Last month at ten in the evening I was sitting at my kid's desk explaining 5/8 - 1/6. By the third run-through I had subtracted the denominators too and arrived at 4/2. My kid looked at me, I looked at him, and neither of us knew where it went wrong. That problem lost marks in the end, and the reason written on it was "result not reduced to lowest terms". Around that time I went through the fraction homework from grade five to grade six and found the problem was not arithmetic ability at all. It was three ideas I had never explained properly: where the common denominator comes from, when to reduce, and why mixed numbers do not take part in addition and subtraction. This piece follows the order in which I relearned it myself.
The common denominator is not always the two denominators multiplied together
My kid's first approach was to multiply the two denominators directly. That is not wrong, but it makes the numbers bigger at every later step. The instructions for the Fraction Calculator give the addition rule as a/b + c/d = (ad+bc)/(bd), which is exactly the multiply-the-denominators route, with the result reduced afterwards. For the same problem, 3/4 + 5/6, I worked it both ways and put them side by side to see the difference.
| Route to a common denominator | Intermediate steps | Sum | Still needs reducing? |
|---|---|---|---|
| Multiply the denominators: 4×6=24 | 18/24 + 20/24 | 38/24 | Yes, divide top and bottom by 2 to get 19/12 |
| Least common denominator: the LCM of 4 and 6 is 12 | 9/12 + 10/12 | 19/12 | No, 19 and 12 are coprime |
Both routes give exactly the same answer, 19/12, which as a mixed number is 1 7/12. The only difference is whether you take an extra lap through reducing. I have him do both, because when he meets a pair like 8 and 6 later, multiplying gives 48 while the least common denominator is only 24. Double the numbers, double the chances of a slip.
Take 5/8 - 1/6. With the least common denominator it is 15/24 - 4/24 = 11/24. Force 8×6=48 and it becomes 30/48 - 8/48 = 22/48, which looks scarier, but dividing both by 2 still brings you back to 11/24. I used the Common Denominator Calculator to have him practise converting three fractions at once: 1/6, 3/8, 5/12 have a common denominator of 24, which converts to 4/24, 9/24 and 10/24, and those add up to exactly 23/24. The LCM of 6, 8 and 12 is 24, not the 576 you get by multiplying them all the way through. That is where the word "least" actually pays off.
When to reduce: I stick to two fixed points
The "not reduced to lowest terms" mark came from exactly this. My method has two points: before writing anything, check whether the fractions can be reduced; before handing in the answer, check again.
Early reduction pays off most in multiplication. 5/6 × 3/10 straight through is 15/60, which still needs reducing; cancel 6 against 3 and 5 against 10 first and you land on 1/4 in one step. Early reduction helps in addition and subtraction too: if a fraction is not in lowest terms to begin with, the denominator you convert to gets inflated for no reason.
The after-the-fact check is the one most often skipped. Stopping at 4/12 for 7/12 - 1/4 is a classic lost mark; the GCD of 4 and 12 is 4, so the answer has to be written as 1/3. One clumsy trick I gave him: if the numerator and denominator added together, or the numbers in the result, look even or divide by 3 or 5, try one division. When I am genuinely unsure, I run things like 18/24, 45/60 and 84/126 through the Fraction Simplifier in one batch and see them divide by 6, 15 and 42 to give 3/4, 3/4 and 2/3, so he can notice for himself that the number you divide by is the GCD.
Mixed numbers stay out of it, and borrowing is where subtraction falls apart
Almost every problem I got wrong myself involved mixed numbers. The rule is blunt: convert mixed numbers to improper fractions before adding or subtracting, then convert back afterwards. The calculator takes a numerator and a denominator, so I always convert mixed numbers on paper before typing them in.
