GED Mathematical Reasoning: Multiplying and Dividing Fractions

Multiplying fractions

To multiply fractions, the process is quite straight forward: Multiply the numerators together. Then, multiply the denominators together. And finally, simplify if possible.

 

Cross canceling

‘Cross canceling’ is something done BEFORE multiplying and, in many cases, results in the final answer being written in lowest terms. To ‘cross cancel,’ look at the numbers appearing diagonally from one another and ask yourself: Can they be reduced by a common factor?

fractions8

In this example, notice that the 3’s appearing diagonally from one another can be reduced by a factor of 3. When we divide each 3 by 3, the result is 1.

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We then rewrite the problem using the new, ‘cross canceled’ values and it becomes \frac{1}{1} \times \frac{1}{4}. From here, we multiply the numerators and denominators together. For the numerator: 1 times 1 is 1. For the denominator: 1 times 4 is 4. Notice that the result is the same and it is already reduced to lowest terms.

 

Dividing fractions

Division with fractions actually entails multiplying the fractions, after performing two small preliminary steps.

To divide with fractions, KEEP the first fraction (the dividend) the same. Then, CHANGE the division sign to multiplication. Next, FLIP the second fraction (the divisor) – by interchanging the numerator and denominator – and then multiply. Some students remember this process using the catchy phrase “keep change flip.” Formally, the division process with fractions is called “multiplying by the reciprocal.”

 

Example 1

A shortbread recipe calls for \frac{3}{4} pound of butter, but Jamie would like to make only \frac{1}{3} of a recipe. What fraction of a pound of butter (in lowest terms) should Jamie use?

The answer choices are:

  1. \frac{4}{7} pound
  2. \frac{3}{12} pound
  3. \frac{1}{4} pound
  4. \frac{1}{3} pound

We must determine how much \frac{1}{3} of \frac{3}{4} pound of butter is equal to. Recall that the word “of” implies multiplication. So to help Jamie determine how much butter is needed, we need to multiply \frac{1}{3} by \frac{3}{4}

So to multiply \frac{1}{3} by \frac{3}{4}, we’ll multiply the numerators together – 1 times 3 – and put the result of 3 in the numerator of the answer. Next, we’ll multiply the denominators together – 3 times 4 – and put the result of 12 in the denominator of the answer.

This gives us a fraction of: \frac{3}{12}

\frac{1}{3} \times \frac{3}{4} = \frac{1 \times 3}{3 \times 4} = \frac{3}{12}

Finally, we’ll reduce the fraction to lowest terms if it’s possible. In this case, it is possible to reduce by a factor of 3. When we divide both the numerator and denominator by 3, we get the final result of \frac{1}{4} pound, answer (3).

\frac{3 \div 3}{12 \div 3} = \frac{1}{4}

 

Example 2

Georgia has 7\frac{1}{2} pounds of ground hamburger meat. If she divides this into patties each weighing \frac{3}{4} pound, how many hamburgers can she make?

  1. 9 hamburgers
  2. 10 hamburgers
  3. 8\frac{1}{4} hamburgers
  4. 11 hamburgers

You may have quickly determined that we’ll need to divide 7\frac{1}{2} by \frac{3}{4} since we are splitting a quantity of meat into smaller portions. Of course, the word “divide” in the problem is a good clue, too! It’s important to note that to multiply or divide with a mixed number, we first need to change it into an improper fraction.

So we’ll multiply 2 times 7 and add 1 to arrive at the numerator of 15. And we’ll write 15 over the existing denominator of 2. Therefore, 7\frac{1}{2} is equal to the improper fraction of \frac{15}{2}.

Now we’re ready to divide. \frac{15}{2} divided by \frac{3}{4} can be written as you see here:

\frac{15}{2} \div \frac{3}{4}

Next, we’ll “KEEP CHANGE FLIP.”

We keep the first fraction the same, change the division to multiplication, and flip the fraction of \frac{3}{4} to be \frac{4}{3}. The result of the “keep change flip” process is shown here:

\frac{15}{2} \times \frac{4}{3}

Now, we multiply!

First, I’ll illustrate the traditional approach. We’ll multiply 15 times 4 to get a numerator result of 60. Then we’ll multiply 2 times 3 to get a denominator of 6.

\frac{15}{2} \times \frac{4}{3} = \frac{15 \times 4}{2 \times 3} = \frac{60}{6}

To reduce our result, what is the largest number that divides evenly into both 60 and 6? The answer is 6. When we divide the numerator and denominator by 6, we get a result of \frac{10}{1}. Recall that the fraction bar is another notation for division. So  also means “ten divided by one” and 10 divided by 1 is equal to just 10.

As a “fun fact”: Anytime the denominator of a fraction is 1, the final result in lowest terms will be simply the numerator. This is because any number divided by 1 is equal to the number itself. So the answer to the example is answer (2), 10 hamburgers.

Let’s see how solving this problem using the ‘cross canceling’ technique will lead us to the same result.

Please note: the ‘cross cancel’ trick ONLY works when MULTIPLYING fractions.

So, after we rewrite the division problem to multiply by the reciprocal or “keep change flip,” notice that we can ‘cross cancel’ in both directions.

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We can divide 15 and 3 by a common factor of 3. And we can divide 2 and 4 by a common factor of 2.

\frac{15}{2} \times \frac{4}{3} = \frac{5}{1} \times \frac{2}{1} = \frac{5 \times 2}{1 \times 1} = \frac{10}{1}

When we multiply across, we multiply 5 times 2 for a numerator of 10. And 1 times 1 for a denominator of 1. The result of the multiplication, then, is:  \frac{10}{1} which simplifies to just 10 – the same result!

 

Example 3

Recall that Jamie’s shortbread recipe calls for \frac{3}{4} pound of butter. If Jamie would like to triple this recipe, how much butter should she use?

  1. 3\frac{3}{4} pound
  2. 2\frac{1}{4} pound
  3. 1\frac{1}{4} pound
  4. 2\frac{1}{3} pound

 

The word “triple” implies that we need to multiply the amount of butter by three.

You may be wondering: How do we multiply a fraction by a whole number? The answer is: We turn the whole number into a fraction by placing it over one! So to multiply  by 3, we’ll rewrite 3 as \frac{3}{1}. This gives us \frac{3}{4} times \frac{3}{1}.

\frac{3}{4} \times 3 = \frac{3}{4} \times \frac{3}{1}

Notice that, in this example, cross canceling is not an option since the diagonals have no common factor with which to reduce by. So we’ll follow the traditional method for multiplying fractions. When we multiply the numerators of 3 and 3 together, we get 9 which is the numerator of our result. And when we multiply the denominators of 4 and 1 together, we get 4 which is the denominator of the result. So the product of \frac{3}{4} and 3 is \frac{9}{4}

\frac{3}{4} \times 3 = \frac{3}{4} \times \frac{3}{1} = \frac{3 \times 3}{4 \times 1} = \frac{9}{4}

When dealing with recipe measures, it is more helpful to see the result as a mixed number so let’s transform \frac{9}{4} into a mixed number by first dividing 4 into 9. The result is 2, which is the whole number portion of our mixed number. When we divide 9 by 4, the remainder is 1, which will be the numerator of the fraction portion of the mixed number. And recall that we keep the same denominator. Therefore our final answer is 2\frac{1}{4} pound of butter.

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