Chapter 6 of 14 · Fractions
Chapter 5. Comparing two fractions
Four methods of comparison, from the fastest to the most general, and the rule that says which one to use according to what you have in front of you.
Comparing without calculating, when that is possible
Comparing two fractions consists in saying which one is the larger, that is to say which one lies further to the right on the number line. There is a general method for this that always works, and three shortcuts that work only in certain cases. The shortcuts are not tricks: they are direct readings of the meaning of a fraction, and they call for less writing, so they make fewer mistakes.
The good habit consists in first looking at what you have in front of you, and in bringing out the general method only if no shortcut applies. A calculation avoided is a calculation got right.
This chapter assumes the previous one is held: the general method consists in rewriting both fractions with the same denominator, and that rewriting is exactly the gesture settled in chapter 4.
What this chapter makes you able to do
Five gestures: the methods of comparison, from the fastest to the most general, then their use on a list.
- Compare two fractions with the same denominator.
- Compare two fractions with the same numerator.
- Compare a fraction with 1, then with 1/2, so as to decide without calculating.
- Compare any two fractions by rewriting them with the same denominator.
- Arrange a short list of fractions in increasing order.
What has to have been worked before this chapter
The three direct readings
First case, the simplest: the two fractions have the same denominator. The parts are then the same size, and it is enough to count how many of them are taken. Between 3/7 and 5/7, the second is the larger, because five identical parts go past three identical parts. It is a comparison of whole numbers in disguise.
Second case: the two fractions have the same numerator. The same number of parts is taken, but the parts are of different sizes. The larger the denominator, the finer the sharing, and therefore the smaller the part. Between 3/5 and 3/8, the larger is 3/5: three big parts go past three small parts. This rule comes as a surprise, because it turns the order of the denominators around, and that is precisely why it deserves to be checked once on a strip of paper.
Third case: the two fractions lie on either side of a known benchmark. The most useful benchmarks are 1 and 1/2. A fraction is smaller than 1 when its numerator is smaller than its denominator; it is larger than 1 in the opposite case. It is larger than 1/2 when twice its numerator goes past its denominator, that is to say when twice this fraction goes past the whole. So 5/9 is larger than 1/2, since twice 5 is 10, which goes past 9; and 4/9 is smaller, since twice 4 is 8, which does not reach 9. Between 4/9 and 6/11, then, we decide without a single shared calculation.
Choosing your method by looking at the two writings
A shortcut is enough
- Same denominators: we compare the numerators.
- Same numerators: the larger denominator gives the smaller fraction.
- The two lie on either side of a benchmark, 1 or 1/2: the one that goes past the benchmark is the larger.
The general method is needed
- Nothing in common between the two writings, and both fractions on the same side of the benchmarks.
- The two fractions are close together, and a rough benchmark does not decide.
- More than two fractions have to be arranged: one single common denominator then serves the whole list.
Worked example. Comparing 5/6 and 7/9
The problem: say which of these two fractions is the larger, with justification.
- Look for a shortcut
The denominators differ, and so do the numerators. Both fractions are smaller than 1 and larger than 1/2. No shortcut decides: the general method is needed. - Find a common denominator
We look for a number that is a multiple of both 6 and 9. The multiples of 6 are 6, 12, 18, 24; the multiples of 9 are 9, 18, 27. The first one shared by both is 18. We shall therefore compare in eighteenths. - Rewrite the first fraction
To go from 6 to 18, we multiply by 3. The numerator follows: 5 times 3 gives 15. So 5/6 = 15/18. - Rewrite the second fraction
To go from 9 to 18, we multiply by 2. The numerator follows: 7 times 2 gives 14. So 7/9 = 14/18. - Conclude on the two comparable writings
The two fractions now count parts of the same size: 15 eighteenths against 14 eighteenths. So 5/6 is larger than 7/9, and the gap between the two is worth one single eighteenth, which explains why no rough benchmark was enough.
Arranging a list
Arranging several fractions in increasing order calls for the same method, applied once for the whole list: we look for a denominator that suits them all, we rewrite everybody, then we arrange the numerators. The result is then given with the starting writings, and not with the working writings, otherwise we answer a question other than the one asked.
A simple way of finding a common denominator for several fractions consists in multiplying the denominators together: the product is always a multiple of each one. That denominator is rarely the smallest possible, but it works every time, and a slightly heavier calculation is better than a search that comes to nothing.
Take the method up again on a fresh list: that is the exercise along the way below, and its answer is at the end of the volume.
Exercise along the way, chapter 5. Arranging a list of three fractions
Set in the course of the chapter: apply the method of arranging, once for the whole list.
- Arrange the list 2/3, 3/4 and 5/8 in increasing order, rewriting the three fractions with the denominator 24.
The dominant trap of the chapter: comparing the numerators with each other and the denominators with each other, as though they were two independent numbers. That leads to claiming that 3/8 goes past 2/3, because 3 goes past 2 and 8 goes past 3. The check that saves you fits in one sentence: place each fraction in relation to 1/2 before concluding. Here 3/8 is smaller than 1/2 and 2/3 is larger: the answer was therefore wrong before any calculation.
Exercise 5.1. Comparing by direct reading
Compare each pair, stating which of the three shortcuts you are using. No common denominator is needed here.
- Compare 4/9 and 7/9.
- Compare 2/5 and 2/11.
- Compare 8/7 and 6/7, then say which of the two goes beyond the whole.
Exercise 5.2. Comparing with 1 and with 1/2
For each question, first place each fraction in relation to the benchmark asked for, then conclude.
- Among 3/8, 9/8 and 8/8, which one is smaller than 1, which one is worth 1, which one goes beyond 1?
- Compare 5/12 and 7/10 using the benchmark 1/2.
- A fraction has 14 as its denominator. What is the smallest whole numerator that makes it larger than 1/2?
Exercise 5.3. General method and arranging
This exercise crosses this chapter with chapter 4. Write out the intermediate rewritings.
- Compare 3/4 and 5/7 by rewriting them with the same denominator.
- Arrange 1/2, 5/8 and 3/5 in increasing order.
- Two workshops have each used a reel of thread of the same length: the first used 5/6 of it, the second 9/10. Which one used more?
Self-check, chapter 5
Answer without turning back, then compare with the answer.
- With equal numerators, which fraction is the larger, and why?
- Which question do we ask ourselves first in front of two fractions to be compared?
- How can a common denominator for two fractions be obtained for certain, even without looking for the smallest one?
What to remember from this chapter
- With equal denominators, the larger fraction is the one whose numerator is the larger.
- With equal numerators, the larger fraction is the one whose denominator is the smaller, because its parts are bigger.
- The benchmarks 1 and 1/2 often decide with no calculation at all.
- The general method consists in rewriting both fractions with the same denominator, then comparing the numerators.
- The product of the denominators always gives a common denominator, even if it is not the smallest one.