The Mystery Number for the Math Fair
The math club is creating a "Guess My Number" booth for the school fair. The mystery number is a two-digit whole number. The club writes three clues on a poster: (1) The tens digit is 3 more than the ones digit. (2) The sum of the digits is 11. (3) The number is less than 80. Two students disagree: Jada says the number must be 74, while Omar says there is not enough information because many two-digit numbers fit clue 3. The club also wants to add a fourth clue to make the answer unique if needed. They must check whether clues 1 and 2 already give one answer, and whether clue 3 rules out any possible answer. Finally, they want to compare the mystery number with the product of its digits using <, >, or =.
1. Use clues 1 and 2 to find the mystery number. Show your reasoning with a variable or a table.
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Let the ones digit be x. Then the tens digit is x + 3.
Clue 2 says the digit sum is 11:
(x + 3) + x = 11
2x + 3 = 11
2x = 8
x = 4
So the ones digit is 4 and the tens digit is 4 + 3 = 7. The number is 74.
Check: 7 is 3 more than 4, and 7 + 4 = 11. Clue 3 also fits because 74 < 80.
2. Omar says clue 3 makes the answer not unique. Is he right? Explain.
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No. Clues 1 and 2 together give exactly one number, 74. Clue 3 is extra information that only confirms 74 is allowed. It does not create multiple answers. Omar is thinking that clue 3 alone allows many numbers, but the question uses all the clues together.
3. Compare the mystery number to the product of its digits. Write a true statement using <, >, or =.
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The digits are 7 and 4. Their product is 7 × 4 = 28. The mystery number is 74.
Since 74 is greater than 28, the true statement is:
74 > 28
4. If the club wanted to add a fourth clue that keeps 74 as the only answer, what is one clue that would work?
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Many clues could work, as long as they do not allow another number. For example: "The number is even." 74 is even, and no other number fits clues 1 and 2 anyway. A clue like "The ones digit is 4" would also work. The important point is that the fourth clue must be true for 74 and must not contradict the earlier clues.