Fractions & Multiplications

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Introduction to Fractions & Multiplications

We have already seen how to add and subtract proper fractions as well as improper fractions/mixed fractions. How does one tackle the problem of multiplying two fractions? Well, multiplication is a much easier method. You just need to be aware of what to do when the fractions are proper and improper. Let us start with the relatively easier topic; multiplication of proper fractions with whole numbers.

The Big Idea: Fractions & Multiplications

Multiplication of proper fractions with whole numbers

The “Multiplication by repeated addition” concept doesn’t really follow through naturally when it comes to multiplying one whole number with a fraction. There’s nothing to worry about though. The way to visualise multiplication of whole numbers with fractions is to think of it this way:

You are finding that particular fraction of the whole number. For example:

\(\begin{align}2 \times \frac{1}{3};\end{align}\)

This can be thought of as \(\begin {align}\frac13\end {align}\)rd portion of 2;

So, it becomes;

\(\begin {align}\frac23;\end {align}\) Thus, visually:

Multiplication of proper fractions

The traditional approach of multiplication being repeated addition doesn’t really translate to the multiplication of two proper fractions. That doesn’t mean you have to reduce the operation to “following steps”. There is a very simple way to picture the whole operation. Let’s consider the following scenario:

You want to prepare a dish that requires some rice. The bags of rice with you are \({3 \over 4}\)ths full. You took \({1 \over 5}\)ths from it and everything worked out perfectly. Now, you want to repeat the dish, but this time you have bags full of rice, how much rice do you take?

Well, you could first split the full back into \({3 \over 4}\) and then take \({1 \over 5}\) of it, or simply perform the operation:

\(\begin{align}\frac{3}{4} \times \frac{1}{5}\end{align}\)

Here’s how you visualize it:

Each bag of rice originally:

Now you want to find out how much \({1 \over 5}\)ths of this quantity will be when compared to the whole (because always remember, fractions represent a part of a whole). So, it becomes:

You get the second figure by dividing the first into 5 equal (horizontal) parts and just considering one of the 5 new subdivisions (as the fraction we are interested in \({1 \over 5}\)). Therefore, the answer is \({3 \over 20}\)

See, it’s not that complicated to paint a picture of the multiplication of fractions!

Multiplication of mixed fractions

Awesome! Now you know how to multiply two proper fractions, however, the question still remains as to how to visualise the multiplication of mixed fractions? The curious thing is, the method described above works perfectly for mixed fractions as well. Have a look for yourself:

Suppose now, instead of \({3 \over 4}\) kg, each packet holds \({5 \over 3}\) kg of rice and you want to find out how much rice is there in \(\begin {align}1\frac14\end {align}\) packets, let’s visualise this together!

\(\begin{align}\left( {1 \times \frac{5}{3}} \right) + \left( {\frac{1}{4} \times \frac{5}{3}} \right)\end{align}\)

When it comes to the multiplication of fractions, one can employ this nifty little trick. It would seem that the operation works perfectly if we simply multiply the numerators and the denominators.

So,

\(\begin{align}\frac{5}{3} + \frac{5}{{12}}\end{align}\) is what we get as the final problem statement.

Then, simply carry out the addition:

\({25 \over 12}\)

Visually:

Remember, each shape represents 1 kg of rice.

How is it important?

The Beauty of cancellation

In the above example none of the four numbers involved were divisible by each other or by a common number. So you got your final number in just two steps, with no further step needed. What would be the change if the two fractions involved were \(3 \over 4\) and \(8 \over 11\). When you write the two fractions with the operator (x) between them, this is what they would look like:

\(\begin{align}\frac{3}{4} \times \frac{8}{{11}}\end{align}\)

Now the 8 in the second fraction’s numerator and the 4 in the first fraction’s denominator are divisible by each other, aren’t they? So, this is how you could simplify the expression, by dividing both 8 and 4 by 4:

\(\begin{align}   & \frac{3}{4} \times \frac{8}{{11}}\\     &  = \frac{3}{1} \times \frac{2}{{11}}\\    &  = \frac{6}{{11}}   \end{align}\)

