Estimating and checking

Idea: Round, then pay back

Round to something easy, compute, then repay exactly what you changed.

• $497+286=500+286-3=783$
• $803-296=803-300+4=507$
• $39\times 47=40\times 47-47=1833$

Example: Repay the rounding

Compute $1998\times 7$.

1. Round: $2000\times 7=14000$
2. Repay the error times the other factor: $2\times 7=14$
3. $14000-14=13986$

Idea: Know which way you pushed

Rounding a factor up raises the product. Push opposite ways and errors partly cancel.

• $100\times 70=7000$, while $103\times 68=7004$
• $400\times 20=8000$ lands above $412\times 19=7828$
• Small percent errors roughly add: $\frac{8000}{7828}\approx 1.02$

Idea: Order of magnitude

Get the power of $10$ right first: multiply the short numbers, add the exponents.

• $3\times 10^5\times 4\times 10^3=1.2\times 10^9$
• A day has $86400$ seconds, about $90000$ for a quick look

Quick check: Events in a day

Events arrive at $250$ a second. How many blocks of $1000000$ arrive in a day of $86400$ seconds?

Idea: Check before you speak

Say you reach $47\times 86=4042$. Run fast checks.

• Last digit: $7\times 6=42$ ends in $2$
• Size: $3200<47\times 86<4500$
• By $9$: remainders $2\times 5=10$ leave $1$, so does the digit sum

Common mistake: A passed check is no proof

$4024$ has the same digit sum as $4042$: the check by $9$ cannot see swapped digits.

Instead Use at least $2$ different checks. Only a failed check proves something.

Example: Read a sequence by size

What comes after $5,15,45$?

1. Gaps grow in proportion: ratio $3$
2. $45\times 3=135$

Idea: With the clock running

Say a rough size before you compute.

• After computing, compare with the rough size
• Then test the last digit