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e) Akili, Njozi and Aziz ring their bells. Akili rings his after 2 minutes, Njozi after every 3 minutes and Aziz after 4 minutes. After how many minutes will all the bells ring together? f) The school

Mathematics

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2 parts
e

12 minutes12 \text{ minutes}

f

20 male teachers20 \text{ male teachers}

Step-by-step solution

4 steps

Part e· 2 steps

Akili, Njozi and Aziz ring their bells. Akili rings his after 2 minutes, Njozi after every 3 minutes and Aziz after 4 minutes. After how many minutes will all the bells ring together?

  1. Find the Least Common Multiple (LCM) of the ringing intervals.

    The bells ring every 2, 3, and 4 minutes. We need to find the LCM of 2, 3, and 4.

  2. List multiples or use prime factorization.

    Multiples of 2: 2, 4, 6, 8, 10, 12, ...
    Multiples of 3: 3, 6, 9, 12, ...
    Multiples of 4: 4, 8, 12, ...
    The smallest common multiple is 12.

    Alternatively, using prime factorization:
    2=212 = 2^1
    3=313 = 3^1
    4=224 = 2^2
    LCM = 22×31=4×3=122^2 \times 3^1 = 4 \times 3 = 12.

    All the bells will ring together after 12 minutes\boxed{12 \text{ minutes}}.

Part f

20 male teachers20 \text{ male teachers}

The school has 80 teachers if 34\frac{3}{4} are female teachers. How many male teachers are there in that school?

  1. Determine the fraction of male teachers.

    If 34\frac{3}{4} of the teachers are female, then the remaining fraction are male.
    Fraction of male teachers = 1−34=44−34=141 - \frac{3}{4} = \frac{4}{4} - \frac{3}{4} = \frac{1}{4}.

  2. Calculate the number of male teachers.

    Total number of teachers = 80.
    Number of male teachers = 14×80\frac{1}{4} \times 80.
    Number of male teachers = 804=20\frac{80}{4} = 20.

    There are 20 male teachers\boxed{20 \text{ male teachers}} in that school.

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