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Aisha and Zaharah visit their parents. Aisha visits every 3 days and Zaharah visits every 5 days. They last visited together on June 1st, 2025. They want to hold a party on a day they both visit. Aish

Mathematics

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4 parts
a

16th June 2025\text{16th June 2025}

b

45,500\text{45,500}

c

120451204_5

d

2 hours and 24 minutes\text{2 hours and 24 minutes}

Step-by-step solution

12 steps

Item 2:

Part a· 3 steps

Help Aisha to determine the date in the near future on which they can hold the party.

  1. Determine the frequency of Aisha's and Zaharah's visits.

    Aisha visits every 3 days.
    Zaharah visits every 5 days.

  2. Find the least common multiple (LCM) of their visiting frequencies to determine when they will visit together again.

    The LCM of 3 and 5 is 3×5=153 \times 5 = 15.
    They will visit their parents together every 15 days.

  3. Calculate the date of their next combined visit.

    Their first visit together was on 1st June 2025.
    The next visit will be 15 days after 1st June 2025.

    1st June+15 days=16th June1 \text{st June} + 15 \text{ days} = 16 \text{th June}

    The date they can hold the party is 16th June 2025.

Part b

45,500\text{45,500}

Determine the amount Aisha borrowed.

  1. Determine the ratio of their contributions based on their ages.

    Aisha is twice as old as Zaharah.
    Let Zaharah's age be ZZ. Then Aisha's age is 2Z2Z.
    The ratio of Aisha's contribution to Zaharah's contribution is 2Z:Z=2:12Z : Z = 2 : 1.

  2. Calculate Aisha's total contribution.

    Zaharah contributes 65,000.
    Since Aisha's contribution is twice Zaharah's contribution:

    Aisha’s contribution=2×Zaharah’s contribution\text{Aisha's contribution} = 2 \times \text{Zaharah's contribution} Aisha’s contribution=2×65,000=130,000\text{Aisha's contribution} = 2 \times 65,000 = 130,000

  3. Calculate the amount Aisha borrowed.

    Aisha borrowed 35% of her contribution.

    Amount borrowed=35% of 130,000\text{Amount borrowed} = 35\% \text{ of } 130,000 Amount borrowed=35100×130,000\text{Amount borrowed} = \frac{35}{100} \times 130,000 Amount borrowed=35×1300=45,500\text{Amount borrowed} = 35 \times 1300 = 45,500

    The amount Aisha borrowed is 45,500.

Part c

120451204_5

Help Zaharah to find the actual 4-digit PIN to use.

  1. Convert the given base ten number to base five.

    The PIN is the base five representation of 179 (base 10).
    Divide 179 by 5 repeatedly and record the remainders.

    179÷5=35 remainder 4179 \div 5 = 35 \text{ remainder } 4 35÷5=7 remainder 035 \div 5 = 7 \text{ remainder } 0 7÷5=1 remainder 27 \div 5 = 1 \text{ remainder } 2 1÷5=0 remainder 11 \div 5 = 0 \text{ remainder } 1

  2. Read the remainders from bottom to top to get the base five number.

    The PIN is 120451204_5.
    The actual 4-digit PIN to use is 1204_5.

Part d

2 hours and 24 minutes\text{2 hours and 24 minutes}

Determine the time they will both take to wash the plates.

  1. Determine the individual work rates.

    Aisha can wash the plates in 4 hours. Her rate is 14\frac{1}{4} of the work per hour.
    Zaharah can wash the plates in 6 hours. Her rate is 16\frac{1}{6} of the work per hour.

  2. Calculate their combined work rate.

    When working together, their rates add up.

    Combined rate=Aisha’s rate+Zaharah’s rate\text{Combined rate} = \text{Aisha's rate} + \text{Zaharah's rate} Combined rate=14+16\text{Combined rate} = \frac{1}{4} + \frac{1}{6}

    Find a common denominator, which is 12.

    Combined rate=312+212=512 of the work per hour\text{Combined rate} = \frac{3}{12} + \frac{2}{12} = \frac{5}{12} \text{ of the work per hour}

  3. Calculate the time taken to complete the work together.

    Time taken = Total workCombined rate\frac{\text{Total work}}{\text{Combined rate}}. Since total work is 1 (representing all plates):

    Time taken=1512=125 hours\text{Time taken} = \frac{1}{\frac{5}{12}} = \frac{12}{5} \text{ hours}

  4. Convert the time to hours and minutes.

    125 hours=225 hours\frac{12}{5} \text{ hours} = 2 \frac{2}{5} \text{ hours}

    To convert 25\frac{2}{5} hours to minutes:

    25×60 minutes=2×12 minutes=24 minutes\frac{2}{5} \times 60 \text{ minutes} = 2 \times 12 \text{ minutes} = 24 \text{ minutes}

    The time they will both take to wash the plates is 2 hours and 24 minutes.

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