Ballpark
About how big is 4,872 × 316? Round each number to one figure and multiply — the options start powers of ten apart and close in.
How it plays
No time to calculate: round every number to its first figure, multiply (or divide), and pick the option nearest.
Levels add numbers, digits and division, and bring the options closer together — from ten times apart toward less than twice.
Keys
Space starts and skips ahead · 1–9 pick · Enter submits · Esc stops. On a phone, tap.
What it trains
This practises estimating the size of a calculation. It improves estimating like this; it claims nothing more.
Learn
How to get better at ballpark: the method, the techniques, what each new level asks of you, and the traps the wrong answers are made of.
Your approach to each question
- Round every number to its first figure: 4,872 becomes 5,000 and 316 becomes 300. Note, as you go, which way each one moved.
- Multiply the leading figures, then add up the zeros: 5 × 3 = 15, with 3 + 2 zeros, is 1,500,000.
- If there is a ÷, divide its leading figure into the rest and take its zeros away instead of adding them.
- Find the two options your estimate falls between and take the nearer one by ratio, leaning the way your rounding drifted. Here the exact answer is 1,539,552, and the right option reads 1,500,000.
Techniques and tricks
One figure, then count the zeros
Keep only the first figure of each number and the count of zeros behind it. Multiply the figures; the zeros simply add.
e.g. 4,872 × 316 ≈ 5,000 × 300 = 15 with five zeros = 1,500,000 (exactly 1,539,552)
Count the digits first
A product of a d-digit and an e-digit number has d + e digits if the leading figures multiply to 10 or more, and d + e − 1 if they do not. While the options are ten times apart, that alone picks the answer.
e.g. 66 × 11: 7 × 1 = 7 is under 10, so 2 + 2 − 1 = 3 digits. It is 726.
Two figures for a leading 1
Rounding 149 to 100 cuts it by a third, the biggest error one-figure rounding can make. When a number starts with 1, keep two figures.
e.g. 149 × 251: 100 × 300 = 30,000 is well short; 150 × 250 = 37,500 (exactly 37,399)
Round a pair in opposite directions
When two numbers would both round up (or both down), push one the other way so the errors cancel instead of stacking.
e.g. 650 × 350: not 700 × 400 = 280,000, but 700 × 300 = 210,000 or 600 × 400 = 240,000. Exactly 227,500.
Track the drift in per cent
Say roughly how far each rounding moved its number, and add those up. The total tells you whether the truth is above or below your estimate.
e.g. 4,872 → 5,000 is about +3%; 316 → 300 is about −5%. Net about −2%, so the truth is a little above 1,500,000.
Divide by the leading figures
For a ÷, divide the leading figures and subtract the divisor’s zeros. Turn the quotient into a friendly decimal: 7 ÷ 4 = 1.75.
e.g. 6,644 × 1,025 ÷ 3,925 ≈ 7,000 × 1,000 ÷ 4,000 = 7 ÷ 4 with three zeros = 1,750 (exactly about 1,735)
Read the options as a ladder
The options are the true value times one fixed step — ×10, then ×4, ×2.5 and ×1.8 — so they climb in equal ratios. Your estimate picks the right rung if it is within the square root of the step: about ×1.34 at the closest.
e.g. With rungs 1,200 · 2,100 · 3,800 · 6,900 and an estimate of 4,200, the nearest by ratio is 3,800.
As the levels climb
- A third number joins first, and with it division: half the time the last number divides. Subtract its zeros, and divide its leading figure into the product of the others.
- Then each number gains a digit, up to nine digits. More digits only means more zeros, so count them in threes — thousand, million, billion — instead of one at a time.
- The options close in: ten times apart, then four, then 2.5, then 1.8. At ×10 the digit count is enough; from ×2.5 on, keep two figures for leading 1s and balance your roundings, or the error carries you a rung off.
- The clock starts at 4 seconds plus 1 for each number and closes to 55%: about 4.4 seconds for four numbers. Fold each leading figure into one running product as you read, rather than rounding everything first.
Traps to avoid
- A zero miscounted. While the options are a power of ten apart, one lost zero usually lands you exactly on a wrong option. Check against the digit count before you tap.
- Leading 1s rounded down. Two numbers like 1,450 and 1,480 rounded to 1,000 each put you more than twice too low — a whole rung out once the options are ×2.5 apart.
- Every rounding in the same direction. Four numbers each rounded up by 20% overshoot by about two times (1.2⁴ ≈ 2.07). Push at least one the other way.
- Hunting for your own estimate among the options. The right option is the exact answer to two figures, not your one-figure sum, so it rarely matches. Take the nearest rung.
- Division done the wrong way round: the divisor’s zeros added, or the small figure divided by the big one. Say it as “this many thousands, shared among this many”.
How to practise
- There are no settings here: the ladder sets the pace. When you miss, read the working — it puts the one-figure sum beside the exact value, so you can see which rounding cost you.
- Learn the leading-figure products and quotients until they are instant: 3 × 4 = 12, 7 × 8 = 56, 9 ÷ 4 = 2.25, 7 ÷ 3 ≈ 2.3. They are the whole calculation here.
- When the options reach ×2.5 and ×1.8 and your misses climb, slow down for a few rounds and say the drift aloud for each number before you multiply.
Levels
Each question has a clock that comes from the work it asks for. There are 16 levels with the default settings, and the counter stops at the real top. Your level is judged on your last ten answers, every ten: 90% moves you up, below 80% moves you down, which holds most people near four in five right.
All of it is in the settings beside Start — how many answers a judgement looks at and how often it recalibrates, the thresholds, the time, which questions, how many options, and whether you see each verdict at once or review them together at the judgement. You can also jump to any level.
More number sense
All games →Digits and logs
How many digits has 2¹⁰⁰? What does 7⁴⁰ start with? Work it out from log 2, log 3 and log 7 — no calculator, and the answer is exact.Last digits and remainders
The last digit of 17²⁰²⁶, 3¹⁰⁰ mod 7, a product mod 9: powers fall into short cycles, and the cycle gives the answer without the number.Energy you can feel
A kettle in kWh, a phone battery in Wh, a snack in kJ and in kettle boils, the cost of lifting a weight: energy, worked out from a few exact definitions.Shrink the universe
If the Sun were a 1 m ball, Earth would be a 9 mm bead 107 m away. Scale the solar system, the Moon and the nearest star, and feel the sizes.Read the chart honestly
Bars that start at 40, "up 5%" that means 5 points, a bigger share of a smaller town: say what a chart really supports, from its numbers.Everyday maths
Percentages, discounts, change, splitting a bill, clock times, speed, price per gram, exchange rates and interest — the arithmetic of an ordinary day.