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How Many Digits?
Age
14 to 16
Short
Challenge Level
Secondary curriculum
Problem
Solutions
Answer
: 35
Using estimation
$38 ~659 ...\approx 4\times10^{19}$ ($=4000...0$ (19 zeroes))
$342 ~973 ... \approx 3\times10^{14}$ ($=300...0$ (14 zeroes))
$4\times10^{19}\times 3\times10^{14}=12\times10^{33}$, which has 33 zeroes so 35 digits.
But the first number was rounded up, the second number was rounded down. We need to round both up or both down.
$342 ... \approx 4\times10^{14}$ (rounded up)
$4\times10^{19}\times 4\times10^{14}=16\times10^{33}$ which also has 35 digits.
$38 ...\approx 3.8\times10^{19}$ (rounded down)
$3.8\times10^{19}\times 3\times10^{14}=11.4\times10^{33}$ which also has 35 digits.
Comparing to an easier product
$38 ~659 ...=3.865...\times10^{19}$
$342 ~973 ... =3.429...\times10^{14}$
Product $=3.865...\times10^{19}\times 3.429...\times10^{14}$ has 34 digits if $3.865...\times 3.429...\lt10$, and 35 digits if it is $\gt10$
$3.333...\times3=10$ and $3.333...\lt3.8...$, $3\lt3.4...$ so $3.865...\times 3.429...\gt10$
You can find more short problems, arranged by curriculum topic, in our
short problems collection
.