sums will set you free
how to teach your child numbers arithmetic mathematics
fractions, decimals, percentages and ratios 2
Fractions, decimals and percentages are particular types of division sum, and are all effectively the ‘same’ thing. It is important to internalise this realisation right from the start. That is why here, as with the first page on fractions, decimals and percentages, all three topics are discussed on the same page. Once you understand one of these topics, you are in a position to understand them all. They are just different ways communicating and writing the same manner of information.
And when you divide integers by fractions, the inverse
occurs - the value becomes greater.
Ordinary counting, addition and subtraction are one-dimensional.
Multiplication and division with fractions are two-dimensional.
Note that the method for multiplying fractions that follows can be used for any fraction multiplication.
The illustration on the left shows forty-nine blocks
in a seven by seven square.
Thus, three sevenths by (or times) four sevenths is the
same as (equals) twelve forty-ninths:
Five tenths by five tenths are twenty-five hundredths,
Of course, twenty-five hundredths is an equivalent fraction to one quarter, illustrated in the photo above. Twenty-five hundredths can be reduced to one quarter by cancelling, by dividing both the top and bottom parts of the fraction 25/100 by 5:
And now an example of multiplying fractions when the bottom numbers are different:
The examples of multiplication with fractions so far have illustrated multiplying fractions by fractions. When multiplying a fraction by a whole number (an integer), such as the sum 1/3 x 3, convert the whole number into a fraction - a whole number is a fraction where the bottom number is one, a number divided by one remains the same number.
So 1/3 x 3 can be multiplied like this:
Here’s an example, nine tenths divided by three
tenths, 9/10 ÷ 3/10.
Now, there are many permutations of dividing whole numbers by a fraction, a fraction by a whole number, a fraction by another fraction, and then there’s numbers that are combined whole numbers and fractions as well. So how to not be completely confused as what to do?
With this following example, 2 ÷ 2/5, firstly the whole number 2 is converted to 10 fifths (2 x 5). Next the division is done to the top part and to the bottom part. The photo of blocks illustrated how 2/5 divides five times into 2, or 10/5.
Often, a easier way to do divisions involving fractions is to invert the fraction doing the dividing, and then multiply the first fraction by the inverted fraction. Now, explaining this in words is not easy to understand, so after giving a short explication why this process works, we will give a couple of worked examples.
And why does multiplying with the inverted dividing fraction work? Well, multiplying is the inverse of dividing, just as subtraction is the inverse (the opposite) of addition. So when a fraction is inverted (or turned upsidedown) the action being done (dividing) with that fraction is also inverted (to become multiplying).
Examples of a fraction divided by a fraction and a mixed fraction divided by a mixed fraction:
Until we reach putting in details in this section, 1/10 is written as .1, that is a point (or dot) before the one, and one quarter is written as .25, that is 25/100, and so on.
Originally, the decimal was called the decimal fraction (the Latin word for ten being decem). Decimals are a convenient way of writing and using tenths, or other fractions divisible by ten, such as hundredths, thousandths and so on.
.1 (or more commonly, of less accurately, 0.1) is the
same as one tenth, 1/10.
Notice that with the fractional
and decimal parts of numbers, all the exciting action
takes place in
Fractions are, in part, called rational numbers, not because they are particularly sane (although they are) but because they consist of ratios - ratio-nal. The integers also come within the class of rational numbers, for any integer can be expressed as a fraction or ratio. For example, 2 can be expressed as 2/1, or as 6/3, or even as 50/25, while 1734 can be written as 1734/1.
Any fraction can be converted into a decimal form (13/19 = 13 ÷ 19 = .6841...) and any decimal can be converted into a fraction ( .731 = 731/1000), but there comes a time when this starts to become a bit more difficult, and even mathematicians have problems keeping their heads straight. If you do want to go deep-diving, see comparing predicates, relational strengths and irrational numbers.
The diagram above gives help for converting a decimal into a fraction. As you can see, the decimal number goes on the top of the fraction, and the bottom part is dictated by how many places there are after the decimal point (how many numbers to the right of the dot). So .3 = 3/10 and .03 = 3/100, while .38 = 38/100.
Multiplications and divisions involving decimals are like doing those sums with integers, but you must make sure that the decimal point is in the right position. Such sums are easy to check using a calculator, or the educational counter below.
The full version with more detailed instructions, go to the introduction page.
Here is how to practise sums with decimals, for example .25 x 4,
The counter counts up: 0, .25, .5, .75, 1. Thus .25 x 4 = 1.
Now help the learner to try other multiplication sums. Each time, click the red Reset button to return Manual Steps (the red number) to zero.
Below is a concise version of the abelard.org eduacational maths counter. For an expanded version with more detailed instructions, go to how to teach your child number arithmetic mathematics - introduction.
Until we reach putting in details in this section, 100% (one hundred percent) is the whole cake. 1% is 1/100th (one hundredth) of the cake.
1/10th (one tenth) is written as 10%, that is a percent sign (%) after the ten, and one quarter is written as 25% and so on.
Multiplications and divisions involving percentages are like doing those sums with integers and fractions.
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