Compare Numbers 10⊡ and 140: Determining the Correct Inequality Sign

Q: Why can't the answer be '=' or '>'?

Since $ 10\Box $ can only range from 100 to 109, and even the largest value 109 is less than 140, there's no way for $ 10\Box $ to equal or exceed 140.

Q: What does the □ symbol represent?

The $ \Box $ represents any single digit from 0 to 9. It's the tens digit in the three-digit number $ 10\Box $.

Q: Do I need to check all 10 possible digits?

Not individually! Instead, find the range by checking just the smallest (□ = 0) and largest (□ = 9) possibilities. This covers all cases efficiently.

Q: What if the hidden digit was in a different position?

The position matters! If it were $ 1\Box0 $, the range would be 100-190. Always identify which digit place the □ represents first.

Q: Could there be 'no correct answer'?

Only if the comparison were impossible to determine. Since we can prove \( 10\Box for all possible digits, '

Number Comparison with Hidden Digits

There is an unknown digit behind the $\Box$

Determine the correct sign for the following expression:

$10\Box?140$

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Step-by-step written solution

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Understand the problem

There is an unknown digit behind the $\Box$

Determine the correct sign for the following expression:

$10\Box?140$

Step-by-step solution

To solve this problem, we must determine which inequality sign fits the comparison $10\Box?140$ .

Step 1: Consider the expression $10\Box$ as $100 + \Box$ , where $\Box$ is any digit from 0 to 9.
Step 2: Calculate the range of values for $10\Box$ :
If $\Box = 0$ , then $10\Box = 100 + 0 = 100$ .
If $\Box = 9$ , then $10\Box = 100 + 9 = 109$ .
Step 3: Compare all possible values of $10\Box$ (100 to 109) with 140.
Step 4: Since 109, the largest possible value, is less than 140, for any digit $\Box$ , $10\Box$ will always be less than 140.

Therefore, the correct inequality is $10\Box < 140$ . The proper sign for the expression is $<$ .

Final Answer

Key Points to Remember

Essential concepts to master this topic

Range Analysis: Find minimum and maximum possible values first
Technique: $10\Box$ ranges from 100 to 109
Check: Compare largest possible value 109 < 140 ✓

Common Mistakes

Avoid these frequent errors

Checking only one possible digit value
Don't test just one digit like □ = 5 and conclude 105 < 140! This ignores other possibilities and may miss cases where the inequality could change. Always check the full range from minimum (□ = 0) to maximum (□ = 9) values.

Practice Quiz

Test your knowledge with interactive questions

Arrange the numbers in ascending order (smallest to largest):

$$ 85,69,88,61,90 $$

FAQ

Everything you need to know about this question

Why can't the answer be '=' or '>'?

Since $10\Box$ can only range from 100 to 109, and even the largest value 109 is less than 140, there's no way for $10\Box$ to equal or exceed 140.

What does the □ symbol represent?

The $\Box$ represents any single digit from 0 to 9. It's the tens digit in the three-digit number $10\Box$ .

Do I need to check all 10 possible digits?

Not individually! Instead, find the range by checking just the smallest (□ = 0) and largest (□ = 9) possibilities. This covers all cases efficiently.

What if the hidden digit was in a different position?

The position matters! If it were $1\Box0$ , the range would be 100-190. Always identify which digit place the □ represents first.

Could there be 'no correct answer'?

Only if the comparison were impossible to determine. Since we can prove $10\Box < 140$ for all possible digits, '<' is definitely correct.

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