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Simplifying a Fraction Quickly: Clear the 2s and 5s First

Reduce in a fixed order — halve, clear the 5s, then use digit sums for 3 and 9 — and you cut the numbers down before the expensive tests start.

By Published 6 min read

Reign Creative LLC makes IQ Test & Brain Training Games, which this article links to, so we have a commercial interest in it. How we write and check articles.

Key takeaways

  • Simplifying is faster when the tests run in a fixed order: halve while both numbers are even, then clear the 5s, then apply digit sums for 3 and 9, and only then hunt for 7, 11 and 13.
  • The order matters because each cancellation shrinks the numbers the next test has to work on, so the cheap tests do their work while the numbers are still large.
  • Cancelling before dividing rather than after turns an awkward division into an easy one, which is the same saving in a different place.
  • You have finished when the two numbers share no common factor, and the cheap way to confirm that is to test only the primes up to the square root of the smaller number.
  • When inspection fails entirely, the Euclidean algorithm finds the greatest common divisor by repeated division with remainder and never needs a single guess.

Simplifying a fraction quickly is a question of order, not of insight. Halve while both numbers are even, then divide by 5 while both end in 0 or 5, then test 3 and 9 with digit sums, and only then start hunting for 7, 11 and 13. Each step shrinks the numbers the next step has to work on, which is why doing the cheap tests first is faster than doing them at all.

The order is the whole technique. Most people reduce a fraction by looking for the biggest factor they can spot, which means staring at large numbers and frequently missing. Running the tests in sequence means you are always working on the smallest numbers available.

The sequence, and why it is a sequence

  1. Both even? Halve both. Repeat until at least one is odd. This is a glance, not a calculation.
  2. Both ending in 0 or 5? Divide both by 5. Repeat.
  3. Digit sums both divisible by 3? Divide both by 3. Repeat, and use 9 when both digit sums are divisible by 9.
  4. Only now look for 7, 11 and 13, which require actual division to test.

Steps 1 and 2 are visual and free. Step 3 costs one addition per number. Step 4 is the only one that costs a division, and by the time you reach it the numbers are usually small enough that you can see the answer anyway. The reasons these tests are valid at all — why the digit sum decides 3 and 9, why the last digit decides 2 and 5 — are worked out in why the divisibility rules work, and NRICH's article on divisibility tests sets out the same arguments.

Worked example: 168/420

  • Both even, so halve: 84/210. Both still even, so halve again: 42/105. Now 105 is odd; stop halving.
  • Does 5 divide both? 105 ends in 5, but 42 does not. No.
  • Digit sums: 4 + 2 = 6, and 1 + 0 + 5 = 6. Both divisible by 3, so divide by 3: 14/35.
  • Now 14 and 35 are small enough to see. Both divide by 7: 2/5.

Check: 168 divided by 420 is 0.4, and 2/5 is 0.4.

Notice that the original numbers shared a factor of 84, which almost nobody spots by inspection. The sequence found it in four painless moves instead of one hard one.

Worked example: 450/1200

  • Both even: 225/600. Now 225 is odd; stop.
  • Both end in 0 or 5: divide by 5 to get 45/120, and again to get 9/24.
  • Digit sums 9 and 6 are both divisible by 3: 3/8.

Check: 450 divided by 1,200 is 0.375, and 3/8 is 0.375.

Cancel before you divide, not after

The same saving applies to a plain division problem, because every division is a fraction in disguise. The techniques library in our app states the move as: before dividing, cancel common factors from both numbers — 180 ÷ 12 becomes 90 ÷ 6 after halving, which becomes 45 ÷ 3, which is 15.

It works on ugly numbers too. For 2,400 ÷ 150, dividing both by 50 gives 48 ÷ 3 = 16, and 150 × 16 does indeed come to 2,400. The alternative is long division with a three-digit divisor.

This is the point at which a fraction stops being a shape on a page and starts being a number you can push around. The What Works Clearinghouse practice guide on fractions instruction puts this idea near the centre of its recommendations: help students recognise that fractions are numbers, and help them understand why the computation procedures make sense rather than treating them as rituals. More division methods, including the ones that produce quotients rather than decisions, are in how to divide in your head.

Knowing when you have finished

A fraction is in lowest terms when the two numbers share no common factor other than one. Testing that exhaustively sounds expensive, but it is not, because of a standard shortcut: you only need to test primes up to the square root of the smaller number. Any factor larger than that square root would have to pair with a factor smaller than it, and you have already ruled those out.

91/143. Both odd, so no halving. Neither ends in 0 or 5. Digit sums are 10 and 8, so no 3. Try 7: 91 = 7 × 13, but 143 is not a multiple of 7 — 7 × 20 is 140, leaving a remainder of 3. Try 11: 143 = 11 × 13, but 91 is not a multiple of 11. Try 13: both are. Dividing gives 7/11.

