Answers to the exercises

The fraction is 4/6, and it reads four sixths. The row of the table: 4/6; four sixths; the whole is shared into six equal parts, four of them are taken.

We are looking at whether the word third applies to this piece. We use the condition that the parts be equal, which is the only condition required by the definition of a fraction: a third is one of the three parts of a sharing into three equal parts. Here the three pieces are not the same size, so the sharing is not into equal parts and none of the pieces can be called a third. The answer is no: the loaf has been cut into three pieces, which is not the same thing as shared into three thirds. The big piece is indeed a part of the loaf, but no fraction points to it as long as there is no equal sharing.

We are looking for the number of parts and their name. We use the fact that folding produces parts that lie exactly on top of one another, therefore equal parts. The first fold gives two equal parts; folding each of them in two gives two times two parts, that is to say four parts, and they all lie exactly on top of one another. The whole is the entire sheet, shared into four equal parts: each part is therefore a quarter of the sheet, which is written 1/4.

We are looking for the name of a part. We use the rule which says that the denominator names the part: a whole shared into six equal parts gives sixths. The problem states that the plots have the same area, so the condition of equality is met. One plot is one sixth of the garden, which is written 1/6. The reference whole is the whole garden, and that has to be said, because one sixth of the garden means something only in relation to it.

We are looking for the fraction that describes this taking. We use the two roles: the number of parts of the sharing goes under the bar, the number of parts taken goes above it. The sharing is into ten equal parts, so the denominator is 10 and each part is one tenth. Three of those parts are taken, so the numerator is 3. The answer is 3/10, which reads three tenths of the ribbon.

We are looking for the fraction eaten. We use the same pattern, noting first that the squares are identical: the sharing into twelve is therefore indeed a sharing into equal parts, and each square is one twelfth of the bar. Five squares eaten make five twelfths. The answer is 5/12. The reference whole is the entire bar, and not what is left of it.

We are looking for a fraction of an hour. We use the fact that the whole can be a duration just as well as an object: here the whole is the entire hour. It is shared into four equal parts, so each part is a quarter of an hour, which matches the fifteen minutes stated. Three of those parts give 3/4, that is to say three quarters of an hour, which is forty-five minutes.

We are looking for the whole, then the fraction. We use the rule that a fraction always relates to a named whole. The whole is here the entire class, that is to say twenty pupils: it is a collection, and nothing forbids a whole to be a collection rather than a single object. The sharing is therefore into twenty equal parts, each pupil being worth one twentieth of the class. Five absent pupils represent five of those parts, that is to say 5/20 of the class.

We are looking for the explanation of an apparent contradiction. We use the fact that a fraction does not state a quantity on its own, but a quantity in relation to a whole. Both people did indeed eat 1/2, but not 1/2 of the same whole: one of them had a large pizza, the other a small one. Both claims are therefore true at the same time, and there is no contradiction. This example shows why the first gesture in front of a fraction is to name the reference whole.

We are looking for two fractions of the same whole. We use the condition of equality, met here because the glasses are identical and filled to the brim: the litre is therefore indeed shared into five equal parts. One glass holds one fifth of the litre, which is written 1/5. Three glasses hold three of those parts, which is written 3/5. The reference whole is the litre poured out, and not the capacity of one glass.

We are looking for the condition that allows the word quarter. We use the definition of a fraction, which requires a sharing into equal parts. What has to be checked is therefore that the four pieces are exactly the same size, which folding or laying one on another makes it possible to control. Without that equality there are four pieces and not four quarters. We also have to know which whole is being talked about, failing which the word quarter points to no quantity at all.

We are looking for the role of each term. We use the two definitions of the chapter. The number written after the slash is the denominator: here 9, and it is the one that names the part, since the whole is shared into nine equal parts called ninths. The number written before the slash is the numerator: here 7, and it is the one that counts, since seven of those parts are taken. The fraction reads seven ninths.

We are looking for the reason for a possible gap between two identical writings. We use the role of the reference whole. A fraction states a quantity in relation to a whole: half a page of an exercise book and half a sports field are both written 1/2, and they do not cover the same surface. The writing is the same because the gesture is the same, but the wholes differ. That is why a problem must always name the whole the fraction relates to.

Six fits three times into twenty-three, since three times six is eighteen, and five are left. So 23/6 = 3 + 5/6, and the bracket is 3 < 23/6 < 4.

We are looking for a writing as a sum of identical parts. We use the fact that the denominator chooses the counting unit and that the numerator counts the repetitions. Here the unit is the fifth, and four of them are taken. The breakdown is written 4/5 = 1/5 + 1/5 + 1/5 + 1/5. It means that four fifths are worth one fifth repeated four times, which is exactly the meaning of the fraction.

We are looking for the same writing with another denominator. We use the same reasoning: the denominator is 8, so the counting unit is the eighth, and the numerator 3 says that three of them are taken. The breakdown is written 3/8 = 1/8 + 1/8 + 1/8. Three eighths are therefore worth one eighth repeated three times.

We are looking for the fraction matching a repetition described in words. We use the reverse reading of the breakdown: the counting unit is the tenth, so the denominator is 10; it is repeated nine times, so the numerator is 9. The answer is 9/10, which reads nine tenths. We note in passing that this fraction is smaller than 1, since ten tenths would be needed to make the whole.

