Depth · one concept from first principles · step 2 of 4
Over 11 means the block is a multiple of 9
The repeating decimal equals z/11 in lowest terms, with x a one-digit odd number. Count the possible natural numbers z.
Number systemOperations
Your answer
How to solve
Strategy Convert to Algebra — Write both sides over the same denominator 99 to get an equation in whole numbers. It says the two-digit number xy is a multiple of 9, which a digit-sum list handles quickly.
1STEP 1
Put both sides over 99
A two-digit repeating block sits over 99, and z/11 = 9z/99.
(10x + y)/99 = 9z/99 → 10x + y = 9z
So the two-digit number 10x + y must be a multiple of 9.
8.NS.A.1Convert To Algebrameans .
Why?
Multiplying by lines the endless tail up with the original's, so subtracting the number cancels the tail and leaves , the block over .
Why?
Multiplying by moves every digit two places to the left, so , and its whole-number part is worth .
🧱Place-value groupingTen ones make one ten and ten tens make one hundred — each place bundles ten of the last.
Why?
The block starts right after the point and repeats every places, so a shift of exactly places lays a copy of the same digit under every digit of 's tail.
🧱A cycle returns after a full periodA fixed repeating cycle of length p comes back to the same position every p steps.
Why?
Taking the same away from both sides keeps them equal, and the matching tails take away to nothing, leaving .
🧱Do the same to both sides and equality holdsA balanced scale stays level when you take the same amount off both pans or add the same amount to both.
Why?
after multiplying top and bottom by .
🧱Scaling top and bottom keeps a fraction's valueMultiplying or dividing a fraction's top and bottom by the same number only renames it — the pieces change size but the amount you hold does not.
Why?
Two equal amounts cut into the same pieces must count the same number of pieces, so the tops agree.
🧱A fraction is equal shares of one wholeThe bottom number names how many equal pieces one whole was cut into and the top counts them, which is why same-sized pieces add just by counting.
2STEP 2
Use the digit-sum test
A number is a multiple of 9 exactly when its digit sum is. With x ≤ 9 and y ≤ 9, the digit sum x + y is 9 or 18.
x + y = 9 or x + y = 18
Only two digit sums are possible.
4.OA.B.4Convert To Algebra3STEP 3
List the odd x
x + y = 9 gives (1, 8), (3, 6), (5, 4), (7, 2), (9, 0); x + y = 18 gives (9, 9), but = 1, so it is out.
(x, y) = (1, 8), (3, 6), (5, 4), (7, 2), (9, 0)
Each pair gives a different block, so a different z.
4.OA.B.4Make A Systematic List4STEP 4
Read off z
Divide each block by 9: 18, 36, 54, 72, 90 give z = 2, 4, 6, 8, 10. None is a multiple of 11.
z = 2, 4, 6, 8, 10
There are 5 possible values.
4.OA.B.4Make A Systematic ListAnswer
5
Spot-check: = 36/99 = 4/11 and = 90/99 = 10/11, both in lowest terms with denominator 11.
Takeaway
Over 11 means over 99 -- so the repeating block has to be a multiple of 9.
- Put both sides over 99
- Use the digit-sum test
- List the odd x
- Read off z
Where next?
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