High School

1. I am a 2-digit number. My tens digit is twice my one's digit. If you reverse my digits, the new number is 27 less than me. What number am I?

2. A clock shows 10:15. What is the angle between the hour and minute hands?

3. Find the missing term in the pattern: A, C, F, J, ?, T

4. Tom's father is twice as old as Tom. 10 years ago, he was three times Tom's age. How old is Tom now?

5. You have a 3 L jug and a 5 L jug. How can you measure exactly 4 L using these two jugs?

Answer :

  1. To find the 2-digit number where the tens digit is twice the one's digit and reversing the digits gives a number 27 less than the original:

    • Let the one's digit be [tex]x[/tex].
    • Then the tens digit is [tex]2x[/tex].
    • The number can be expressed as [tex]10(2x) + x[/tex] or [tex]21x[/tex].
    • When the digits are reversed, the new number is [tex]10x + 2x = 12x[/tex].
    • According to the problem, [tex]21x - 12x = 27[/tex].
    • Solving gives [tex]9x = 27[/tex], so [tex]x = 3[/tex].
    • Thus, the original number is [tex]21 \times 3 = 63[/tex].

  2. To find the angle between the hour and minute hands at 10:15:

    • The minute hand is at 15 minutes, which is at [tex]15 imes 6 = 90[/tex] degrees from the 12.
    • The hour hand at 10:15 is [tex]10 imes 30 + 15 imes 0.5 = 307.5[/tex] degrees from the 12.
    • The difference between them is [tex]307.5 - 90 = 217.5[/tex] degrees.
    • However, the smaller angle between them is [tex]360 - 217.5 = 142.5[/tex] degrees.

  3. The missing term in the pattern A, C, F, J, ?, T:

    • The pattern skips 1, 2, 3, and 4 letters respectively: A (1), C (+2), F (+3), J (+4), now add 5.
    • After J is J+5=O (skipping K, L, M, N).
    • Thus, the missing term is O.

  4. To find Tom's current age:

    • Let Tom's current age be [tex]t[/tex], so his father is [tex]2t[/tex].
    • Ten years ago, Tom was [tex]t - 10[/tex] and his father was [tex]2t - 10[/tex], which was 3 times Tom's age.
    • So, [tex]2t - 10 = 3(t - 10)[/tex].
    • Solving this: [tex]2t - 10 = 3t - 30[/tex] leads to [tex]20 = t[/tex].
    • Thus, Tom is 20 years old.

  5. To measure exactly 4 L using a 3 L jug and a 5 L jug:

    • Fill the 5 L jug completely.
    • Pour from the 5 L jug into the 3 L jug until it's full, leaving 2 L in the 5 L jug.
    • Empty the 3 L jug, and transfer the 2 L from the 5 L jug to the 3 L jug.
    • Fill the 5 L jug again.
    • Pour from the 5 L jug into the 3 L jug until it fills up (1 L needed), thus leaving exactly 4 L in the 5 L jug.

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