19 (a) Fountain A squirts water every 24 minutes. Fountain B squirts water every 42 minutes. They squirt water together at 15:19. Find the next time they squirt water together. (b) A school sends 60 students from Year 8 and 105 students from Year 9 to a museum. The school divides these students into groups using the following rules. - The groups must all be the same size. - All students in any group must be from the same year. - There should be as few groups as possible. Find the size of each group and the total number of groups. Size of each group = Total number of groups =

GCSE OCR Maths Prime Factors, HCF and LCM: 19 (a) Fountain A squirts water

19 (a) Fountain A squirts water every 24 minutes - OCR - GCSE Maths - Question 19 - 2021 - Paper 1

Question 19

19 (a) Fountain A squirts water every 24 minutes. Fountain B squirts water every 42 minutes. They squirt water together at 15:19. Find the next time they squirt wat... show full transcript

Worked Solution & Example Answer:19 (a) Fountain A squirts water every 24 minutes - OCR - GCSE Maths - Question 19 - 2021 - Paper 1

Step 1

Find the next time they squirt water together.

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Answer

To find the next time that fountain A and fountain B squirt water together, we first need to determine the least common multiple (LCM) of their squirt intervals, which are 24 minutes and 42 minutes.

Calculate LCM:

The prime factorization of 24 is: $24 = 2^3 \times 3^1$

The prime factorization of 42 is: $42 = 2^1 \times 3^1 \times 7^1$

The LCM is found by taking the highest power of each prime: $LCM = 2^3 \times 3^1 \times 7^1 = 168$

Thus, they will squirt together again after 168 minutes.

Convert minutes into hours and minutes:
- 168 minutes is 2 hours and 48 minutes.
- Starting from 15:19, add 2 hours 48 minutes:
- 15:19 + 2:48 = 18:07
Therefore, the next time they squirt water together is at 18:07.

Step 2

Find the size of each group and the total number of groups.

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Answer

To find the size of each group and the total number of groups, we need to determine the greatest common divisor (GCD) of the number of students from Year 8 and Year 9, which are 60 and 105 respectively.

Calculate GCD:

The prime factorization of 60 is: $60 = 2^2 \times 3^1 \times 5^1$

The prime factorization of 105 is: $105 = 3^1 \times 5^1 \times 7^1$

The GCD can be taken by the lowest powers of the common primes: $GCD = 3^1 \times 5^1 = 15$

Thus, each group should have 15 students.

Calculate the total number of groups:
- From Year 8: $\frac{60}{15} = 4$ groups
- From Year 9: $\frac{105}{15} = 7$ groups
Therefore, the total number of groups is: $4 + 7 = 11$ groups.

In summary:

Size of each group = 15
Total number of groups = 11

19 (a) Fountain A squirts water every 24 minutes - OCR - GCSE Maths - Question 19 - 2021 - Paper 1

Worked Solution & Example Answer:19 (a) Fountain A squirts water every 24 minutes - OCR - GCSE Maths - Question 19 - 2021 - Paper 1

Find the next time they squirt water together.

Find the size of each group and the total number of groups.

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