This trainer practises exact fraction working without a calculator. You enter two fractions and an operation, choose the step you would take, and the trainer checks it against exact arithmetic and explains where a wrong step goes astray.
What does the trainer do?
You give it:
- a first fraction (numerator and denominator),
- a second fraction,
- an operation: add, subtract, multiply or divide,
- optionally, your chosen step, such as “flip the second fraction, then multiply”.
It works out the exact answer using whole-number arithmetic, compares it with the result of your chosen step, and lists the other possible steps with what each one gives. It finishes with an independent check using decimals.
How do I use it?
- Type the four numbers. They must be whole numbers, and neither denominator can be zero.
- Pick the operation.
- Choose your step, or choose “No choice, just show the correct working” if you want to see the method first.
- Press Check.
- Read the explanation, then try a different step or a new pair of fractions. Use Reset to return to the example.
How do I read the result?
The first line gives the exact answer in lowest terms, for example “3/4 ÷ 2/5 = 15/8”.
If you chose a step, the trainer says whether it is correct. A wrong step shows what it actually gives, then explains why it fails. If a step leads to a zero denominator, it says the value is undefined.
The table lists every step for the operation, the result each one gives and whether it is valid. A wrong step can look reasonable, so seeing all of them side by side helps you spot which idea went wrong.
For division, the tool adds a sense check: dividing a positive number by a positive number smaller than 1 makes it bigger. If your result is smaller than the starting fraction, the trainer flags it.
Example walk-through
The trainer opens on 3/4 ÷ 2/5 with the step “Multiply by the second fraction as it is” selected.
Exact answer: flip the second fraction and multiply. 3/4 × 5/2 = 15/8.
The chosen step: 3/4 × 2/5 = 6/20 = 3/10. The trainer says this is not correct, and explains that dividing by a fraction means multiplying by its reciprocal. It also flags that 3/10 is smaller than 3/4, which cannot happen when you divide by a number smaller than 1.
Other steps in the table: flipping the first fraction instead gives 4/3 × 2/5 = 8/15, which is also wrong. Dividing numerators and denominators separately is valid: 3 ÷ 2 over 4 ÷ 5 is 1.5/0.8, which equals 15/8, though it is awkward by hand.
Independent check: 0.75 ÷ 0.4 = 1.875, and 15/8 = 1.875. They match.
Try addition next: 2/3 + 1/4. The common-denominator step gives 8/12 + 3/12 = 11/12. Adding the tops and bottoms separately gives 3/7, which is wrong, and the decimal check (0.667 + 0.25 ≈ 0.917 against 3/7 ≈ 0.429) shows why.
Assumptions and limits
- It handles exact fractions with whole-number inputs only, up to six digits.
- Some topics and papers allow a calculator and some do not. The tool does not claim every topic or paper prohibits calculators, so check the rules for your own code and paper.
- It checks the steps it knows. A method it does not list is not automatically wrong, so ask a teacher if in doubt.
Where to read more
The lesson on working with fractions without decimal rounding builds the idea, and checking operation order in a multi-step calculation covers the next layer. Comparing exact and approximate answers explains when a decimal check is enough. If calculator habits cost you accuracy, see losing accuracy by using a calculator too early, and for paper rules, checking calculator permissions for the exact code year and paper.
The full route is in the Mathematics learning guide, and the other tools sit alongside this one. Teachers who work on exact arithmetic one-to-one are described on online one-to-one Mathematics tuition.