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Matemáticas

Adding and Subtracting Fractions, Up to Borrowing in Mixed Numbers

Twelve questions from same-denominator sums to mixed numbers needing borrowing, each answer showing the common denominator chosen and the final simplification.

La respuesta corta

Fractions can only be added or subtracted once they name the same size of piece, so rewrite both over a common denominator — usually the LCM of the two — then work with the numerators only. 3/4 − 1/6 becomes 9/12 − 2/12 = 7/12.

Los ejercicios

  1. Ejercicio 1Básico1 puntos

    Work out 3/7 + 2/7.

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    5/7

    1. The denominators already match, so the pieces are the same size.
    2. Add the numerators only: 3 + 2 = 5.
    3. The denominator stays as 7.
    4. 5/7, which will not simplify since 5 and 7 share no factor.
  2. Ejercicio 2Básico2 puntos

    Work out 5/9 + 7/9, giving your answer as a mixed number in its simplest form.

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    1 1/3

    1. 5 + 7 = 12, so the answer is 12/9.
    2. 12/9 is more than one whole: 9/9 + 3/9.
    3. That is 1 3/9.
    4. 3/9 simplifies by 3 to 1/3, giving 1 1/3.
  3. Ejercicio 3Básico2 puntos

    Work out 1/2 + 1/3.

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    5/6

    1. 2 and 3 have LCM 6.
    2. 1/2 = 3/6 (multiply both parts by 3).
    3. 1/3 = 2/6 (multiply both parts by 2).
    4. 3/6 + 2/6 = 5/6, already in lowest terms.
  4. Ejercicio 4Básico2 puntos

    Work out 3/4 − 1/6.

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    7/12

    1. 4 = 2² and 6 = 2 × 3, so the LCM is 12.
    2. 3/4 = 9/12 (× 3) and 1/6 = 2/12 (× 2).
    3. 9/12 − 2/12 = 7/12.
    4. 7 and 12 share no factor, so it is finished.
  5. Ejercicio 5Estándar3 puntos

    Work out 5/6 + 3/8.

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    1 5/24

    1. 6 = 2 × 3 and 8 = 2³, so the LCM is 24 — not 48.
    2. 5/6 = 20/24 (× 4) and 3/8 = 9/24 (× 3).
    3. 20/24 + 9/24 = 29/24.
    4. 29/24 = 1 5/24, and 5/24 will not simplify.
  6. Ejercicio 6Estándar3 puntos

    Work out 7/10 − 4/15.

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    13/30

    1. 10 = 2 × 5 and 15 = 3 × 5, so the LCM is 30, not 150.
    2. 7/10 = 21/30 (× 3) and 4/15 = 8/30 (× 2).
    3. 21/30 − 8/30 = 13/30.
    4. 13 is prime and does not divide 30, so the answer is in lowest terms.
  7. Ejercicio 7Estándar4 puntos

    Work out 2 3/5 + 1 3/4.

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    4 7/20

    1. Add the whole numbers: 2 + 1 = 3.
    2. The LCM of 5 and 4 is 20, so 3/5 = 12/20 and 3/4 = 15/20.
    3. 12/20 + 15/20 = 27/20, which is more than one whole.
    4. 27/20 = 1 7/20, so carry that 1 into the whole numbers.
    5. 3 + 1 = 4, giving 4 7/20.
  8. Ejercicio 8Estándar3 puntos

    Work out 5/12 + 7/18.

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    29/36

    1. 12 = 2² × 3 and 18 = 2 × 3², so the LCM is 2² × 3² = 36.
    2. 5/12 = 15/36 (× 3) and 7/18 = 14/36 (× 2).
    3. 15/36 + 14/36 = 29/36.
    4. 29 is prime, so there is nothing to cancel. Using 12 × 18 = 216 as the denominator would have given 174/216 and a long simplification.
  9. Ejercicio 9Avanzado4 puntos

    Work out 4 1/4 − 1 5/6.

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    2 5/12

    1. The LCM of 4 and 6 is 12: the question becomes 4 3/12 − 1 10/12.
    2. 3/12 is smaller than 10/12, so borrow 1 from the 4.
    3. That 1 is worth 12 twelfths: 4 3/12 = 3 15/12.
    4. 3 15/12 − 1 10/12 = 2 5/12.
    5. Check as improper fractions: 51/12 − 22/12 = 29/12 = 2 5/12.
  10. Ejercicio 10Avanzado4 puntos

    Work out 5/6 + 3/4 − 2/3.

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    11/12

    1. 6, 4 and 3 all divide 12, so use 12 for all three fractions.
    2. 5/6 = 10/12, 3/4 = 9/12, 2/3 = 8/12.
    3. 10/12 + 9/12 = 19/12.
    4. 19/12 − 8/12 = 11/12.
    5. Under a whole, and 11 and 12 share no factor.
  11. Ejercicio 11Avanzado3 puntos

    A recipe needs 2/3 of a cup of sugar. 5/12 of a cup has already gone in. How much more is needed?