We are looking for the position of three fractions in relation to the whole. We use the rule that compares the numerator with the denominator. For 5/6, the numerator is smaller than the denominator: not all the parts are taken, so the fraction is smaller than 1. For 6/6, the two numbers are equal: all the parts of the sharing are taken, so the fraction is worth exactly 1. For 9/6, the numerator goes past the denominator: more than one whole was needed in order to take nine sixths, so the fraction goes beyond 1.

We are looking for a number of parts. We use the fact that one whole is worth three thirds, since the sharing is into three equal parts. One whole strip therefore calls for three thirds, and two whole strips call for twice as many, that is to say six. The answer is six thirds, which is written 6/3. We check that it is consistent: 6/3 has a numerator double its denominator, so this fraction is indeed worth 2.

We are looking at whether the writing can be carried out on a single object. We use the reading by counting: 5/4 calls for five parts, each one being a quarter of a pie. But a single pie shared into quarters supplies only four of them. The answer is therefore no: a second pie, shared into quarters of the same size, is needed in order to take the fifth quarter. The writing 5/4 is perfectly legitimate, it simply describes a quantity that goes beyond the whole, and it is worth one entire pie plus one quarter.

We are looking for a breakdown into a whole number plus a fraction, then a bracket. We use the division of the numerator by the denominator. Three thirds make one whole: we therefore look for how many times 3 fits into 11. Three times 3 is 9, and 2 are left; four times 3 would be 12, which goes too far. So 11/3 = 3 + 2/3. Since the remaining fraction is smaller than 1, the number is larger than 3 and smaller than 4, which gives the bracket 3 < 11/3 < 4.

We are looking for the same thing with another denominator. We use the same division: eight eighths make one whole, so we look for how many times 8 fits into 25. Three times 8 is 24, and 1 is left; four times 8 would be 32, which goes too far. So 25/8 = 3 + 1/8, and the bracket is written 3 < 25/8 < 4. We note that this number is very close to 3, since it goes beyond the whole by one single eighth.

We are looking at what the method gives in a particular case. We use the same division: four times 5 is exactly 20, and nothing is left. The breakdown therefore gives 4 plus zero fifths, that is to say 4 and nothing more. The fraction 20/5 is a whole number, and there is no point in bracketing it between two consecutive whole numbers, since it is one of them. The strict bracket does not exist here, and it is the only case where the method does not produce two bounds.

We are looking for the value of such a fraction. We use the definition of the sharing: the denominator says into how many parts the whole is shared, and the numerator how many of them are taken. When the two numbers are equal, all the parts of the sharing are taken, so the entire whole is taken back. Such a fraction is therefore worth 1, whatever its denominator: 3/3, 8/8 and 100/100 are all worth 1.

We are looking for the operation that produces the bracket. We use the fact that the denominator of parts makes one whole: looking for how many wholes fit inside the fraction amounts to looking for how many times the denominator fits into the numerator. It is therefore the division of the numerator by the denominator. Its quotient gives the whole number of wholes, and its remainder gives the parts that go beyond, which delivers both the breakdown and the bracket.

We are looking for the sense of this expression. We use the breakdown of a fraction into a sum of unit fractions. The denominator fixes into how many parts the whole is shared, therefore the size of the part: with 5, we count in fifths; with 10, in tenths. Every quantity in the problem is then said in that unit, exactly as we count in kilograms or in litres. The numerator, for its part, does nothing but count how many of those units are taken.

We count in quarters: 1/4 is placed one part after 0, and 5/4 is worth 4/4 + 1/4, therefore one part after 1. The gap between 1/4 and 1 is worth three quarters; the gap between 5/4 and 1 is worth one quarter: 5/4 is the closer to 1.

We are looking for a drawing and a number of marks. We use the rule of the chapter: obtaining parts between two whole numbers calls for one mark fewer than there are parts, since the two ends are already there. To obtain five parts, four marks are therefore needed between 0 and 1. The point 3/5 is placed by moving forward three parts from 0, so on the third mark. On the drawing, it falls beyond the middle of the interval, since three parts out of five go past half the way.

We are looking for the position of the point. We use the breakdown before drawing: four quarters make one whole, so seven quarters are worth 1 + 3/4. The point therefore lies between 1 and 2, three quarters of the way along the interval. On the drawing, each unit is shared into four equal parts by three marks, and we move forward seven parts from 0: the first four lead to 1, the next three to the point looked for.

We are looking to compare two distances to the point 1. We use the fact that with an equal denominator the parts are the same size, so it is enough to count. To reach 1, ten tenths are needed. From 4/10, six tenths are missing; from 9/10, only one is missing. The answer is therefore 9/10, which is the closer to 1. On the drawing, it is placed on the ninth mark after 0, just before the point 1.

We are looking for the fraction matching a position. We use the reverse reading of the drawing: the number of parts of the sharing gives the denominator, the number of parts travelled gives the numerator. The unit is shared into eight, so each part is one eighth. The point is reached after five parts, so it points to 5/8. On the drawing, this point lies just beyond the middle of the interval, since five parts out of eight go past half the way.