2 1/3as an improper fraction: 2×3+1=7, so 7/33 2/5: 3×5+2=17, so 17/519/12back to a mixed number: 19 divided by 12 is 1 remainder 7, so 1 7/1225/6: 25 divided by 6 is 4 remainder 1, so 4 1/6
Apply both rules and 2 1/3 + 1 5/6 becomes 7/3 + 11/6, then 14/6 + 11/6 = 25/6 after converting, and finally 4 1/6. Rush it by adding 1/3 + 5/6 and tacking on the whole-number part 3, and 3 1/4 - 1 3/4 will stop you dead: 1/4 is not enough to subtract 3/4. Done honestly it is 13/4 - 7/4 = 6/4, which reduces to 3/2, written back as 1 1/2. There is also the 5 - 2 3/8 type: treat 5 as 4 8/8 and the result is 2 5/8. My kid's usual mistake is to answer 3 5/8, because he subtracted one too many from the whole-number part.
When a fraction and a decimal can be treated as equal
Once he asked why 1/8 can be written as 0.125 but 1/3 never ends, and I could not answer on the spot. The pattern is in the prime factors of the denominator: once a fraction is in lowest terms, a denominator containing only 2 and 5 gives a terminating decimal, 1/8 is 0.125, 7/20 is 0.35, 1/16 is 0.0625; any prime factor other than 2 or 5 in the denominator produces a repeating decimal, 1/3 is 0.333..., with just 3 repeating; 5/12 is 0.41666..., where 41 does not repeat and only the 6 runs on, because 12 is 2 times 2 times 3.
So instead of making him memorise "one third equals 0.33", I explain that 0.33 is only an approximation and that anything used for real accounting has to stay in fraction form. The calculator page shows a single line under the result, "equivalent decimal", and 19/12 gives 1.5833333333..., where a repeating value is cut to 10 places and followed by an ellipsis; there is no percentage row on that page. 1.5833 is my own truncation to four decimals, and 158.33% is that 1.5833 multiplied by 100, which has nothing to do with the tool. I use that number to check whether his written work has drifted: if the whole-number part of the decimal is wrong, it shows immediately.
For comparing fractions I teach cross multiplication, which is faster than converting. For 3/5 against 5/8, 3 times 8 is 24, 5 times 5 is 25, and 25 is larger, so 5/8 is the larger fraction, with no need to hunt for 40 as a common denominator.
Common questions
Q: Do I always have to find the least common multiple?
No. Any common multiple gives the right answer, and multiplying the two denominators always produces one. But the least common denominator keeps the intermediate numbers smallest and saves a reduction, so for homework I make him use the smallest one.
Q: Why do teachers insist the answer be in lowest terms?
Because one value has countless spellings: 4/12, 2/6 and 1/3 are all correct, but only the lowest terms give a single standard. Marking and later calculation both rely on that uniqueness, so do not leave it unreduced.
Q: Can mixed numbers be added or subtracted directly?
For addition, barely; for subtraction, it goes wrong easily. My rule is to convert to improper fractions, work it out, then convert back. When something like 3 1/4 - 1 3/4 cannot be subtracted directly, converting to improper fractions is the only route that avoids worrying about borrowing.
Q: When should a fraction be turned into a decimal?
It depends on the precision required. Fractions like 1/8 and 7/20, whose denominators contain only 2 and 5, lose nothing as decimals; 1/3 and 5/12 turn into approximations, so keep them as fractions whenever you can.
Q: How should mental arithmetic and the calculator split the work?
In my division of labour, mental arithmetic handles the process and the calculator handles acceptance. He writes out the common-denominator and reduction steps by hand first, then I put the same problem into the calculator and compare every step it shows. If the two sides differ, we go back to the paper to find which numerator went wrong, which works better than me lecturing ten times.
That 5/8 - 1/6 ends up as 11/24, which had nothing left to reduce anyway; my kid lost the mark for writing 22/48 and stopping there. Now I explain this problem in a fixed three lines: find the smallest denominator first, check whether the result reduces before you stop, and always convert mixed numbers to improper fractions. I follow those three lines myself, and at ten at night I no longer come out with 4/2.