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    1/4 of a cup

    1. The question is a subtraction: 2/3 − 5/12.
    2. 12 is a multiple of 3, so use 12: 2/3 = 8/12.
    3. 8/12 − 5/12 = 3/12.
    4. 3/12 simplifies by 3 to 1/4 of a cup.
  12. Ejercicio 12Avanzado3 puntos

    Work out 7 − 2 5/9.

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    4 4/9

    1. 7 has no fraction part to subtract from, so borrow 1 from the 7.
    2. That 1 is worth nine ninths: 7 = 6 9/9.
    3. 6 9/9 − 2 5/9: whole numbers 6 − 2 = 4.
    4. Fractions 9/9 − 5/9 = 4/9.
    5. Answer 4 4/9.

Dónde se falla

  • Adding the denominators as well as the numerators, so 1/2 + 1/3 comes out as 2/5.
  • Converting one fraction to the common denominator and leaving the other untouched, as in 9/12 − 1/6.
  • Leaving an improper fraction unconverted, or converting it but not simplifying, so 12/9 becomes 1 3/9 rather than 1 1/3.
  • Subtracting the fraction parts in whichever direction is easier, turning 3/12 − 10/12 into 10/12 − 3/12 and answering 2 7/12.
  • Borrowing 10 rather than the common denominator, so 4 3/12 becomes 3 13/12 instead of 3 15/12.

The denominator names the piece; only the numerator gets added

Three sevenths plus two sevenths is five sevenths for the same reason that three apples plus two apples is five apples. The denominator is the name of the thing being counted, so it does not change. That is the whole justification for not adding denominators, and it is more use to a struggling pupil than being told the answer to 1/2 + 1/3 is not 2/5.

It also explains why unlike denominators need converting first. 1/2 and 1/3 are different objects, and nothing sensible can be done until both are expressed in sixths: 3/6 + 2/6 = 5/6.

Choose the LCM, not the product

Multiplying the two denominators always gives a common denominator, and it is never wrong. It is just larger than necessary, and the extra size shows up as harder multiplication now and a bigger simplification at the end. For 5/6 + 3/8, the product is 48 and the LCM is 24, and 24 halves every number you have to handle.

Find the LCM the way you would for any pair: 6 = 2 × 3 and 8 = 2³, so the LCM is 2³ × 3 = 24. Then ask what each denominator was multiplied by — 4 and 3 here — and apply the same multiplier to each numerator.

Borrowing in mixed numbers takes 12 twelfths, not 10

In 4 1/4 − 1 5/6, the fraction parts become 3/12 and 10/12, and 3/12 is too small to take 10/12 from. Borrowing 1 from the 4 converts it into twelve twelfths, not ten: 4 3/12 becomes 3 15/12, and 15/12 − 10/12 = 5/12, leaving 2 5/12.

The number borrowed always equals the common denominator, which is the one thing worth saying out loud each time. A pupil who imports the decimal habit of borrowing ten will get 2 3/12 here and never see why.

The alternative is to convert both mixed numbers into improper fractions — 51/12 and 22/12 — and subtract. It avoids borrowing entirely at the cost of larger numbers, and for questions like 7 − 2 5/9 it is usually the cleaner route.

Preguntas frecuentes

Do I have to use the lowest common denominator?

No — any common denominator gives a correct answer, and multiplying the two denominators always produces one. The LCM simply keeps the numbers small and the final simplifying short. For 5/12 + 7/18 the LCM is 36 while the product is 216, which is six times more arithmetic for the same result.

Is it better to convert mixed numbers to improper fractions?

For subtraction, usually yes, because it removes borrowing altogether: 4 1/4 − 1 5/6 becomes 51/12 − 22/12 with nothing to borrow. For addition, keeping the whole numbers separate is generally quicker. Both are accepted methods, so let a pupil settle on one and use it consistently.

Why does my child add 1/2 + 1/3 and get 2/5?

Because the halves and thirds are being treated as counters rather than sizes. The fastest correction is physical: half a chapati and a third of a chapati is plainly more than two fifths of one. Once that is felt, the sixths conversion is a technique rather than an arbitrary rule.

How much of this is expected at primary level?

In England, adding and subtracting fractions with different denominators and mixed numbers, using equivalent fractions, is a Year 5 expectation, with harder work continuing in Year 6. The borrowing questions here sit at the top of that range and are usually where a Year 5 or Year 6 pupil needs the most support.

Fuentes

  1. National curriculum in England: mathematics programmes of study — Department for Education

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