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The LibraryODERSA publishing house
An ODERSA resource · Knowledge programmeA book only goes online once it is whole and read by someone else.

Chapter 10 of 14 · Fractions

Chapter 9. Solving problems with fractions

Recognising the meaning at work, estimating before calculating, then checking: the method that makes the eight previous chapters usable.

Knowing how to calculate is not enough

Many people know how to add two fractions and are stopped dead by a three-line problem. This is not a problem of calculation: it is a problem of translation. A problem describes a situation, and you have to decide what the fractions are doing in it before writing anything at all. This chapter settles that decision, which is the real skill of the subject.

Three questions are enough, and they are always asked in the same order. Which whole are we talking about? What is the fraction doing in this situation? What order of magnitude must the result have? Once those three answers are written down, the calculation is almost always obvious, and above all it can be checked.

This chapter calls for no new notion. It gathers everything that has come before, and that is why it comes last.

What this chapter makes you able to do

Four gestures, to be done in this order in front of any problem.

  • Name the reference whole of a problem before calculating.
  • Recognise which of the three uses of a fraction is at work.
  • Estimate the result before calculating it.
  • Check a result by setting it against the estimate made at the start.

What has to have been worked before this chapter

The three uses of a fraction in a problem

The first use is that of the part of a whole: the fraction describes a portion of an object, of a collection or of a quantity. The signs of it are phrases such as half the field, three quarters of the pupils, two fifths of the stock. That is the meaning of chapter 1, and the reference whole is named just after the word of.

The second use is that of measurement: the fraction states a length, a duration, a capacity, expressed in a unit that had to be shared because whole numbers were not enough. The signs of it are units: three quarters of an hour, five eighths of a metre, two thirds of a litre. Such quantities are added and subtracted like numbers, and that is the meaning of chapters 2 and 6.

The third use is that of the operator: the fraction acts on a number in order to produce another one. The signs of it are the same as those of the first use, but the question asked bears on a counted quantity: how many marbles, how many euros, how many metres. That is the meaning of chapter 7, and the calculation is done in two steps.

One and the same problem can mix two uses, and that is the case with problems in several stages. The method does not change: we deal with one stage at a time, renaming the reference whole each time, since it may have changed along the way.

The words of this chapter

Two words of method, used everywhere in this book.

Estimate
Give an order of magnitude of the result before calculating it, leaning on simple benchmarks such as 1/2 and 1. Estimating is not guessing: it is bracketing the result between two bounds you are sure of.
Check
Set the result obtained against the estimate made at the start. A result outside the announced bounds is wrong, and there is no need to do the calculation again in order to know it.

Worked example. A problem in two stages

The problem: a water store holds 60 litres of water. Two fifths of it are used in the morning, then a quarter of what is left in the afternoon. How many litres are left in the evening?

  1. Name the whole of the first stage
    In the morning, the whole is the full store, that is to say 60 litres. It is on that number that the two fifths act.
  2. Estimate before calculating
    Two fifths is a little less than half: a little less than 30 litres will therefore be used in the morning, and a little more than 30 litres will be left. In the afternoon a quarter of that remainder is taken away, so about 8 litres. The final result must come out around 25 litres.
  3. Calculate the first stage
    60 shared into five gives 12, and two times 12 is 24. 24 litres are used in the morning, and 60 minus 24 are left, that is to say 36 litres.
  4. Rename the whole of the second stage
    In the afternoon, the whole is no longer the full store: it is what is left, that is to say 36 litres. This is the point where most mistakes happen, because the whole has changed without the problem saying so twice.
  5. Calculate the second stage, then check
    A quarter of 36 is worth 9 litres. So 36 minus 9 are left, that is to say 27 litres in the evening. This result is close to the 25 litres estimated, and it is smaller than the morning remainder: the check is passed.

Estimating, and why it saves time

An estimate takes a few seconds and catches most mistakes. It rests on three benchmarks already settled: is a fraction smaller or larger than 1/2, smaller or larger than 1, and what happens when it is applied to a quantity. With those three readings, almost every result can be bracketed before being calculated.

Take an addition: 5/8 + 3/5. Each of the two fractions goes past 1/2, so their sum goes past 1. Each is smaller than 1, so their sum is smaller than 2. The result must therefore fall between 1 and 2, and any result outside that range is wrong. This check costs nothing.

Take a subtraction in a problem: if three quarters of a quantity are taken away, one quarter of it is left, therefore far less than half. A result close to the starting quantity points to a mistake of meaning, not a mistake of calculation, and the remedy is to read the problem again rather than to count again.

Do the exercise along the way below before turning the page; its answer is at the end of the volume.

Exercise along the way, chapter 9. Bracketing without calculating

Set in the course of the chapter: answer without setting down the calculation.

  1. Between which whole numbers must the result of 7/6 + 3/4 fall? First break 7/6 down into a whole number plus a fraction, then place what is left in relation to 1.

The costliest mistake of meaning: keeping the same reference whole from one stage to the next. In a problem where a fraction is taken away, then a fraction of the remainder, the second fraction bears on the remainder and not on the starting whole. A simple way of not getting lost consists in writing, at each stage, the sentence the whole is now, followed by the quantity concerned.

Exercise 9.1. Recognising the meaning at work

For each problem, say which of the three uses is at work and name the reference whole. Do not calculate.

  1. A garden of 240 square metres is planted over two thirds of its area. Which use of the fraction is at work, and what is the whole?
  2. One plank measures 5/8 of a metre, another 3/4 of a metre. Which use is at work, and what is going to be done with these two fractions?
  3. Out of the 30 people signed up for a workshop, three fifths came. Which use is at work, and what will the nature of the result be?

Exercise 9.2. Estimating before calculating

First give a bracket for the result without calculating, then calculate and compare.

  1. Between which whole numbers does 5/6 + 7/8 fall? Then work out the exact result.
  2. An 80 litre tank is filled to three quarters. Estimate, then work out the quantity of water it holds.
  3. A person covers 2/5 of a journey of 45 kilometres. Estimate, then work out the distance covered.

Exercise 9.3. Problems in several stages

This exercise crosses the whole book. Write down at each stage what the reference whole is.

  1. A bag holds 36 marbles. A third of them are given away, then half of what is left. How many marbles are left?
  2. A 120 litre vat is filled to two thirds; 1/4 of its total capacity is then added. Does the vat overflow?
  3. Two workshops share an order between them: the first makes 3/8 of the order, the second 2/5. What fraction of the order is left to be made?

Self-check, chapter 9

Answer without turning back, then compare with the answer.

  1. What are the three questions to ask in front of a problem, and in which order?
  2. In a problem in two stages, what does the fraction of the second stage bear on?
  3. What is an estimate for, if the exact calculation has to be done anyway?

What to remember from this chapter

  • In front of a problem, the reference whole is named before a calculation is written.
  • A fraction can be a part of a whole, a measurement, or an operator acting on a quantity.
  • An estimate made before the calculation brackets the result and catches most mistakes.
  • In a problem in several stages, the reference whole changes along the way.
  • A result outside the announced bracket is wrong, and that is known without doing the calculation again.