Problem
A repeating block gives any decimal place
An improper fraction is described by the quotient and remainder of its numerator over 27. Written as a decimal, we need the digit far down in the 18th place.
Number system
Your answer
How to solve
Strategy Look for a Pattern — Nobody divides eighteen places by hand. Once the block that repeats is known, the question becomes 'where in the block does place 18 land?', which is a remainder question.
1STEP 1
Rebuild the fraction
The numerator is the divisor times the quotient plus the remainder.
27 × 1 + 13 = 40
The fraction is 40/27.
6.NS.B.3Identify Subproblems2STEP 2
Divide far enough to see it repeat
Dividing 40 by 27 gives 1 and then the digits 4, 8, 1 before the same remainder comes back and the pattern starts again.
40 ÷ 27 = 1.481481…
The block 481 repeats forever.
7.NS.A.2Solve An Easier Related Problem3STEP 3
Find where the 18th place falls in the block
The block is 3 digits long, so divide 18 by 3. It comes out exactly, which means the 18th place is the last digit of a block.
18 ÷ 3 = 6
It is the third digit of the block.
7.NS.A.2Look For A PatternThe th decimal place holds the last digit of the repeating block.
Why?
The digits repeat in a fixed block of three, so the sequence comes back to the same place in the block every three steps.
🧱A cycle returns after a full periodA fixed repeating cycle of length p comes back to the same position every p steps.
Why?
is six whole blocks with nothing over, so place lands exactly at the end of a block.
🧱Multiplication as equal groupsa times b means a equal groups of b — it is just repeated adding of the same amount.
4STEP 4
Read the digit off
The block is 4, 8, 1, so its third digit is 1.
The 18th decimal place holds 1.
7.NS.A.2Look For A PatternAnswer
1
Count a few by hand: places 1 to 6 are 4, 8, 1, 4, 8, 1, so every place that is a multiple of 3 holds 1 -- and 18 is one of them.
Takeaway
When a decimal repeats, a far-off digit is a remainder question, not a division you have to finish.
- Rebuild the fraction
- Divide far enough to see it repeat
- Find where the 18th place falls in the block
- Read the digit off