After that you multiply the fractions in the earlier way to get \(\begin{align}\frac{{(3 \times 2)}}{{(1 \times 11)}} = \frac{6}{{11}}\end{align}\). You could have done the cancellation at a later stage also, by writing it this way:

\(\begin{align}    & \frac{3}{4} \times \frac{8}{{11}}\\    & = \frac{{(3 \times 8)}}{{(4 \times 11)}}\\    & = \frac{{24}}{{44}}\\    & = \frac{{12}}{{22}}\\    & = \frac{6}{{11}}     \end{align}\)

The core concept behind cancellation is to basically eliminate the common factors, so that the product obtained is in the standard form. The removal of common factors can be done at any time before presenting the final product.

The double overlapping coloured portion represents the product fraction.

A Sign of the Times

In all the examples shown above, we have considered only positive fractions, meaning fractions which are to the right of zero on the number line. But you might sometimes have to multiply two fractions when one or both of the fractions are negative fractions, which means that they might be on the left of zero on the number line. A few examples are \(-{5 \over 7}\) or \(-{3 \over 4}\) . There are a few simple rules you need to keep in mind regarding positive or negative signs:

  • Positive fraction multiplied by positive fraction: result has positive sign

  • Positive fraction multiplied by negative fraction: result has negative sign

  • Negative fraction multiplied by negative fraction: result has positive sig

A Share of the Pie

While multiplying fractions, it is often very convenient to represent the fractions as slices of a circular pie (think of how you slice and divide your pizza with your friends). This helps you get a visual interpretation of the two fractions and their product. Let us start with two very simple fractions, \({1 \over 3}\) and \({3 \over 5}.\) If you simply multiply (and simplify) the multiplication of these fractions according the process explained above, this is what you would get:

\(\begin{align}   &  \frac{1}{3} \times \frac{3}{5}\\    &  = \frac{{(1 \times 3)}}{{(3 \times 5)}}\\   &   = \frac{3}{{15}}\\  &   = \frac{1}{5}     \end{align}\)

Now think of the two fractions as two pizzas or apple pies. The first pizza would be divided into 3 equal slices, of which you would color 1, and the second pizza would be divided into 5 slices, of which you would color 3. This would give you the two fractions \({1 \over 3}\) (one slice out of three) and \({3 \over 5}\) (three slices out of five). You can see the visual representation of the multiplication of these two fractions here.

Common mistakes or misconceptions

  • Children often tend to find the least common multiple of the denominators and multiply the numerators.
    This happens because they are incorrectly applying a recently learnt procedure of addition and subtraction of fractions to multiplication as well.
  • A common misconception is that the product of a whole number and a fraction, and/or the product of two fractions is always greater than either of them.
    This happens because of this idea carries through from multiplication of whole numbers. For whole numbers, except when 0 or 1 are involved, the product is always greater than the two numbers being multiplied. This does not hold for multiplication involving proper fractions. E.g. \(\begin{align}\frac{1}{2}\times\frac{1}{4}=\frac{1}{8}\end{align}\). One-eighth is smaller than both \(\begin{align}\frac{1}{2}\end{align}\) and \(\begin{align}\frac{1}{4}\end{align}\).

Tips and Tricks

  • Generally, students simplify a fraction after multiplication. However, to make calculations easier, check if the two fractions to be multiplied are already in the lowest form. If not, first simplify them and then multiply. Your calculation will be easier and the answer will not change.
    E.g \(\begin{align}\frac{4}{12}\times\frac{5}{13}\end{align}\) will be difficult to directly multiply. But simplifying the first fraction we get \(\begin{align}\frac{1}{3}\times\frac{5}{13}=\frac{5}{39}\end{align}\). 

     
  • Simplification can also be done across the two fractions. E.g. if there is a common factor between the numerator of one of the fractions and the denominator of the other fraction, you can simplify them and proceed.
    E.g. \(\begin{align}\frac{5}{28}\times\frac{7}{9}\end{align}\) can be simplified to \(\begin{align}\frac{5}{4}\times\frac{1}{9}\end{align}\) before multiplying.

 

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