divided by two, you get 6, which is the same as multiplying 3 by 2.

When you divide 3 by one divided by three, you get 9, which is the same as multiplying 3 by 3.

When you divide 3 by one divided by four, you get 12, which is the same as multiplying 3 by 4.

Activity symbol Calculator Exploration: Further Investigation

Let’s investigate further. Try the following calculations, and make a note of the answers.

(a) three division two divided by three and three multiplication three divided by two

(b) three division two divided by five and three multiplication five divided by two

(c) three division three divided by seven and three multiplication seven divided by three

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Answer

In each case, you should obtain the same answer for the division as for the multiplication. For (a), you should get nine divided by two or four one divided by two for (b) 15 divided by two or seven one divided by two; and for (c) the answer is 7.

Look again for a pattern. In each case, you start with 3. Look at the fraction that you divide by, and the fraction that you multiply by. What is the connection in each case?

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Discussion

In each case, the fraction that you multiply by is the fraction that you divide by turned upside down. In other words, the numerator and the denominator switch places.

This is the exciting part of mathematics, spotting patterns and then seeing whether they hold for other cases. We have a conjecture!

If you think you have spotted the pattern, then let's try to write it out. Completing the following sentence:

To divide by a fraction …

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Answer

To divide by a fraction, switch the numerator and the denominator, then multiply.

There is another way to write this. When the numerator and the denominator trade places, the result is called the reciprocal of the original fraction. Dividing by a fraction is the same as multiplying by its reciprocal.

To convince yourself that this is true, try two additional examples on the calculator. Divide the first fraction by the second, and note your answer. Then, multiply the first fraction by the reciprocal of the second fraction.

(a) four divided by seven division 15 divided by two

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Discussion

Remember to put each fraction in parentheses. Watch the screen at the top as you input the calculation to make sure that you are entering it correctly.

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Answer

(a) The calculator doesn’t enter the division very elegantly, but you can tell that it is correct because the middle fraction line is longer than the other two. This means that the whole of the first fraction is divided by the whole of the second fraction.

Multiplying by the reciprocal gives the same answer:

(b) five divided by 13 division 15 divided by 52

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Answer

(b) The division results in:

Meanwhile, multiplication by the reciprocal gives:

To recap:

  • To divide fractions on the calculator, put each fraction in a set of parentheses.
  • To divide by a fraction, multiply by its reciprocal.

6.4 Introduction to Dividing Fractions

6.4.2 Dividing Fractions