Check: 91 divided by 143 is 0.636363…, and 7 divided by 11 is 0.636363…

And you are done, because 7 is prime and does not divide 11. That is the finishing check in its shortest form.

When inspection fails: the Euclidean algorithm

Sometimes there is nothing to see. The reliable fallback is the Euclidean algorithm, which MathWorld describes as repeatedly computing remainders of consecutive terms until one comes out zero — its own worked example runs 42, 30, 12, 6, 0 to give a greatest common divisor of 6.

1071/462. Divide and keep the remainder each time:

  • 1,071 = 2 × 462 + 147
  • 462 = 3 × 147 + 21
  • 147 = 7 × 21 + 0

The last non-zero remainder is 21, so that is the greatest common divisor. Dividing both original numbers by 21 gives 51/22, and that is final: 51 is 3 × 17, and 22 is 2 × 11, so they share nothing.

The algorithm never guesses and never needs a factorisation. It is slower than a glance when a glance works, and faster than everything else when it does not.

Why any of this is worth the trouble

Reducing is not tidiness. A fraction in lowest terms is one you can compare, convert and estimate with — which is why the ordering heuristic here feeds directly into comparing two fractions without a common denominator and into converting between the three notations.

There is also a documented failure mode worth naming. In a study of sixth and eighth graders published in Developmental Psychology, Siegler and Pyke reported that roughly a third of the children based their fraction magnitude estimates primarily on the numerator alone or the denominator alone, and that this pattern went along with poorer fraction arithmetic. That is a finding about how a fraction gets read, not a diagnosis of anyone, but it explains why fluency with the whole object — rather than with its top and bottom separately — is the thing worth building. The wider set of arithmetic methods sits in the mental math training guide, with the rest of the hub under mental math and the broader subject under education and brain training.

Drilling the ordering until it is automatic

IQ Test & Brain Training Games, which is our own app, built by Reign Creative, ships a Fraction Simplifier mode: a multiple-choice format on a 75-second clock with Easy, Medium and Hard settings. Multiple choice suits this particular skill, because the decision you are practising is recognising which reduced form is correct rather than writing one out.

Two entries in the app's techniques library cover the method directly. Fraction Simplification states the ordering explicitly — clear the 2s and 5s first, then use digit-sum rules for 3 and 9 — and Simplify Fractions First covers the cancel-before-you-divide move with the 180 ÷ 12 example above. Each technique carries steps, a worked example, a "best for" note, a named common mistake and a two-question mini quiz, plus a practice link into a specific mode at a specific difficulty.

None of this is an assessment of anything. The app is an educational game that records score, accuracy, streak, response time and personal bests on the drills you play, and those are game statistics rather than measurements of ability.

Good to know

Frequently asked questions

What order should you cancel factors in?
Halve while both numbers are even, then divide by 5 while both end in 0 or 5, then test 3 and 9 using digit sums, then look for 7, 11 and 13. The first two tests are visual and cost nothing, the third costs one addition, and the last requires real division — so running them in that order keeps the expensive work to a minimum.
How do you know when a fraction is fully simplified?
When the two numbers share no common factor other than one. In practice you only need to test prime numbers up to the square root of the smaller of the two, because any larger factor would have to be paired with a smaller one you have already ruled out. If the smaller number is itself prime, you are finished unless it divides the other.
Is it better to cancel before dividing or after?
Before, almost always. Dividing 180 by 12 is awkward; halving both first gives 90 divided by 6, and halving again gives 45 divided by 3, which is 15. The answer is identical, but you never handle a two-digit divisor. Cancelling first also keeps the numbers small enough to stay in working memory.
What do you do when no factor is obvious?
Use the Euclidean algorithm. Divide the larger number by the smaller and keep the remainder, then divide the previous divisor by that remainder, and repeat until the remainder is zero. The last non-zero remainder is the greatest common divisor, and dividing both original numbers by it finishes the job in one step.

Sources

  1. Developing Effective Fractions Instruction for Kindergarten Through 8th Grade (WWC Practice Guide) (opens in a new tab)What Works Clearinghouse, Institute of Education Sciences, U.S. Department of Education — accessed
  2. Euclidean Algorithm (opens in a new tab)MathWorld — A Wolfram Resource — accessed
  3. Divisibility Tests (opens in a new tab)NRICH, Millennium Mathematics Project, University of Cambridge — accessed
  4. Developmental and Individual Differences in Understanding of Fractions (opens in a new tab)Developmental Psychology (American Psychological Association), via PubMed Central — accessed

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