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What is the 100th digit to the right of the decimal point in decimal expression of \(\frac{1}{27}\)?
Let's break down what we're being asked to find in everyday terms. Imagine you have a pie that you want to divide equally among 27 people - each person gets \(\frac{1}{27}\) of the pie. Now, if we want to express this fraction as a decimal (like 0.something), we need to figure out what comes after the decimal point. The question is asking: if we write out this decimal and count 100 places to the right of the decimal point, what single digit will we find there?
This is like asking what the 100th letter would be in a very long word - we need to find the pattern so we don't have to write out all 100 digits manually.
Process Skill: TRANSLATE - Converting the fraction problem into a decimal pattern-finding problem
To convert \(\frac{1}{27}\) to a decimal, we perform long division. Let's think of this as: "How many times does 27 go into 1?" Since 27 is bigger than 1, the answer starts with "0." and then we need to see what comes next.
Let me show you the long division step by step:
- \(1.000000... \div 27\)
- 27 goes into 10 zero times, remainder 10
- 27 goes into 100 three times (\(3 \times 27 = 81\)), remainder 19
- 27 goes into 190 seven times (\(7 \times 27 = 189\)), remainder 1
- 27 goes into 10 zero times, remainder 10 (we're back where we started!)
So far we have: \(0.037...\)
Notice something important: when we got remainder 1, and then tried to divide 10 by 27, we got the exact same situation as when we started! This means the pattern will repeat from here.
Continuing the pattern:
\(\frac{1}{27} = 0.\overline{037}\)
The decimal representation is \(0.037037037...\) where "037" repeats forever.
From our long division, we can see that the decimal representation of \(\frac{1}{27}\) has a repeating pattern. The repeating block is "037" - these three digits repeat over and over again indefinitely.
Let's verify this makes sense:
- Position 1: 0
- Position 2: 3
- Position 3: 7
- Position 4: 0 (starts repeating)
- Position 5: 3
- Position 6: 7
- Position 7: 0
- And so on...
So our repeating cycle has length 3, and the digits in the cycle are 0, 3, 7 (in that order).
Process Skill: INFER - Recognizing that when the remainder repeats in long division, the decimal pattern will repeat
Now, instead of writing out 100 digits, we can use the repeating pattern to find what we need. Since the pattern "037" repeats every 3 positions, we need to figure out where position 100 falls within this cycle.
Think of it this way: if we group the positions into sets of 3:
- Positions 1,2,3: 0,3,7
- Positions 4,5,6: 0,3,7
- Positions 7,8,9: 0,3,7
- And so on...
To find which digit is in position 100, we divide 100 by 3:
\(100 \div 3 = 33\text{ remainder }1\)
This means:
- We complete 33 full cycles of "037"
- Then we need the 1st digit of the next cycle
- The 1st digit in each cycle "037" is 0
Therefore, the 100th digit is 0.
The 100th digit to the right of the decimal point in the decimal expression of \(\frac{1}{27}\) is 0.
Looking at our answer choices: (A) 0, (B) 2, (C) 3, (D) 6, (E) 7
The answer is (A) 0.
1. Misunderstanding what "to the right of the decimal point" means: Students might include the units digit (before the decimal) in their counting, thinking the first digit is the units place rather than starting from the first digit after the decimal point.
2. Not recognizing that fractions with denominators create repeating decimals: Students may attempt to calculate all 100 digits manually instead of looking for a repeating pattern, making the problem unnecessarily complex and time-consuming.
3. Confusion about the relationship between remainders and repeating cycles: Students might not understand that when performing long division, a repeated remainder indicates the start of a repeating cycle in the decimal expansion.
1. Arithmetic errors during long division: Students commonly make calculation mistakes when dividing, such as incorrectly computing \(7 \times 27 = 189\) or misplacing digits, leading to an wrong repeating pattern.
2. Incorrectly identifying the repeating cycle: Students might identify the wrong repeating block (like "37" instead of "037" or "370" instead of "037") by not carefully tracking when the remainder pattern repeats.
3. Miscalculating the modular arithmetic: When finding \(100 \div 3\), students might incorrectly compute the remainder or misunderstand what the remainder represents in terms of position within the cycle.
1. Using the wrong remainder interpretation: Students might correctly find that \(100 \div 3 = 33\text{ remainder }1\), but then incorrectly conclude this means the 100th digit is the same as the 33rd digit, rather than the 1st digit of the repeating pattern.
2. Off-by-one errors in cycle positioning: Students may confuse whether remainder 1 corresponds to the 1st, 2nd, or 3rd position in the cycle "037", potentially selecting 3 or 7 instead of 0.