We are looking for the fraction of a point lying beyond 1. We use the breakdown into a whole number plus a fraction, read backwards. The point is worth two wholes plus two thirds, which is written 2 + 2/3. To say it as a single fraction, we convert the wholes into thirds: two wholes are worth six thirds, to which the two thirds of the overshoot are added, that is to say eight thirds. The answer is 8/3, and we check that this number is indeed between 2 and 3.

We are looking for the writing of the number 1 in sixths. We use the rule which says that a fraction is worth the whole when all the parts of the sharing are taken. The unit is shared into six parts, so six parts have to be travelled in order to reach 1. The answer is 6/6. It is one writing among others of the same number: the point 1 carries 1, 6/6 or 12/12 just as well, which will be worked in the next chapter.

We are looking for a fraction lying strictly between two numbers. We use the drawing: on a number line marked in quarters, the point 1/2 falls on the second mark and the point 1 on the fourth, so the third mark, which carries 3/4, lies between the two. The method is general: to find a number between two fractions, we share the unit more finely and take an intermediate graduation. Other answers work, such as 5/8 or 7/10, and it is precisely the fact that there is an endless number of them that sets fractions apart from whole numbers. The next chapter will explain why 1/2 and 2/4 point to the same point.

We are looking to compare two distances to 0. We use the second break of the chapter: with an equal numerator, a large denominator gives small parts. Sharing the unit into a hundred gives tiny parts, and only one of them is taken: the point is therefore almost stuck to 0. Sharing the unit into two gives very big parts, and one of them alone already leads halfway to 1. The answer is 1/100, and no calculation was needed.

We are looking for the nature of a mistake. We use the role of the denominator, which is not a quantity but a number of parts. The pupil compares the denominators as though they were two quantities, whereas a larger denominator means a finer sharing, therefore smaller parts. With the same numerator 1, one part is taken in both cases: the one from the sharing into nine is smaller than the one from the sharing into two. What has to be looked at is therefore the size of the part, and not the value of the number written underneath. The answer is that 1/9 is smaller than 1/2.

We are looking for a number of marks. We use the rule of the drawing: the points 0 and 1 are already set down, so there is always one intermediate mark fewer than there are parts to obtain. To obtain seven equal parts, six marks are therefore needed between 0 and 1. Drawing seven of them is a frequent mistake, which would give eight parts and eighths instead of sevenths.

We are looking for the check that protects against a faulty placing. We use the breakdown of chapter 2: the question is how many whole ones fit inside this fraction. It is answered by dividing the numerator by the denominator, and it gives at once the interval where the point must fall. For 3/4, no whole one fits, so the point lies between 0 and 1, and not between 3 and 4.

We are looking for one of the three breaks of the chapter. We use any one of them. A number no longer has a next one: after 1/2, no number immediately above it can be named, since between 1/2 and 3/4 there still lies 5/8, and so on without end. The two other acceptable answers are that the number written with the largest digits is not necessarily the largest, as 1/100 compared with 1/2, and that multiplying no longer always makes things bigger.

To go from 3 to 12, we multiply by 4. The numerator follows: 2 times 4 is 8. So 2/3 = 8/12.

We are looking for three writings of the same number. We use the rule of multiplying both terms by one and the same number other than zero. Multiplying by 2 gives 4/10; by 3, 6/15; by 4, 8/20. The three fractions 4/10, 6/15 and 8/20 are equivalent to 2/5 and point to the same point on a number line. Other answers work, since there is an endless number of writings: what matters is that the same factor be applied top and bottom.

We are looking for the numerator that goes with an imposed denominator. We use the question of the factor: what does 4 have to be multiplied by in order to obtain 20? By 5, since five times 4 is 20. The same factor applies to the numerator: one times 5 gives 5. The answer is 5/20. The move is possible because 20 is a multiple of 4.

We are looking for the same thing with a factor of ten. We use the same method: to go from 10 to 100, we multiply by 10, since ten times 10 is 100. The numerator follows: three times 10 gives 30. The answer is 30/100. This kind of conversion will serve throughout chapter 8, where the denominators are ten, one hundred or one thousand.

We are looking for the simplest form. We use the division of both terms by a common divisor, repeated for as long as it is possible. Both numbers are even, so we divide by 2: 8 divided by 2 gives 4, and 12 divided by 2 gives 6, that is 4/6. Both are still even, so we divide by 2 again: we obtain 2/3. Finally, 2 and 3 have no common divisor other than 1, so the simplest form is 2/3.

We are looking for the simplest form. We use the common divisors that are easy to spot: the two numbers end in 0 and in 5, so they both divide by 5. We obtain 6 and 9, that is 6/9. Those two numbers divide by 3, since 6 is worth two times 3 and 9 is worth three times 3: we obtain 2/3. No common divisor remains, so the simplest form is 2/3. We could have gone faster by dividing directly by 15, which gives the same result.

We are looking for the simplest form and for its nature. We use the division of both terms by 9, since 27 is worth three times 9. We obtain a numerator 3 and a denominator 1, that is 3/1. A denominator equal to 1 means that the whole has not been shared at all: the fraction is therefore worth the whole number 3. This result is a reminder that certain fractions are whole numbers in disguise, something already met in chapter 2 with 20/5.

