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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 7 of 14 · Fractions

Chapter 6. Adding and subtracting

Only parts of the same size are added: this chapter settles addition over equal denominators, then the move to a common denominator.

Adding is bringing identical parts together

Two fifths plus one fifth make three fifths, and that sentence is understood with no rule at all: two parts are brought together with one part, all of them the same size, and three of them are obtained. Nothing has changed about the size of the parts, so the denominator does not move. Only the count of the parts changes, so only the numerator moves.

What is being added are fifths, exactly as kilograms are added to kilograms. Nobody would add three kilograms and two litres to obtain five of something: a shared quantity is needed first. Fractions obey the same requirement, and the whole difficulty of the chapter lies there.

The consequence can be guessed: when the denominators differ, nothing can be added until parts of the same size have been made. The method of chapter 4 then serves to rewrite both fractions in a shared counting unit, and the addition becomes a simple count again.

What this chapter makes you able to do

Four gestures, only one of which calls for a calculation beforehand.

  • Add and subtract two fractions with the same denominator.
  • Explain why the denominator is not added.
  • Put two fractions over the same denominator so that they can be added.
  • Add a fraction and a whole number.

What has to have been worked before this chapter

The case of equal denominators

When the two fractions have the same denominator, we add the numerators and keep the denominator. Three sevenths plus two sevenths make five sevenths, which is written 3/7 + 2/7 = 5/7. Subtraction follows exactly the same logic: five sevenths minus two sevenths make three sevenths.

Two results deserve to be looked at rather than merely written down. The first is the one where the total reaches the denominator: 5/6 + 1/6 gives 6/6, that is to say the whole, that is to say 1. The second is the one where the total goes beyond the denominator: 5/6 + 4/6 gives 9/6, a fraction greater than the whole, which is broken down where needed into 1 + 3/6, then simplified into 1 + 1/2. These two cases are not exceptions, they are ordinary additions whose result deserves to be read.

Adding a whole number and a fraction calls for nothing more. Two wholes plus three fifths are written 2 + 3/5, and that writing is already a result. Everything can also be said in fifths: two wholes are worth ten fifths, so the total is worth thirteen fifths.

Worked example. Working out 2/3 + 1/4

The problem: add these two fractions, and give the result in its simplest form.

  1. Note that direct addition is impossible
    The parts are not the same size: thirds on one side, quarters on the other. They cannot be counted together as long as they differ.
  2. Look for a shared counting unit
    A number that is a multiple of 3 and of 4 is needed. The product of the two denominators, 12, always works, and here it is also the smallest. We shall therefore count in twelfths.
  3. Rewrite both fractions in twelfths
    To go from 3 to 12, we multiply by 4: 2/3 becomes 8/12. To go from 4 to 12, we multiply by 3: 1/4 becomes 3/12.
  4. Add the parts, now comparable
    Eight twelfths plus three twelfths make eleven twelfths. The calculation is written 8/12 + 3/12 = 11/12, and the denominator does not move.
  5. Check the result by an order of magnitude
    2/3 goes past 1/2, and 1/4 is smaller than 1/2: the total must therefore go past 1/2 without reaching 1. Eleven twelfths is a little less than the whole: the result is plausible. Finally we check that it does not simplify, which is the case, 11 having no divisors other than 1 and itself.

Choosing a common denominator without a long search

Three ways of finding a common denominator live side by side, and they all give a correct result. The first consists in looking at whether one of the two denominators is already a multiple of the other: for 1/2 and 3/8, eight is a multiple of two, so everything is written in eighths and the work is almost done. The second consists in listing the multiples of each until a shared one is found. The third consists in multiplying the two denominators together, which always works.

The only drawback of the third method is that it sometimes gives large numbers, and therefore a result to be simplified at the end. That is not a fault: an exact result in a long writing is worth more than a wrong result in a short one. Simplifying is done afterwards, calmly, with the gesture of chapter 4.

Now carry a calculation through yourself: that is the exercise along the way below, and its answer is at the end of the volume.

Exercise along the way, chapter 6. The calculation 3/4 + 1/6

Set in the course of the chapter: choose your common denominator yourself.

  1. Work out 3/4 + 1/6 by choosing your common denominator, then compare what the product of the denominators and the smallest common multiple give: the final result must be the same.

The most common form of the mistake announced in chapter 1 happens here: adding the numerators with each other and the denominators with each other, which would give 1/2 + 1/3 equals 2/5. The check that gets rid of it fits in one sentence: 1/2 is already half the whole, so adding something must go past the half. Yet 2/5 is smaller than 1/2. The result was therefore wrong before being checked in detail, and that reading takes two seconds.

Exercise 6.1. Adding and subtracting over an equal denominator

Work these out, and write the result in its simplest form where that is possible.

  1. Work out 3/8 + 4/8.
  2. Work out 7/10 minus 3/10.
  3. Work out 5/6 + 1/6, and say what that result is worth.

Exercise 6.2. Putting over the same denominator

Write out the rewriting of each fraction before calculating. The reasoning counts as much as the result.

  1. Work out 1/2 + 3/8.
  2. Work out 2/5 + 1/3.
  3. Work out 5/6 minus 1/4.

Exercise 6.3. Additions that go beyond the whole

This exercise crosses this chapter with chapter 2: break down the results that are greater than the whole.

  1. Work out 5/6 + 4/6, then break the result down into a whole number plus a fraction smaller than the whole.
  2. Work out 3/4 + 3/4, and say between which whole numbers the result lies.
  3. A barrel is filled to 2/5 of its capacity, then the equivalent of 3/4 of that capacity is poured into it. Does the barrel overflow?

Self-check, chapter 6

Answer without turning back, then compare with the answer.

  1. Why is the denominator not added when two fractions are added?
  2. What is the first gesture in front of an addition of fractions whose denominators differ?
  3. Which quick check makes it possible to spot an addition result that is plainly wrong?

What to remember from this chapter

  • Only parts of the same size are added: the denominator must be the same before any calculation.
  • With an equal denominator, we add the numerators and keep the denominator.
  • When the denominators differ, we rewrite both fractions with a common denominator, then we add.
  • The product of the denominators always gives a common denominator, at the cost of simplifying the result afterwards.
  • An order of magnitude taken before the calculation spots most mistakes without having to start again.