Division, fractions and approximation under time pressure
Division is where time goes. Divide by 5, 25 and 125 by multiplying instead, compare fractions without dividing, and approximate when the options are far apart.
Why this topic matters
Most students lose time dividing. Dividing by 25 is harder than multiplying by 4, so do the multiplication instead.
Simplification and approximation questions ('approximately what value should replace ?') appear in most placement tests and bank-style papers. They reward three habits: turning division into multiplication, comparing fractions without long division, and rounding when the options allow it.
Divide by 5, 25 and 125 by multiplying
5 = 10/2, so dividing by 5 is the same as multiplying by 2 and dividing by 10. 25 = 100/4 and 125 = 1000/8 work the same way.
Dividing by 10, 100 or 1000 only moves the decimal point, so all the real work is a doubling or a × 4 or a × 8.
- 4375 × 8 = 35,000.125 × 8 = 1000.
- 35000 / 1000 = 35.
Compare fractions without dividing
To compare a/b with c/d, cross-multiply: compare a × d with b × c. Whichever side has the bigger product has the bigger fraction.
For several fractions, convert to percentages with the fraction table, or compare how far each is from 1 (1 - a/b).
- 13 × 21 = 273 and 17 × 16 = 272.Cross products.
- 273 > 272, so 13/17 is bigger (only just).
BODMAS without mistakes
Order: Brackets, then 'of', then Division and Multiplication from left to right, then Addition and Subtraction from left to right.
Division does NOT come before multiplication. They share one level and you go left to right. 12 / 3 × 2 = 4 × 2 = 8, not 12 / 6 = 2.
- 48 / 4 = 12, then 12 × 3 = 36.Division and multiplication left to right.
- 2 × 6 = 12.
- 36 + 5 - 12 = 29.
Approximation: when the options let you round
If the answer options are more than about 10% apart, you do not need the exact value. Round every number to the nearest friendly value, compute, and pick the nearest option.
Round in opposite directions when you can (one up, one down) so the errors cancel.
- 400 × 25 / 50.Friendly numbers.
- = 200.
Exam-hall time plan
Give each question at most 90 seconds on the first pass. If you are not near the answer by then, mark it and move on.
Look at the options before calculating: the unit digit, the size or the form of the answer often rules out two of them.
Do the easy topics of your paper first (percentages, averages, series), then return to the long ones (DI, puzzles).
| Situation | What to do |
|---|---|
| Options far apart | Approximate |
| Options close together | Compute exactly, check the unit digit |
| 'Find the number' type | Try the options from the middle |
| No starting value given | Take 100 or the LCM |
| Stuck after 90 s | Mark, skip, come back |
Formula summary
Common mistakes
Before you move on, you should be able to...
- divide by 5, 25 and 125 without long division
- compare two fractions by cross-multiplying
- apply BODMAS left to right for / and x
- decide from the options whether to approximate
Practice questions (9)
Try each one before opening the solution. Or practise them one by one so your score is saved.
easy Divide 2,375 by 25.
- 2375 × 4 = 9500.Dividing by 25 is multiplying by 4 and dividing by 100.
- 9500 / 100 = 95.
Answer: 95
easy Which is larger, 5/7 or 7/10?
- Cross-multiply: 5 × 10 = 50 and 7 × 7 = 49.Compare a × d with b × c.
- 50 > 49, so 5/7 is larger.
Answer: 5/7
easy Divide 7,000 by 125.
- 7000 × 8 = 56,000.125 × 8 = 1000.
- 56000 / 1000 = 56.
Answer: 56
medium Arrange 3/5, 5/8, 4/7 in increasing order.
- As percentages: 3/5 = 60%, 5/8 = 62.5%, 4/7 = 57.1%.Converting to % is quick when you know the table.
- 4/7 < 3/5 < 5/8.
Answer: 4/7, 3/5, 5/8
medium Simplify: 18 + 6 / 3 × 2 - 4
- Division and multiplication left to right: 6 / 3 = 2, then 2 × 2 = 4.BODMAS: D and M before A and S, in reading order.
- 18 + 4 - 4 = 18.
Answer: 18
medium Approximately what is 49.98% of 801.2 + 24.9% of 399.7? (a) 480 (b) 500 (c) 520 (d) 560
- 50% of 800 = 400 and 25% of 400 = 100.The options are 20 apart, so rounding is safe.
- 400 + 100 = 500.
Answer: 500
medium What is 3/7 as a decimal, to 6 places?
- 1/7 = 0.142857 repeating.Sevenths all use the same cycle 142857.
- 3/7 starts at the 4: 0.428571.
Answer: 0.428571
hard Approximately 1,249 / 24.96 + 19.97 × 5.02 = ? (a) 120 (b) 150 (c) 170 (d) 200
- 1250 / 25 = 50. 20 × 5 = 100.Round each number to the nearest friendly one.
- 50 + 100 = 150.
Answer: 150
hard Which is the largest: 7/9, 11/14, 15/19?
- 7/9 = 0.778, 11/14 = 0.786, 15/19 = 0.789.Close fractions: divide to 3 places or cross-multiply pairwise.
- 15/19 is the largest.
Answer: 15/19
Class plan (2 hours, for trainers)
A tested order for teaching this topic in one 2-hour session. Present mode follows the same order.
| 10 min | Warm-up: 5 divisions by 25 and 125, long way vs the trick. |
| 20 min | Division by multiplying. |
| 20 min | Comparing fractions: cross-multiplying and the % table. |
| 15 min | BODMAS traps. |
| 20 min | Approximation: 10 questions with options. |
| 10 min | The exam-hall time plan. |
| 20 min | Practice set, timed. |
| 5 min | Recap. |