We are looking to compare two writings. We use simplifying, which brings each fraction back to its simplest form: two numbers are equal if and only if those forms coincide. Dividing 6 and 9 by 3 gives 2/3. Dividing 4 and 6 by 2 also gives 2/3. The two fractions therefore point to the same number, and they fall at the same point on a number line. The answer is yes.

We are looking for the same comparison. We use simplifying: 4/8 divides by 4 and gives 1/2. The fraction 3/7 does not simplify, 7 having no divisors other than 1 and itself. It remains to be seen whether 3/7 is worth 1/2: for that, two times 3 would have to make 7, but two times 3 is 6, which does not reach 7. The fraction 3/7 is therefore slightly smaller than 1/2, and the answer is no: the two writings do not point to the same number.

We are looking for the nature of the mistake and for a way of making it visible. We use the distinction between multiplying and adding: only the multiplication of both terms keeps the value, because it cuts each part into pieces without taking anything away or adding anything. Adding the same number top and bottom matches no cutting at all, so it keeps nothing. To show it, we place the two fractions on a number line marked in fifteenths: 2/3 is worth 10/15 and 4/5 is worth 12/15, therefore two distinct points, two fifteenths apart. The answer is that the equality written down is false.

We are looking for the rule of the chapter. We use the idea that the value is kept when the sharing changes without the quantity moving. The two gestures allowed are therefore to multiply the numerator and the denominator by one and the same whole number other than zero, and to divide them by one and the same whole number other than zero when the division comes out exactly on both sides. No other gesture keeps the value, and in particular adding the same number to both terms changes it.

We are looking for the stopping test of simplifying. We use the definition of the simplest form: it is the one that no common division can shorten any further. So we know we have arrived when the numerator and the denominator no longer have any common divisor other than 1. In practice, we try the small divisors in order, 2, then 3, then 5, and we stop when none of them divides both numbers at once.

We are looking at whether a rewriting with an imposed denominator is possible. We use the condition of the chapter: the denominator aimed at must be a multiple of the starting denominator, because the only gesture allowed is the multiplication of both terms by one and the same whole number. But 7 is not a multiple of 5: no whole number multiplied by 5 gives 7. The answer is no, there is no whole number of sevenths worth three fifths.

In twenty-fourths: 2/3 = 16/24, 3/4 = 18/24 and 5/8 = 15/24. The numerators arrange as 15, 16, 18, so 5/8 < 2/3 < 3/4. The answer is given with the starting writings, as the method requires.

We are looking for the larger of the two. We use the shortcut of equal denominators: both fractions count ninths, therefore parts of exactly the same size. The comparison then comes down to that of the numerators, as for whole numbers. Seven parts go past four parts, so 7/9 is larger than 4/9. No common denominator has to be looked for, it is already there.

We are looking for the larger of the two. We use the shortcut of equal numerators: two parts are taken in both cases, but the parts are not the same size. The sharing into five gives bigger parts than the sharing into eleven, since the larger the denominator, the finer the sharing. Two big parts go past two small parts, so 2/5 is larger than 2/11.

We are looking for the larger one, then for a position in relation to 1. We first use the shortcut of equal denominators: both count sevenths, so 8/7 is larger than 6/7. We then use the comparison of the numerator with the denominator: in 8/7 the numerator goes past the denominator, so this fraction goes beyond the whole and is worth 1 + 1/7; in 6/7 the numerator is the smaller, so it stays below 1. It is therefore 8/7 that goes beyond the whole.

We are looking to classify three fractions in relation to the whole. We use the comparison of the numerator with the denominator, every denominator here being 8. In 3/8, the numerator is smaller than 8, so the fraction is smaller than 1. In 8/8, the two numbers are equal, so the fraction is worth exactly 1. In 9/8, the numerator goes past 8, so the fraction goes beyond 1 and is worth 1 + 1/8.

We are looking for the larger one without putting them over the same denominator. We use the benchmark 1/2, by comparing double the numerator with the denominator. For 5/12, two times 5 is 10, which does not reach 12: the fraction is therefore smaller than 1/2. For 7/10, two times 7 is 14, which goes past 10: the fraction is therefore larger than 1/2. The two fractions lie on either side of the benchmark, so 7/10 is larger than 5/12, and no shared calculation was needed.

We are looking for a threshold. We use the same test: the fraction goes beyond 1/2 when double the numerator goes past the denominator. With 7, the double is 14, therefore exactly the denominator: the fraction 7/14 is worth precisely 1/2 and does not go beyond it. With 8, the double is 16, which goes past 14: the fraction 8/14 is therefore larger than 1/2. The answer is 8.

We are looking for the larger of two fractions that no shortcut separates: the denominators and the numerators differ, and both fractions go past 1/2 without reaching 1. We use the general method. The product of the denominators, 28, works as a common denominator. To go from 4 to 28, we multiply by 7: 3/4 becomes 21/28. To go from 7 to 28, we multiply by 4: 5/7 becomes 20/28. The parts are now the same size: 21 twenty-eighths go past 20 twenty-eighths, so 3/4 is larger than 5/7.

