Math Vale

🍰 Fraction Isles · Fractions & Proportion

Equivalent & Simplifying Fractions

Read a fraction as a division and as a part of a whole, generate equivalent fractions, and reduce to lowest terms.

In short

  • The denominator names the size of the pieces and the numerator counts them; a fraction is also just a division.
  • Multiplying or dividing both parts by the same non-zero number gives an equivalent fraction, because you are multiplying by 1.
  • Lowest terms means dividing both parts by their greatest common factor, so nothing but 1 divides them both.
  • Improper fractions and mixed numbers are two spellings of the same number: divide to go one way, multiply and add to come back.
  • For a fraction of a set, the denominator is the whole set, never the leftovers.

What the two numbers mean

A fraction is written a/b, and the two numbers do completely different jobs.

  • The denominator b says how many equal pieces one whole has been cut into. It names the size of a piece.
  • The numerator a says how many of those pieces you have. It is a count.

So 3/8 means "three of the pieces you get when a whole is cut into eight equal ones". Cut a bar into 8 strips, shade 3, and the picture is the fraction.

A fraction is also a division: 3/8 is exactly 3 divided by 8, which is what makes 8/8 equal to 1 and 8/4 equal to 2.

The word "equal" matters. Pieces of a fraction bar must be the same size, or the fraction does not describe anything.

Equivalent fractions

Cut every piece of a bar in half and you have twice as many pieces, each half the size. The shaded amount has not moved, so the fraction still names the same number:

1/2 = 2/4 = 3/6 = 4/8 = 50/100

The rule behind the picture is: multiplying (or dividing) the numerator and the denominator by the same non-zero number does not change the value.

3/4 = (3 x 5)/(4 x 5) = 15/20

That works because you are really multiplying by 5/5, which is 1, and multiplying by 1 changes nothing.

Watch what does *not* work: adding. 3/4 is not 4/5, even though 1 was added to both. Adding the same amount to top and bottom always moves a fraction towards 1.

Lowest terms

Every fraction has one form where the numerator and denominator share no common factor bigger than 1. That form is called lowest terms, or "fully simplified".

To get there, divide top and bottom by their greatest common factor (GCF).

18/24: the GCF of 18 and 24 is 6, so 18/24 = 3/4.

You may simplify in stages instead — 18/24 = 9/12 = 3/4 — and you will land in the same place. Using the GCF just gets you there in one step.

Why bother? Because 3/4 is easier to picture, easier to compare with other fractions, and it is the form everyone agrees to write answers in.

A useful check: if the numerator and denominator are both even, you can always divide by 2 at least once.

Improper fractions and mixed numbers

When the numerator is at least as big as the denominator the fraction is worth one whole or more. That is an improper fraction, and there is nothing wrong with it.

17/5 means seventeen fifths. Five fifths make one whole, so:

17 = 5 + 5 + 5 + 2, which is 3 wholes and 2 fifths left over.

17/5 = 3 2/5, a mixed number. To convert, divide the numerator by the denominator: the quotient is the whole part and the remainder stays over the same denominator.

Going the other way, 3 2/5 = (3 x 5 + 2)/5 = 17/5. Multiply the whole number by the denominator, add the numerator, and keep the denominator unchanged.

Mixed numbers are easier to picture; improper fractions are easier to calculate with. Both are the same number, so use whichever suits the job.

Fractions of a set

A fraction does not need a bar or a pie. It works just as well on a group of separate things.

If 18 of the 24 arrows in a quiver are fletched, then 18/24 of them are fletched, which is 3/4.

The method is always the same: part on top, whole underneath, then simplify.

The trap here is comparing the part with the *rest* instead of with the whole. 18 fletched and 6 unfletched is a ratio of 18 to 6, but the fraction fletched is 18/24. Ask yourself "out of how many altogether?" before you write the denominator.

A number line is the third picture worth keeping: mark 0 and 1, cut the gap into 24 equal steps, and 18/24 lands at exactly the same point as 3/4.

