Introduction / Context:
This is a digit and operation puzzle that asks you to reach 100 in a restricted way. You must use exactly four instances of the digit 9 and one instance of the digit 1, and you are allowed to use only one arithmetic symbol. That symbol might be plus, minus, multiply, or divide, but you cannot combine several different operators. The challenge is to arrange the digits into one or more multi digit numbers and then connect them with a single operator so that the final result is exactly 100. This tests creativity, number sense, and careful reading of constraints.
Given Data / Assumptions:
- You must use the digits 9, 9, 9, 9, and 1 exactly once each.
- You may form multi digit numbers such as 99 or 199.
- You are allowed only one arithmetic symbol between numbers.
- The expression must evaluate exactly to 100.
- Standard arithmetic rules for addition, subtraction, multiplication, and division apply.
Concept / Approach:
The key idea is to think about how close you can get to 100 using multi digit numbers made from the given digits. Subtraction is particularly useful because it can turn a difference of two multi digit numbers into an exact target. If you arrange three of the digits as 199 and the remaining two as 99, then 199 minus 99 is exactly 100. This uses the digits 1, 9, 9, 9, and 9 and only a single subtraction sign, satisfying all the conditions laid out in the puzzle.
Step-by-Step Solution:
Step 1: Try to form numbers that are reasonably close to 100 so that a simple operation can reach 100.
Step 2: Consider forming 199 from digits 1, 9, and 9, leaving 9 and 9 to form 99.
Step 3: Test the expression 199 - 99.
Step 4: Calculate 199 - 99. The difference is 100.
Step 5: Check the constraints: the digits used are exactly four 9s and one 1, and only one subtraction symbol appears.
Step 6: Verify that the other answer options either use more than one operation symbol or do not give exactly 100.
Verification / Alternative check:
For a quick verification, add 99 and 100 to see that their sum is 199. This confirms that 199 is exactly 100 more than 99, so 199 minus 99 must be 100. It is also helpful to check that you did not accidentally introduce extra digits or operations. In this expression, the multi digit numbers 199 and 99 are built strictly from the four 9s and one 1. No extra symbol beyond the single minus sign appears, so the solution is fully valid under the puzzle rules.
Why Other Options Are Wrong:
The expression 91 + 9/9 uses both addition and division, so it does not respect the one symbol rule. The expression 9 * 9 + 1 + 1 uses multiplication and two additions, again violating the constraint on the number of symbols. The expression 99 + 1/9 uses both addition and division and also does not evaluate to 100. Thus, even if some of these get numerically close to 100, they do not follow the strict requirement of using only one arithmetic symbol.
Common Pitfalls:
A typical mistake is to overlook the restriction on the number of operations and to focus only on reaching 100. Many clever looking expressions that equal 100 use multiple operations or parentheses, which are not allowed here. Another pitfall is to think only in terms of single digit numbers and forget that multi digit numbers like 99 or 199 are permitted. Once you remember that 199 and 99 are possible and consider subtraction, the path to the correct answer becomes straightforward.
Final Answer:
The correct expression is
199 - 99, which uses four 9s, one 1, and exactly one arithmetic symbol to make 100.
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