We are looking for an arrangement, therefore for a comparison of three fractions at once. We use one single common denominator for the whole list: 40 works, since it is a multiple of 2, of 8 and of 5. We rewrite each one: 1/2 becomes 20/40 by multiplying by 20, 5/8 becomes 25/40 by multiplying by 5, and 3/5 becomes 24/40 by multiplying by 8. Arranging the numerators gives 20, then 24, then 25. The increasing order is therefore written with the starting fractions: 1/2, then 3/5, then 5/8.

We are looking for the larger of two fractions. The reference whole is the same for both workshops, since the problem states that the reels have the same length: comparing the fractions is therefore enough to answer. We use the general method: a common denominator for 6 and 10 is 30, which is a multiple of both. To go from 6 to 30, we multiply by 5: 5/6 becomes 25/30. To go from 10 to 30, we multiply by 3: 9/10 becomes 27/30. Twenty-seven thirtieths go past twenty-five thirtieths, so the second workshop used more.

We are looking for the rule of the second shortcut. We use the link between the denominator and the size of the part: the larger the denominator, the finer the sharing, and therefore the smaller the part. With equal numerators, the same number of parts is taken in both fractions; the one whose parts are the bigger therefore wins, that is to say the one whose denominator is the smaller. So 3/5 is larger than 3/8.

We are looking for the recommended order of work. We use the principle that a calculation avoided is a calculation got right. The first question is therefore whether a shortcut applies: are the denominators equal, are the numerators equal, or do the two fractions lie on either side of a benchmark such as 1 or 1/2. We bring out the general method, that is to say the putting over the same denominator, only if none of those three shortcuts decides.

We are looking for a method that always works. We use the fact that the product of two numbers is a multiple of each of them. It is therefore enough to multiply the two denominators together: the result works as a common denominator in every case. It is not always the smallest possible, which gives larger numbers and sometimes a result to be simplified, but it saves a search that comes to nothing.

With the smallest common multiple, 12: 3/4 = 9/12 and 1/6 = 2/12, so 9/12 + 2/12 = 11/12. With the product of the denominators, 24: 3/4 = 18/24 and 1/6 = 4/24, so 22/24, which simplifies by 2 into 11/12. Both roads give the same number, only the working writing changes.

We are looking for a sum. We use the rule of equal denominators: both fractions count eighths, therefore parts of the same size, and it is enough to count how many there are in all. Three eighths plus four eighths make seven eighths, which is written 3/8 + 4/8 = 7/8. The denominator does not move, since the size of the parts has not changed. The result does not simplify, 7 having no divisors other than 1 and itself.

We are looking for a difference. We use the same rule, applied to subtraction: tenths are taken away from tenths, so we subtract the numerators and keep the denominator. Seven tenths minus three tenths make four tenths, that is 4/10. This result simplifies: dividing both terms by 2 gives 2/5, which is the simplest form.

We are looking for a sum and for its reading. We use the rule of equal denominators: five sixths plus one sixth make six sixths, that is 6/6. We then use the rule of chapter 2: when the numerator equals the denominator, all the parts of the sharing are taken, so the fraction is worth the whole. The result is therefore worth exactly 1. It is the simplest example of an addition whose result deserves to be read rather than left as it stands.

We are looking for a sum whose denominators differ. We first use the remark that saves work: 8 is already a multiple of 2, so it is enough to rewrite the first fraction in eighths. To go from 2 to 8, we multiply by 4, so 1/2 becomes 4/8. The calculation becomes 4/8 + 3/8, that is to say an addition over an equal denominator, which gives 7/8. Quick check: 1/2 is already half, and a little less than half is added, so the result must be close to 1 without reaching it, which is the case.

We are looking for a sum with no obvious common denominator. We use the product of the denominators: five times three is 15, which is a multiple of both. To go from 5 to 15, we multiply by 3: 2/5 becomes 6/15. To go from 3 to 15, we multiply by 5: 1/3 becomes 5/15. The addition then gives 6/15 + 5/15 = 11/15. The result does not simplify, and it is smaller than 1, which is consistent since both starting fractions were smaller than 1/2.

We are looking for a difference with different denominators. We use the putting over the same denominator, exactly as for an addition: a common multiple of 6 and 4 is 12. To go from 6 to 12, we multiply by 2: 5/6 becomes 10/12. To go from 4 to 12, we multiply by 3: 1/4 becomes 3/12. The subtraction gives 10/12 minus 3/12, that is 7/12. The result does not simplify, and it stays larger than 1/2, which is plausible since we started from 5/6 and took away a quarter.

We are looking for a sum, then for its breakdown. We use the rule of equal denominators: five sixths plus four sixths make nine sixths, that is 9/6. We then use the division of chapter 2: six sixths make one whole, and three sixths are left. So 9/6 = 1 + 3/6. The remaining fraction simplifies by dividing both terms by 3, which gives 1 + 1/2. The result is therefore worth one whole and a half.