Worked examples

Example 1

Fill in the missing number: 7/9 = ?/45

  1. Look at the part you can compare: the denominator went from 9 to 45.
  2. 45 / 9 = 5, so the whole fraction was multiplied by 5 (that is, by 5/5, which is 1).
  3. Do the same to the numerator: 7 x 5 = 35.
  4. 7/9 = 35/45, so the missing number is 35.
  5. Check by simplifying back: 35 and 45 are both multiples of 5, and 35/45 = 7/9.

Example 2

Write 42/56 in lowest terms.

  1. Find the greatest common factor of 42 and 56. Both are even, so 2 works; both are also divisible by 7.
  2. 42 = 14 x 3 and 56 = 14 x 4, so the GCF is 14.
  3. Divide both parts by 14: 42 / 14 = 3 and 56 / 14 = 4.
  4. 42/56 = 3/4.
  5. Confirm it is finished: 3 and 4 share no factor bigger than 1, so this is lowest terms.

Example 3

Write 29/6 as a mixed number.

  1. Divide the numerator by the denominator: 29 / 6 = 4 remainder 5.
  2. The 4 counts the whole units; 6 sixths make each one.
  3. The remainder 5 stays over the original denominator: 5/6.
  4. 29/6 = 4 5/6.
  5. Check by going back: 4 x 6 + 5 = 29, so the conversion is right.

Practice problems, with solutions

Three problems of increasing difficulty, each with the full working. In the game these are generated fresh every time; these three are fixed so this page always shows the same ones.

Problem 1

Difficulty 1 of 5

Fill in the missing number so the two fractions are equal. 1/5 = ?/15

Answer: 3

  1. 5 becomes 15, so the whole fraction was multiplied by 3.
  2. The other number must change the same way: 1 x 3 = 3.
  3. 1/5 = 3/15, and the missing number is 3.
  4. Check: both fractions simplify to 1/5.

Problem 2

Difficulty 3 of 5

Write 9/63 in lowest terms.

Answer: 1/7

  1. Find the greatest common factor of 9 and 63: it is 9.
  2. 9 / 9 = 1 and 63 / 9 = 7.
  3. 9/63 = 1/7.
  4. 1 and 7 share no factor bigger than 1, so this is lowest terms.

Problem 3

Difficulty 4 of 5

A quest banner is cut into 16 equal strips, and 15 of them are dyed gold. What fraction of the banner is dyed gold? Give your answer in lowest terms.

Answer: 15/16

  1. The banner has 16 equal strips, so the denominator is 16.
  2. 15 of them are dyed gold, so the fraction is 15/16.
  3. 15 and 16 have no common factor, so 15/16 is already in lowest terms.

Common mistakes

  • Adding the same number to the top and the bottom instead of multiplying: writing 3/4 = 4/5.
  • Changing only one part of the fraction, so 3/4 becomes 3/12 instead of 9/12.
  • Stopping half way when simplifying, leaving 9/12 when 3/4 was asked for.
  • Swapping the whole number and the remainder when converting, writing 17/5 as 2 3/5 instead of 3 2/5.
  • Using the leftovers as the denominator: calling 18 fletched out of 24 arrows "18/6".

What you should be able to do

  • Name the fraction shown by a diagram or a number line point.
  • Generate equivalent fractions by multiplying or dividing top and bottom.
  • Simplify a fraction to lowest terms using the greatest common factor.
  • Convert between improper fractions and mixed numbers.

Where this fits in the curriculum

Common Core

  • 3.NF.A.1

    Grade 3 — Understand a fraction 1/b as one part of a whole partitioned into b equal parts.

  • 4.NF.A.1

    Grade 4 — Explain why a fraction is equivalent to another by multiplying or dividing numerator and denominator.

  • 4.NF.B.3.B

    Grade 4 — Decompose a fraction, and convert between mixed numbers and improper fractions.

    The Common Core never asks for "lowest terms" as a goal in itself — simplifying is treated as one direction of the equivalence in 4.NF.A.1.

Ontario

  • G5.B1.3

    Grade 5 — Represent equivalent fractions from halves to twelfths, using appropriate tools and drawings.

  • G8.B1.4

    Grade 8 — Use equivalent fractions to simplify fractions, when appropriate, in various contexts.

    Ontario names "simplify" surprisingly late: the equivalence work starts at Grade 5 but simplifying to lowest terms is only expected at Grade 8, so this skill straddles four grades.

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