Cluster estimation

Cluster estimation can be used to estimate sums and products when the numbers you are adding or multiplying cluster near or is close in value to a single number.
Example # 1:

Estimate 699 + 710 + 695 + 705 + 694 + 715

Carefully examine all the numbers above. You should notice that they all cluster around 700

Therefore, 700 + 700 + 700 + 700 + 700 + 700 will give us a good estimate for the answer.

Instead of adding 700 six times, just do 6 × 700.

700 × 6 = 4,200 is a good estimate of the sum

In fact, the real answer 699 + 710 + 695 + 705 + 694 + 715 = 4,218. Thus 4,200 is indeed close to the real answer.

Example # 2:

Estimate 257 + 247 + 255 + 245

Carefully examine all the numbers above. You should notice that they all cluster around 250

Therefore, 250 + 250 + 250 + 250 will give us a good estimate for the answer.

Instead of adding 250 four times, just do 4 × 250.

250 × 4 = 1,000 is a good estimate of the sum

In fact, 257 + 247 + 255 + 245 = 1004. Thus 1000 is indeed close to the real answer.

Example # 3:

Estimate 23 × 18 × 22 × 17

This time, you are estimating a multiplication problem. However,you will still use cluster estimation to estimate the product.

Just notice that all numbers above cluster around 20

Therefore, 20 × 20 × 20 × 20 will give us a good estimate for the answer.

Multiply the 2s to get 16.

Then, just put four zeros after 16 to get 160,000

The real answer is 23 × 18 × 22 × 17 = 154,836.

It is not a very good estimate, but we can live with that

In general, addition will give better estimates

Example # 4:

Estimate 8 × 11 × 12

Just notice that all numbers above cluster around 10

Therefore, 10 × 10 × 10 will give us a good estimate for the answer.

10 × 10 × 10 = 1000

The real answer is 8 × 11 × 12 = 1056.