We are looking for a sum and for its bracket. We use the rule of equal denominators: three quarters plus three quarters make six quarters, that is 6/4. We then use the breakdown: four quarters make one whole, and two quarters are left, so 6/4 = 1 + 2/4, that is to say 1 + 1/2 after simplifying. The result is therefore larger than 1 and smaller than 2, which is written 1 < 6/4 < 2.

We are looking to compare a sum with the whole. We use the putting over the same denominator, the reference whole being the capacity of the barrel. A common multiple of 5 and 4 is 20: 2/5 becomes 8/20 by multiplying by 4, and 3/4 becomes 15/20 by multiplying by 5. The sum is worth 8/20 + 15/20, that is 23/20. Since the numerator goes past the denominator, this quantity goes beyond the whole: the barrel overflows, and the overflow is worth 3/20 of its capacity.

We are looking for the reason behind a rule. We use the meaning of the denominator, which is not a quantity but the statement of the size of the parts. Adding two fractions amounts to bringing parts together; if those parts are the same size, that size does not change because more of them are gathered. Only their number changes, so only the numerator moves. Adding the denominators would amount to claiming that bringing sixths together makes twelfths, which the figure denies at once.

We are looking for the opening gesture. We use the requirement of a shared counting unit: only parts of the same size can be counted together. The first gesture therefore consists in rewriting both fractions with the same denominator, by looking for a common multiple of the two denominators. Only then does the addition become a simple count of the numerators.

We are looking for a cheap check. We use the benchmarks 1/2 and 1. We place each fraction in relation to them, then deduce a range for the sum: two fractions each going past 1/2 give a sum going past 1, and two fractions smaller than 1 give a sum smaller than 2. A result outside that range is wrong, with no need to do the calculation again. It is this check that gets rid of the mistake of adding the denominators as well.

Two sevenths is smaller than 1: the result will be smaller than 63. Two sevenths is also smaller than half, since two times 2 is less than 7: the result will stay below half of 63. The calculation confirms it: 63 shared into seven gives 9, and two times 9 is 18, well below those bounds.

We are looking for a quantity, not a fraction. We use the two-step calculation: divide by the denominator, multiply by the numerator. The whole is here 48 apples. We divide by 4, which gives 12: a quarter of 48 is therefore worth 12. The numerator being 1, there is only one part to take, and the answer is 12 apples. Check: four times 12 does indeed make 48.

We are looking for a length. We use the two steps, beginning with the division since it comes out exactly: 40 shared into five gives 8, so one fifth is worth 8 metres. We then take three of those parts: three times 8 is 24. The answer is 24 metres. Check: three fifths go past half of 40, which is 20, and stay below 40, which is the case.

We are looking for a duration. We use the fact that a whole can be a duration, provided it is expressed in a unit that shares well: one hour is worth 60 minutes. We divide by 3, which gives 20: one third of an hour is worth 20 minutes. We take two of those parts: two times 20 is 40. The answer is 40 minutes. Check: two thirds go past half an hour, that is 30 minutes, which is the case.

We are looking for a quantity by an imposed road. We use the two steps in the order asked for: 100 shared into four gives 25, so a quarter of 100 is worth 25. We then take three of those parts: three times 25 is 75. The answer is 75. This road keeps the numbers small, since we never go beyond 100, whereas multiplying first would have taken us through 300.

We are looking to deal with a case where the division does not come out exactly. We use the freedom of order between the two steps. Dividing 10 by 3 does not give a whole number, so we begin by multiplying: two times 10 is 20. It then remains to share 20 into three equal parts, which does not come out exactly either. We then change the counting unit: each unit is worth three thirds, so 20 units are worth sixty thirds, and sharing sixty thirds into three parts gives twenty thirds per part, that is 20/3. Breaking it down, six times 3 is 18 and 2 are left, so 20/3 is worth 6 + 2/3. The answer is 20/3, that is to say six units and two thirds.

We are looking for the same result by two roads. We use both orders. First road, the division first: 24 shared into eight gives 3, then five times 3 is 15. Second road, the multiplication first: five times 24 is 120, then 120 shared into eight gives 15. Both roads give 15, which confirms that they are equivalent. The first never handled a number beyond 24, the second went through 120: that is why we prefer to divide first when the division comes out exactly.

We are looking for the product of a fraction by a whole number. We use the rule of the chapter: multiplying by a whole number amounts to repeating the fraction, so only the numerator is multiplied and the denominator stays unchanged. Four times 2/7 gives 8/7. We then use the comparison of the numerator with the denominator: 8 goes past 7, so the result goes beyond the whole. Breaking it down, 8/7 is worth 1 + 1/7.

We are looking for a product, for its breakdown and for its simplification. We first use the rule of the product: three times 5/6 gives 15/6, the denominator staying 6. We then use the division of chapter 2: two times 6 is 12, and 3 are left, so 15/6 = 2 + 3/6. We finally use the simplifying of chapter 4: 3 and 6 both divide by 3, so 3/6 is worth 1/2. The result is written 2 + 1/2, that is to say two units and a half.

We are looking for a total quantity. We use the rule of the product, the situation being a repetition: six dishes call for six times 3/4 of a litre, so 18/4 of a litre. We break it down: four times 4 is 16, and 2 are left, so 18/4 = 4 + 2/4. The remaining fraction simplifies into 1/2. The answer is four litres and a half. Check: each dish calls for a little less than one litre, so six dishes must call for a little less than six litres, which is the case.

