Find the least common denominator for fractions with the given denominators of 6, 8 and 10. Explain also what the LCM for these numbers would be.
(c).The average person drinks about 3 1/5 cups of coffee per day. If a person works 5 days a week for 50 weeks a year. Find how many cups of coffee that person will drink in 51¨ü years of lifetime working.
(d) . A worker has 2 1/6 h of overtime on Tuesday, 1 ¨ú h on Wednesday, and 2 5/6 h on Friday. What is the total overtime for the week?
(e) Divide 35529 by the number 573 and find the remainder r<10.
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To find the least common denominator (LCD) for fractions with the given denominators of 6, 8, and 10, we can follow these steps:
Step 1: List the prime factors of each denominator.
- For 6: 2 * 3
- For 8: 2 * 2 * 2
- For 10: 2 * 5
Step 2: Identify the common prime factors and their highest powers.
- The common prime factors are 2 and 3.
- The highest power of 2 is 3 (from 8).
- The highest power of 3 is 1 (from 6).
- There are no common factors beyond 2 and 3.
Step 3: Multiply the common prime factors by their highest powers.
- 2^3 * 3^1 = 8 * 3 = 24
Hence, the least common denominator for fractions with denominators 6, 8, and 10 is 24.
Moving on to the next question:
(c) To find how many cups of coffee a person will drink in 51 years of working, we can use the given information:
- Average cups of coffee per day: 3 1/5
- Days worked per week: 5
- Weeks worked per year: 50
Step 1: Find the number of cups of coffee consumed in a week:
- Cups per day: 3 1/5 = (5 * 3) + 1 = 16/5
- Cups per week: 16/5 * 5 = 16 cups
Step 2: Find the number of cups of coffee consumed in a year:
- Cups per year: 16 cups * 50 weeks = 800 cups
Step 3: Find the number of cups of coffee consumed in 51 years:
- Cups in 51 years: 800 cups * 51 years = 40,800 cups
Therefore, the person will drink 40,800 cups of coffee in 51 years of working.
Let's move on to the next question:
(d) To find the total overtime for the week, we need to add up the hours of overtime for Tuesday, Wednesday, and Friday:
- Overtime on Tuesday: 2 1/6 hours = (2 * 6 + 1) / 6 = 13/6 hours
- Overtime on Wednesday: 1 1/2 hours = (1 * 2 + 1) / 2 = 3/2 hours
- Overtime on Friday: 2 5/6 hours = (2 * 6 + 5) / 6 = 17/6 hours
Step 1: Add up the overtime hours:
- Total overtime: 13/6 hours + 3/2 hours + 17/6 hours = (13 + 18 + 17) / 6 = 48/6 = 8 hours
Therefore, the total overtime for the week is 8 hours.
Next, we move on to the last question:
(e) To divide 35529 by the number 573 and find the remainder r < 10, we can perform the division and check the remainder:
Step 1: Perform the division:
- 35529 ÷ 573 = 61 remainder 6
Step 2: Check the remainder:
- Since the remainder (6) is less than 10, the condition r < 10 is satisfied.
Therefore, the remainder (r) when dividing 35529 by 573 is 6, and it is less than 10.