Working out a fraction of an amount, a common percent or the order of two fractions in your head comes down to six habits. For 3/8 of 72 you divide by 8 and multiply by 3 (27), and for 20% of 35 you think of one fifth (7). The table shows which habit to use and where it stops being enough. If you are looking for the cut-off points 1/2 and 1, see benchmark fractions, which covers that topic separately. For the exact result there is the fraction calculator.
| Strategy | Mental example | When it is not enough |
|---|---|---|
| Divide and multiply | 2/3 of 90 is 90 ÷ 3 = 30 and 30 × 2 = 60 | The amount is not a multiple of the denominator |
| Percent as a fraction | 25% of 48 is 1/4 of 48 = 12 | Percents that do not match a simple fraction |
| Cut-off at 1/2 | 5/11 and 4/9 are both below one half | Both fall on the same side and you need to refine |
| Distance to 1 | 7/8 leaves 1/8 and 8/9 leaves 1/9, so 8/9 is greater | Fractions far from 1 or greater than 1 |
| Round to estimate | 7/8 + 5/9 is more than 1, because each addend is over one half | If the margin matters, an estimate does not settle the sum |
| Same denominator or numerator | 3/10 < 7/10 and 3/5 > 3/7 | Numerators and denominators both differ |
Each row solves a different problem. Sharing, measuring and discounting usually come down to the first one, a fraction of an amount.
A fraction of an amount: divide and multiply
To find a/b of an amount, divide by the denominator and multiply by the numerator. Dividing first keeps the numbers small, and the result is the same as multiplying first: (24 ÷ 8) × 5 = 15 and (24 × 5) ÷ 8 = 15. With 2/3 of 90 you get 90 ÷ 3 = 30 and 30 × 2 = 60, and with 3/5 of 40 you do 8 × 3 = 24. If the amount is not a multiple of the denominator, an intermediate fraction appears: 3/4 of 10 is 7.5, and that is where it may be better to switch to paper. The fraction of a number tool does the calculation with any number.
Worked example: 5/6 of 36
You have already driven 5/6 of a 36-mile trip. How many miles have you driven?
- Divide by the denominator. 36 ÷ 6 = 6, which is 1/6 of the trip.
- Multiply by the numerator. 6 × 5 = 30.
- Add the unit. That is 30 miles.
You have driven 30 miles.
To check, you would have 1/6 left, which is 6 miles, and 30 + 6 = 36.
Percents that are simple fractions
Many percents are easier to work out as a fraction. 50% is one half, 25% is one fourth, 75% is three fourths, 20% is one fifth and 10% is one tenth. 25% of 48 is a fourth of 48, which is 48 ÷ 4 = 12, and 75% of 48 is three of those parts, 36. 10% of 250 comes from moving the decimal point one place and is 25. The percentage calculator finds the percent of any amount, and the fraction to percent calculator converts any fraction.
Comparing without a common denominator
With the same denominator, the larger numerator wins, as in 3/10 < 7/10. With the same numerator, the smaller denominator wins, as in 3/5 > 3/7. If both change, compare each fraction with one half by doubling the numerator: if the double is less than the denominator, the fraction is below one half. In 5/11 the double of 5 is 10, less than 11, and in 4/9 the double of 4 is 8, less than 9. Both are below one half. If one landed above and the other below, they would already be in order.
When both are on the same side, measure the gap to one half. 5/11 is short of it by 1/22 and 4/9 by 1/18. The smaller gap belongs to 5/11, so it is closer to one half and is the greater one. The benchmark fractions guide develops these cut-off points, and the compare fractions calculator gives the exact decision with cross products.
Distance to 1 for nearly complete fractions
Between two proper fractions close to 1, the one that leaves the smaller gap wins. 7/8 is short by 1/8 and 8/9 is short by 1/9. Since 1/9 is less than 1/8, 8/9 is closer to the whole and is greater. With 11/12 and 7/8 the same happens, since the gaps are 1/12 and 1/8, and the gap of 11/12 is smaller. Subtracting from 1 reverses the order, which is why this works. With fractions greater than 1 you measure the excess instead, and then the one that goes further over wins. To see these gaps on a line, use the fraction number line.
Estimating a sum before calculating it
Before adding, check whether each term is over one half. If both are, the sum is more than 1. For 7/8 + 5/9, the first addend is close to 1 and the second is a little more than 1/2, so the sum is around 1.5. With a common denominator, 63/72 + 40/72 = 103/72 ≈ 1.43, with an error of less than 0.001. The estimate lands close and runs a bit high, because 7/8 was rounded up to 1. If you round every term in the same direction, the error builds up. To see how much each rounding changes, look at the round to a fraction tool.
A five-minute routine
Give one minute to each block and the last two minutes to reviewing your reasoning. First, two fractions of an amount. Then, two percents. Last, one comparison and one sum estimate. Answer each one in your head before checking. If you want more exercises with answers, the fraction worksheet generator creates new sheets, and fraction practice checks your answers. If you are preparing the routine for a class, the teacher resources hub gathers classroom materials.
Six challenges for the routine
- Find 3/4 of 52.
- Find 2/3 of 45.
- Find 25% of 84.
- Find 75% of 60.
- Which is greater, 2/5 or 3/7?
- Is 7/9 + 4/7 greater than 1?
Show the answers
- 39, because 52 ÷ 4 = 13 and 13 × 3 = 39.
- 30, because 45 ÷ 3 = 15 and 15 × 2 = 30.
- 21, because it is a fourth of 84 and 84 ÷ 4 = 21.
- 45, because it is three fourths, 60 ÷ 4 = 15 and 15 × 3 = 45.
- 3/7. Both are below one half; 2/5 is short by 1/10 and 3/7 by 1/14, and the smaller gap belongs to 3/7.
- Yes. 7/9 is over one half and so is 4/7, so the sum is more than 1. The exact value is 85/63.
Sources
The Wikipedia article Mental calculation describes mental calculation as arithmetic done by the mind, without aids such as paper or a calculator, and notes that people use it when computing tools are not available (accessed October 17, 2026). The article does not cover strategies with fractions. The calculations on this page can be repeated in the fraction calculator. The strategies are elementary arithmetic reasoning, with no outside data.