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.