We are looking for the method of the chapter. We use the direct translation of what a fraction means: we divide the quantity by the denominator in order to obtain one part, then we multiply by the numerator in order to take several of them. The two steps can be swapped without changing the result: we begin with the division when it comes out exactly, because the numbers stay small, and with the multiplication when it does not.

We are looking for the term affected. We use the meaning of the operation: multiplying by a whole number amounts to repeating the fraction several times, which increases the number of parts taken without changing their size. Only the numerator is therefore multiplied, and the denominator stays unchanged. So three times 2/5 gives 6/5, and not 6/15.

We are looking for the reason behind an effect contrary to the habit of whole numbers. We use the comparison of the fraction with 1: three quarters is smaller than 1, so only a part of the starting quantity is kept. Taking three quarters of 40 gives 30, which is smaller than 40. The general rule can be read with no calculation: a fraction smaller than 1 makes the quantity smaller, a fraction equal to 1 leaves it unchanged, and a fraction larger than 1 makes it bigger.

Seven tenths are worth seventy hundredths. So 7/10 + 4/100 = 70/100 + 4/100 = 74/100, that is to say seventy-four hundredths, which is written 0.74.

We are looking to sort three fractions according to their denominator. We use the definition: a decimal fraction has ten, one hundred, one thousand, and so on, as its denominator. The fraction 3/10 is one of them, since its denominator is ten. The fraction 25/100 is one of them too, its denominator being one hundred. The fraction 4/7 is not, since 7 is neither ten, nor one hundred, nor one thousand, and none of those numbers is a multiple of 7: it will therefore never be rewritten in tenths or in hundredths.

We are looking for the decimal notation of a decimal fraction. We use the correspondence between the denominator and the place value: a denominator of ten points to tenths, therefore to the first place to the right of the decimal point. Nine tenths are therefore written 0.9. The zero written before the decimal point says that there is no whole unit, which is consistent since 9/10 is smaller than 1.

We are looking for the road back. We use the place value of the last digit written: the 3 occupies the second place to the right of the decimal point, that is to say the hundredths. The number is therefore worth sixty-three hundredths, which is written 63/100. We check the sense: 0.63 is smaller than 1 and a little larger than half, and 63/100 does indeed go past 50/100.

We are looking for a rewriting with an imposed denominator. We use the rule of chapter 4: to go from 10 to 100, we multiply by 10, so the numerator follows and 4 becomes 40. The answer is 40/100. The meaning can also be read directly: each tenth is worth ten hundredths, so four tenths are worth forty hundredths.

We are looking for the same conversion from one place value to the next. We use the same factor: to go from 100 to 1000, we multiply by 10, so the numerator goes from 7 to 70. The answer is 70/1000. The meaning is the same as before, one place further along: each hundredth is worth ten thousandths, so seven hundredths are worth seventy thousandths.

We are looking for a total expressed in a single place value. We use the conversion of tenths into hundredths: three tenths are worth thirty hundredths, since each tenth is worth ten hundredths. We then add the two hundredths already there, which gives thirty-two hundredths, that is 32/100. In decimal notation, that is written 0.32.

We are looking for a sum of two decimal fractions in different place values. We use the bringing to the same place value: five tenths are worth fifty hundredths, so 5/10 becomes 50/100. The addition is then made over an equal denominator: 50/100 + 30/100 = 80/100. In decimal notation, the result is worth 0.8, and it can also be written 8/10 by going back to the tenths.

We are looking for a sum and then for its breakdown. We use the bringing to the same place value: seven tenths are worth seventy hundredths, so 7/10 becomes 70/100. The addition gives 250/100 + 70/100 = 320/100. We then use the breakdown: one hundred hundredths make one unit, so three hundred hundredths make three units, and twenty hundredths are left. The result is written 3 + 20/100, that is to say 3.2 in decimal notation.

We are looking for a total quantity and for its comparison with the whole. We use the bringing to the same place value: four tenths are worth forty hundredths, so 4/10 becomes 40/100. The addition gives 75/100 + 40/100 = 115/100. Since the numerator goes past the denominator, the quantity goes beyond one litre. Breaking it down, one hundred hundredths make one unit and fifteen hundredths are left, so the total is worth 1 + 15/100, that is 1.15 litres.

We are looking for the test. We use the definition of the chapter: it is the denominator, and it alone, that decides. A fraction is decimal when its denominator is ten, one hundred, one thousand, and so on. The numerator does not enter into the test: 3/10 and 250/100 are both decimal, the second one being worth more than the whole.

We are looking for the role of the decimal point. We use the construction of the chapter: decimal notation extends the place value table to the right of the units, each place being worth ten times less than its neighbour on the left. The decimal point therefore marks the place of the units digit, by sitting just after it. It does not separate two independent numbers: in 3.18 there are three units, one tenth and eight hundredths, that is to say one single number.

