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#27 Dancing decimals

Set by Mary Ellis

1/3 can be written as 0.33333… (with a string of 3s going on forever) and 4/9 can be written as 0.44444… . We call these recurring decimals.

Use a calculator to see what 1/11, 2/11 and 3/11 are as decimals. Can you predict what 7/11 and 10/11 will be?

There is a different type of pattern in the sevenths: calculate 1/7 and 2/7, then try to predict what 3/7, 4/7, 5/7 and 6/7 will be.

Can you also find any patterns in the multiples of 1/13? (It is similar to the sevenths, but you might need to calculate more than two of the fractions to be able to predict the rest.)

Solution next week

#26 Up and down

Solution

30 × 30 is bigger. Starting with 30 × 30, we can remove one 30 to get 29 lots of 30. Now, we can add another 29 to get 31 lots of 29, or 31 × 29. Since we removed 30 and added 29, we have made the product smaller.

The same argument works for numbers other than 30. Algebraically, (n+1)(n-1) = n² – 1, which is always less than n².

If instead of 1, we chose to add and subtract k, the product is (30+k)(30-k) = 30² – k², so the difference will be k².