Chapter 5 of 14 · Fractions
Chapter 4. Two ways of writing, one number
One and the same point on the line carries an endless number of writings: this chapter settles equivalent fractions, simplifying, and the move from one denominator to another.
The same point, two labels
Draw a number line from 0 to 1 and share the unit into two: the mark in the middle carries 1/2. Start again from zero and share the same unit into four: the second mark carries 2/4, and it falls at exactly the same place. The two writings point to the same point, therefore to the same number. We say that they are equivalent, and we write 1/2 = 2/4.
This fact is the pivot of all the calculation that follows. Adding two fractions, comparing them, converting them into decimal notation: each of these gestures begins by rewriting a fraction in another form, without changing its value. Whoever does not hold this chapter calculates at random thereafter.
The mechanism is easy to see on the figure: sharing each part into two doubles the number of parts taken and also doubles the total number of parts. The quantity, for its part, has not moved a millimetre. It has been cut finer, nothing has been added or taken away.
What this chapter makes you able to do
Four gestures, which will serve every chapter of calculation.
- Make a fraction equivalent to a given fraction.
- Check whether two fractions written differently point to the same number.
- Simplify a fraction down to its simplest form.
- Rewrite a fraction with an imposed denominator, when that is possible.
What has to have been worked before this chapter
The rule, and the reason for the rule
Multiplying the numerator and the denominator by one and the same whole number other than zero gives a fraction equivalent to the first. Dividing them both by one and the same whole number other than zero, when the division comes out exactly on both sides, also gives an equivalent fraction. Those are the only two gestures allowed, and they work in both directions.
The reason can be read on the figure. Multiplying the denominator by 3 amounts to cutting each part into three: the parts become three times smaller, so three times as many of them are needed to cover the same quantity, which is what multiplying the numerator by 3 does. The two effects cancel out exactly, and the point does not move.
What is forbidden follows from the same reading: adding the same number top and bottom does not keep the value. Going from 1/2 to 2/3 by adding 1 everywhere gives two different numbers, as the number line shows in a second. Multiplication keeps the proportions, addition does not.
A fraction therefore admits an endless number of writings: 1/2, 2/4, 3/6, 4/8, 50/100, and so on without end. Among all those writings, one alone can no longer be simplified: it is the simplest form, and it is the one given as a result when nothing requires otherwise.
The words of this chapter
Three words, one of which describes a gesture and two of which describe a state.
- Equivalent fractions
- Two fractions that point to the same number, therefore to the same point on a number line. Each is obtained from the other by multiplying or dividing both terms by one and the same number other than zero.
- Simplify
- Divide the numerator and the denominator by one and the same number, so as to obtain a shorter writing of the same number. Simplifying never changes the value: it changes the counting unit used to state it.
- Simplest form
- The writing of a fraction that no common division can shorten any further. 3/4 is in its simplest form; 6/8 is not, since both terms divide by 2.
Worked example. Simplifying 18/24
The problem: give the simplest form of the fraction 18/24, justifying each step.
- Look for a common divisor
Both numbers are even, so they both divide by 2. It is the easiest divisor to spot, and it is always a good one to begin with. - Divide both terms by 2
18 divided by 2 gives 9, and 24 divided by 2 gives 12. The fraction becomes 9/12, equivalent to the one before. - Start again for as long as it is possible
9 and 12 both divide by 3, since 9 is worth three times 3 and 12 is worth four times 3. We obtain 3 and 4, therefore the fraction 3/4. - Check that nothing more can be done
3 divides only by 1 and by 3; 4 does not divide by 3. No common divisor other than 1 remains: 3/4 is the simplest form. - Check the value
The check is made by walking the road back the other way: 3 multiplied by 6 gives 18, and 4 multiplied by 6 gives 24. Both terms have indeed been multiplied by the same number, which confirms the equality 18/24 = 3/4.
Rewriting with an imposed denominator
The gesture that is the reverse of simplifying will serve at every addition. It consists in imposing the denominator and looking for the numerator that goes with it. To write 3/5 in fifteenths, we look at what 5 has to be multiplied by in order to obtain 15: by 3. The numerator must therefore be multiplied by 3 as well, which gives 9. So 3/5 = 9/15.
This move is possible only if the denominator aimed at is a multiple of the starting denominator. 3/5 can be written in fifteenths, in twentieths or in hundredths, because 15, 20 and 100 are multiples of 5. It cannot be written in sevenths, because 7 is not a multiple of 5: there is no whole number of sevenths worth three fifths.
Now try the gesture yourself: that is the exercise along the way below, and its answer is at the end of the volume.
Exercise along the way, chapter 4. Writing 2/3 in twelfths
Set in the course of the chapter: the reasoning is the one used for 3/5 in fifteenths.
- Write 2/3 in twelfths. Look first for what 3 has to be multiplied by in order to reach 12, then apply the same factor on top.
Two mistakes look alike and do not have the same remedy. The first consists in adding the same number top and bottom: that changes the value, and it is spotted on the number line. The second consists in multiplying only one of the two terms: that changes the value too, and it is a simple slip. The rule is therefore said in three words, to be repeated before each gesture: both, together, by multiplying.
Exercise 4.1. Making equivalent fractions
Write the fractions asked for and state each time which number you multiplied by.
- Write three fractions equivalent to 2/5.
- Write 1/4 with the denominator 20.
- Write 3/10 with the denominator 100.
Exercise 4.2. Simplifying
Give the simplest form of each fraction, writing out the successive divisions.
- Simplify 8/12.
- Simplify 30/45.
- Simplify 27/9, and say what is particular about the result.
Exercise 4.3. Deciding whether two writings are equivalent
This exercise crosses this chapter with chapter 3: justify each answer, do not settle for a yes or a no.
- Do the fractions 6/9 and 4/6 point to the same number?
- Do the fractions 3/7 and 4/8 point to the same number?
- A pupil writes 2/3 = 4/5, explaining that 2 was added on top and 2 underneath. Where is the mistake, and how can it be shown on a number line?
Self-check, chapter 4
Answer without turning back, then compare with the answer.
- What are the only two gestures that turn a fraction into an equivalent fraction?
- How do we know that a fraction has reached its simplest form?
- Can 3/5 be written in sevenths? Justify your answer.
What to remember from this chapter
- Multiplying or dividing both terms of a fraction by one and the same number other than zero does not change its value.
- Adding the same number to both terms changes the value: that gesture does not exist.
- A fraction has an endless number of writings, and one single simplest form.
- To simplify is to divide both terms for as long as a common divisor exists.
- A fraction can be rewritten with an imposed denominator only if that denominator is a multiple of the denominator of its simplest form: 18/24 can be written in quarters because its simplest form is 3/4, and 3/5 cannot be written in sevenths.