We are looking for the limit of decimal notation. We use the condition for rewriting given in chapter 4: going from 3 to another denominator requires that denominator to be a multiple of 3. But neither ten, nor one hundred, nor one thousand is a multiple of 3, and the same holds for every power of ten. No whole number of tenths, of hundredths or of thousandths is therefore worth exactly one third, and that is why decimal notation covers only a part of the fractions.

7/6 is worth 1 + 1/6, so the result goes beyond 1. What is added to that whole number is worth 1/6 + 3/4: now 1/6 is smaller than 1/4, and 1/4 + 3/4 is worth exactly 1, so 1/6 + 3/4 stays below 1. The result therefore falls between 1 and 2.

We are looking to identify the use and the whole. We use the three cases of the chapter. The phrase planted over two thirds describes a portion of the whole: it is the use part of a whole, and the reference whole is the entire garden, that is to say 240 square metres. Since that whole is a counted one, the same fraction will become an operator as soon as the planted area in square metres is asked for, which would be worked out in two steps.

We are looking for the use and for the likely continuation of the problem. We use the signs given by the units: both fractions bear on metres, so they state measurements, and the reference whole is the unit of length itself, the metre. It is the use measurement. Two measurements in the same unit are added or compared like numbers, which will call for putting them over the same denominator.

We are looking for the use and for the nature of the result expected. We use the fact that the whole is a counted collection: the 30 people signed up. The fraction acts on that number in order to produce another one, so it is the use operator. The result will not be a fraction but a number of people, obtained in two steps: divide 30 by 5, then multiply by 3.

We are looking for a bracket and then for a result. We first use the benchmarks: two times 5 is 10, which goes past 6, so 5/6 is larger than 1/2; two times 7 is 14, which goes past 8, so 7/8 is larger than 1/2 as well. The sum therefore goes beyond 1. Each of them staying smaller than 1, the sum stays smaller than 2: the result falls between 1 and 2. For the exact calculation, a common multiple of 6 and 8 is 24: 5/6 becomes 20/24 and 7/8 becomes 21/24, so the sum is worth 41/24, that is to say 1 + 17/24. The announced bracket is confirmed.

We are looking for an estimate and then for a quantity. We use the benchmark 1/2 in order to estimate: three quarters go past half, so the result goes past 40 litres; three quarters stay below the whole, so the result stays below 80 litres. For the calculation, we apply the two steps: 80 shared into four gives 20, and three times 20 is 60. The tank holds 60 litres, which falls well inside the range announced.

We are looking for an estimate and then for a distance. We use the benchmark 1/2: two times 2 is 4, which does not reach 5, so 2/5 is smaller than 1/2 and the distance covered is less than half the journey. It is therefore less than half the journey, that is to say less than 22.5 kilometres, and it is not zero. For the calculation, we divide first since that comes out exactly: 45 shared into five gives 9, and two times 9 is 18. The distance covered is 18 kilometres, which respects the estimate.

We are looking for a final quantity in two stages. We use the rule of the whole that changes. First stage: the whole is the bag, that is to say 36 marbles, and a third of them are given away, so 36 shared into three, that is 12 marbles; 36 minus 12 are left, that is to say 24 marbles. Second stage: the whole is now what is left, that is to say 24 marbles, and half of them are given away, that is 12 marbles; 24 minus 12 are left, that is to say 12 marbles. The answer is 12 marbles. The mistake to avoid would have been to take half of the 36 marbles of the start.

We are looking to compare a total with the whole. We use here a whole that does not change, because the problem states that the quarter bears on the total capacity. First stage: two thirds of 120, that is 120 shared into three which gives 40, then two times 40 which is 80 litres. Second stage: a quarter of 120, that is 30 litres. The total is worth 80 plus 30, that is to say 110 litres, which stays below 120. The vat does not overflow, and there are even 10 litres of room left.

We are looking for a fraction that is left. We use the putting over the same denominator, then the subtraction from the whole, the reference whole being the entire order. A common multiple of 8 and 5 is 40: 3/8 becomes 15/40 and 2/5 becomes 16/40. Together, the two workshops have made 15/40 + 16/40, that is to say 31/40 of the order. The entire order is written 40/40, so 40/40 minus 31/40 are left, that is to say 9/40 of the order to be made.

We are looking for the method of the chapter. We use the announced order, which is not interchangeable. First, which whole are we talking about, since no fraction means anything without its whole. Next, what the fraction is doing in this situation, that is to say whether it is a part of a whole, a measurement or an operator. Finally, what order of magnitude the result must have, which supplies the check. The calculation comes only after those three answers.

We are looking for the whole of the second stage. We use the rule of the whole that changes along the way: unless the problem says otherwise, a fraction applied after a first operation bears on what is left, and not on the starting quantity. This is the point where most mistakes on problems happen. The remedy consists in writing, at each stage, the sentence the whole is now, followed by the quantity concerned, then in checking what the problem says exactly.

We are looking for the use of a gesture that seems redundant. We use the definition of the check: an estimate brackets the result between two safe bounds, before any calculation. It therefore serves to spot a wrong result without having to do the calculation again, which saves time rather than costing it. It also serves to tell a mistake of calculation from a mistake of meaning: a result far from the estimate generally signals that the wrong whole or the wrong operation was taken, and the remedy is then to read the problem again.