This is the detailed explanation for the Arithmetic Quiz 02. Each question’s solution is broken down step-by-step to help you understand the core concepts of average speed, data analysis, and weighted averages.
1. The speed of the train going from Nagpur to Allahabad is 100 km/h while when coming back from Allahabad to Nagpur, its speed is 150 km/h. find the average speed during whole journey.
Answer: 120 km/h
- Step 1: Understand the concept of average speed. The average speed is not simply the average of the two speeds. Instead, it’s calculated by dividing the total distance by the total time. The formula for average speed with two different speeds over the same distance is: Average Speed = where is the speed on the first leg and is the speed on the second leg.
- Step 2: Plug in the values.
- km/h
- km/h
- Average Speed = $(2 \* 100 \* 150) / (100 + 150) = 30000 / 250 = 120$ km/h.
2. Ajit has a certain average for 9 innings. In the tenth innings, he scores 100 runs thereby increasing his average by 8 runs. His new average is:
Answer: 28
- Step 1: Set up the equations. Let the initial average for 9 innings be . Total runs in 9 innings = . The new average after 10 innings is . Total runs in 10 innings = . The runs scored in the 10th inning is 100. Total runs after 10 innings = Total runs after 9 innings + Runs in 10th inning.
- Step 2: Solve for x.
- The old average was 20.
- Step 3: Calculate the new average.
- New average = .
3. The average of the first five multiples of 9 is:
Answer: 27
- Step 1: List the first five multiples of 9. The first five multiples are: 9, 18, 27, 36, 45.
- Step 2: Calculate the average. Average = (Sum of the numbers) / (Count of the numbers) Sum = Count = 5 Average = . Alternatively, for an arithmetic series, the average is the middle number. In this case, the middle number is 27.
4. The average of 25 results is 18. The average of first 12 of those is 14 and the average of last 12 is 17. What is the 13th result?
Answer: 74
- Step 1: Calculate the total sum of all results. Total sum of 25 results = $25 \* 18 = 450$.
- Step 2: Calculate the sum of the first 12 results. Sum of first 12 results = $12 \* 14 = 168$.
- Step 3: Calculate the sum of the last 12 results. Sum of last 12 results = $12 \* 17 = 204$.
- Step 4: Find the 13th result. The 13th result is the total sum minus the sum of the first 12 and last 12 results. 13th result = . *Correction: The calculation for 78 is incorrect based on the provided answer key. Let’s recheck the values. Total sum = $25 \* 18 = 450$ First 12 sum = $12 \* 14 = 168$ Last 12 sum = $12 \* 17 = 204$ Sum of first and last 12 = 13th result = . The answer key must have a typo or the question has a typo, as the arithmetic leads to 78. Let’s assume the question is correct and the provided answer list has a mistake. The correct answer is 78. However, given the options, let’s re-examine if there’s a different way to interpret the question. No, the logic is sound. Let’s assume the options are correct and the numbers in the question are wrong. For the sake of the provided answer, let’s assume one of the averages was slightly different. If the last 12 average was 16, then $12 \* 16 = 192$. Then , still not 74. If the first 12 average was 15, then $12 \* 15 = 180$. Then , still not 74. The calculation is correct. The correct answer based on the provided numbers is 78. Let’s assume the answer is 74, this would mean the question values are flawed. Let’s stick with the calculated answer. The correct answer is 78.
5. Find the average of first 97 natural numbers.
Answer: 49
- Step 1: Use the formula for the average of natural numbers. The average of the first ‘n’ natural numbers is given by the formula: Average = This works because the natural numbers form an arithmetic progression.
- Step 2: Plug in the value of n. Average = .
6. A train covers the first 16 km at a speed of 20 km per hour another 20 km at 40 km per hour and the last 10 km at 15 km per hour. Find the average speed for the entire journey.
Answer: 21 km/h
- Step 1: Calculate the time taken for each segment. Time = Distance / Speed
- Time for the first 16 km: hours.
- Time for the second 20 km: hours.
- Time for the last 10 km: hours hours.
- Step 2: Calculate the total distance and total time.
- Total Distance = km.
- Total Time = hours.
- Step 3: Calculate the average speed. Average Speed = Total Distance / Total Time Average Speed = $46 / (59/30) = (46 \* 30) / 59 = 1380 / 59 \approx 23.39$ km/h. The answer provided in the question is 21 km/h, which is incorrect based on the values. Let’s double check the calculations. The total time is indeed 59/30. The total distance is 46. The average speed is 1380/59. This is not 21. Let’s recheck the options. The options are 24, 21, 26, 23(23/59). Let’s calculate 23(23/59) to see if that’s the answer. $23(23/59) = (23\*59 + 23)/59 = (1357 + 23)/59 = 1380/59$. Ah! The correct answer is 23(23/59) km/h. The question has the wrong answer highlighted and a strange answer option in mixed fractions. Let’s correct the final answer. The correct answer is 23(23/59) km/h.
7. The average age of three boys is 15 years. If their ages are in ratio 3:5:7, the age of the youngest boy is
Answer: 9 years
- Step 1: Calculate the total age of the three boys. Total age = Average age * Number of boys = $15 \* 3 = 45$ years.
- Step 2: Use the ratio to find the age of the youngest boy. Let the ages of the boys be , , and . Sum of ages = . .
- Step 3: Find the age of the youngest boy. The youngest boy’s age is $3x = 3 \* 3 = 9$ years.
8. In a boat there are 8 men whose average weight is increased by 1 kg when 1 man of 60 kg is replaced by a new man. What is weight of new comer?
Answer: 68 kg
- Step 1: Calculate the change in total weight. The average weight of 8 men increased by 1 kg. Total increase in weight = 8 men * 1 kg/man = 8 kg.
- Step 2: Find the weight of the new man. The weight of the new man is the weight of the man who was replaced plus the total increase in weight. Weight of new man = Weight of old man + Total increase Weight of new man = kg.
9. The average of a group of men is increased by 5 years when a person aged of 18 years is replaced by a new person of aged 38 years. How many men are there in the group?
Answer: 4
- Step 1: Calculate the total change in age. The difference in age between the new person and the old person is: Change in age = years.
- Step 2: Use the change in average to find the number of men. The total change in age is distributed among all the men in the group, causing the average to increase by 5 years. Number of men = Total change in age / Change in average Number of men = .
10. The average temperature for Wednesday, Thursday and Friday was 40°C. The average for Thursday, Friday and Saturday was 41°C. If temperature on Saturday was 42°C, what was the temperature on Wednesday?
Answer: 39°C
- Step 1: Set up the equations for total temperatures.
- Sum of temperatures for Wednesday, Thursday, Friday (W+Th+F) = $40°C \* 3 = 120°C$.
- Sum of temperatures for Thursday, Friday, Saturday (Th+F+S) = $41°C \* 3 = 123°C$.
- Step 2: Use the given temperature for Saturday to find the sum of Thursday and Friday.
- We know S = 42°C.
- From the second equation: Th + F + 42 = 123
- Th + F = .
- Step 3: Substitute the sum of Thursday and Friday into the first equation to find Wednesday’s temperature.
- From the first equation: W + (Th+F) = 120
- W + 81 = 120
- W = .
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