Problem
Some decimals repeat forever
A division gives a decimal that never ends. We need two digits from far down in it.
Number system
Your answer
How to solve
Strategy Look for a Pattern — Twenty places of long division is not the intended work. Find the block that repeats and its length, then ask where each place lands inside a block -- a remainder question, done twice.
1STEP 1
Clear the decimal from the divisor
Multiplying both numbers by 10 leaves the quotient unchanged and makes the division easier to see.
14 ÷ 2.7 = 140 ÷ 27
Same quotient, whole numbers.
6.NS.B.3Work Backwards2STEP 2
Divide until it repeats
The digits 1, 8, 5 come out and then the same remainder returns, so the pattern starts over.
140 ÷ 27 = 5.185185…
The block 185 repeats, three digits long.
7.NS.A.2Solve An Easier Related ProblemThe digits repeat forever in a block of three.
Why?
Dividing by can only ever leave one of a fixed set of remainders, so sooner or later a remainder comes back that has been seen before.
🧱One-to-one correspondenceIf each thing here pairs with exactly one thing there, the two groups are the same size.
Why?
Once the same remainder returns, every following step repeats what it did before, so the digits cycle with a fixed period.
🧱A cycle returns after a full periodA fixed repeating cycle of length p comes back to the same position every p steps.
3STEP 3
Locate the 15th place
Divide 15 by the block length of 3. It comes out exactly, so the 15th place is the last digit of a block.
15 ÷ 3 = 5
The 15th digit is 5.
7.NS.A.2Look For A Pattern4STEP 4
Locate the 20th place
This time there is a remainder of 2, so the 20th place is the second digit of a block.
20 = 3 × 6 + 2
The 20th digit is 8.
7.NS.A.2Look For A PatternAnswer
15th place: 5, 20th place: 8
Count out the first few: places 1 to 6 are 1, 8, 5, 1, 8, 5, so every place that is a multiple of 3 holds 5, and every place two past a multiple holds 8.
Takeaway
A far-off digit of a repeating decimal is decided by a remainder, not by finishing the division.
- Clear the decimal from the divisor
- Divide until it repeats
- Locate the 15th place
- Locate the 20th place