I’m trying to get better at estimating products in my head, and my brain keeps doing that thing where it wants all the numbers to be neat fives and tens. For example, with 398 × 52, I keep bouncing between different roundings and second-guessing myself. If I go to 400 × 50 I get 20,000, which feels pleasantly tidy, but I’m not sure if that’s the best direction. If I do 400 × 52 I get 20,800, and 398 × 50 would be 19,900. Now I’m staring at three different estimates and wondering which one is the sensible choice for a quick, reasonably tight estimate. My current attempt is 400 × 50 = 20,000, because it balances one up and one down, but I don’t know if that actually makes it closer on average or if I just like the zeros too much. I think my confusion is about when to round up versus down to avoid a big bias, and whether I should be compensating (like, nudge one number up and the other slightly down) to keep the product from drifting. Is there a simple rule of thumb for picking the rounding direction for products like this? And is there a quick way to guess how far off my estimate might be, percentage-wise, without doing the full multiplication? Any help appreciated!
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3 Responses
Use percent changes and try to make them cancel; here 398→400 is about +0.5% and 52→50 is about −3.8%, so 400×52 is tighter than 400×50. For a quick error check, add the percent tweaks (about −3.3%), or just add the absolute changes 2 and 2 to get roughly 4% (a rough-and-ready shortcut).
I think the trick is to look at percentage changes rather than how “nice” the numbers look. If you nudge a factor, ask “by what percent did I change it?” Then try to pick roundings so those percent changes roughly cancel: one up, one down, with similar sizes. For 398 × 52, going 398 → 400 is about +0.5% (up by 2 on ~400), while 52 → 50 is about −3.8% (down by 2 on 52). If you do 400 × 50, those don’t balance, so you should expect a total error around −3.3%, and indeed 20,000 is a bit low compared to the true 20,696. If you do 400 × 52, the only change is +0.5%, so 20,800 is about 0.5% high, which is pretty tight. As a quick rule of thumb, add the percent changes (with signs) to estimate your percent error; that’s usually good enough for mental math. And when the percent changes match and cancel, the estimate is perfect… or at least close enough that I stop worrying about it. One more tiny example: 61 × 39 ≈ 60 × 40; that’s −1.6% and +2.6%, net about +1%, so 2,400 is roughly 1% high versus the exact 2,379.
Your inner neat-zeros gremlin is strong, but here’s a simple spell to tame it: think in percentages, not places. When you round a × b to â × b̂, the product’s percentage error is roughly the sum of the percentage errors on each factor: (â − a)/a + (b̂ − b)/b. So the best quick estimate is the “nice” pair whose percentage errors are small and (ideally) cancel. One-up-one-down only helps if their percent shifts are similar in size; otherwise the bigger error wins and drags the product high or low.
Watch it on 398 × 52:
– 398 → 400 is about +0.5% (2/398), 52 → 50 is about −3.85% (−2/52), so 400 × 50 will be about −3.35% low ≈ 20,000 (and indeed the true product is 20,696).
– 400 × 52 has +0.5% + 0% ≈ +0.5% high → 20,800 (only slightly over).
– 398 × 50 stacks −0.5% and −3.85% → about −4.35% low → 19,900.
So the “sensible” pick is 400 × 52, because its combined percent error is closest to zero. A handy bonus tip: if you do round one factor by, say, −4%, try nudging the other by about +4% to compensate; if the nearest “nice” number can’t match that, skip the compensation. More on this style of estimation here: https://www.mathsisfun.com/numbers/estimation.html
Hope this helps!