Basic Numeracy (Class X level): Averages, speed and work

Ratio, averages and mixtures

Average = total ÷ number. Most average questions are easier with totals: change the totals, then divide again. For a ratio a: b, think of a parts and b parts. For mixtures, track only the quantity of one item (milk, water or sugar).

CSAT 2019, Q40: The average marks of 100 students are given to be 40. It was found later that marks of one student were 53 which were misread as 83. The corrected mean marks are Total = 100 × 40 = 4,000. 83 was counted instead of 53: 30 too many. Correct total 3,970, mean 39.7. Answer (b).

CSAT 2018, Q68: A lift has the capacity of 18 adults or 30 children. How many children can board the lift with 12 adults? Prelims 2018 18 adults fill the lift, so 12 adults use two-thirds. One-third is left = one-third of 30 children = 10. Answer (b).

CSAT 2016, Q51: The sum of the ages of 5 members comprising a family, 3 years ago was 80 years. The average age of the family today is the same as it was 3 years ago, because of an addition of a baby during the intervening period. How old is the baby? Prelims 2016 Three years ago the total was 80 for 5 people, average 16. Today those 5 are 3 years older each: 80 + 15 = 95. With the baby there are 6 people and the average is still 16, total 96. Baby = 96 − 95 = 1 year. Answer (b).

CSAT 2020, Q57: A sum of ₹ 2,500 is distributed among X, Y and Z in the ratio 1/2: 3/4: 5/6. What is the difference between the maximum share and the minimum share? Prelims 2020 Multiply 1/2: 3/4: 5/6 by 12: 6: 9: 10, total 25 parts. ₹ 2,500 ÷ 25 = ₹ 100 a part. Biggest 10 parts, smallest 6 parts, difference 4 parts = ₹ 400. Answer (c).

CSAT 2012, Q62: Two glasses of equal volume are respectively half and three-fourths filled with milk. They are then filled to the brim by adding water. Their contents are then poured into another vessel. What will be the ratio of milk to water in this vessel? Prelims 2012 Take each glass as 4 cups. Milk: 2 + 3 = 5 cups. Water: 2 + 1 = 3 cups. Milk: water = 5: 3. Answer (d).

CSAT 2017, Q37: There is a milk sample with 50% water in it. If 1/3rd of this milk is added to equal amount of pure milk, then water in the new mixture will fall down to Prelims 2017 Take 2 litres of the sample: 1 litre water. Add 2 litres of pure milk. Now 1 litre water in 4 litres = 25%. Answer (a).

Time, speed and distance

Distance = speed × time. Two things moving towards each other close the gap at the sum of their speeds; one chasing the other closes it at the difference. Average speed is total distance ÷ total time, not the average of the speeds.

CSAT 2015, Q71: Two cities A and B are 360 km apart. A car goes from A to B with a speed of 40 km/hr and returns to A with a speed of 60 km/hr. What is the average speed of the car? Prelims 2015 Go 360 km at 40: 9 hours. Return 360 km at 60: 6 hours. Total 720 km in 15 hours = 48 km/hr. Answer (b).

CSAT 2014, Q69: Two cars start towards each other, from two places A and B which are at a distance of 160 km. They start at the same time 08: 10 AM. If the speeds of the cars are 50 km and 30 km per hour respectively, they will meet each other at Prelims 2014 Moving towards each other they close 50 + 30 = 80 km every hour. 160 km takes 2 hours. 8:10 + 2 hours = 10:10 AM. Answer (a).

CSAT 2013, Q35: A thief running at 8 km/hr is chased by a policeman whose speed is 10 km/hr. If the thief is 100 m ahead of the policeman, then the time required for the policeman to catch the thief will be The policeman gains 10 − 8 = 2 km an hour, that is 2,000 m in 60 minutes, or 100 m in 3 minutes. Answer (b).

CSAT 2014, Q50: A worker reaches his factory 3 minutes late if his speed from his house to the factory is 5 km/hr. If he walks at a speed of 6 km/hr, then he reaches the factory 7 minutes early. The distance of the factory from his house is Try the options. 5 km: at 5 km/hr it takes 60 minutes, at 6 km/hr 50 minutes. Difference 10 minutes = 3 late + 7 early. Answer (c).

CSAT 2020, Q80: A frog tries to come out of a dried well 4·5 m deep with slippery walls. Every time the frog jumps 30 cm, slides down 15 cm. What is the number of jumps required for the frog to come out of the well? Prelims 2020 Each jump gains 30 − 15 = 15 cm. Careful at the end: once the frog is within 30 cm of the top, one jump takes it out. After 28 jumps it is at 28 × 15 = 420 cm; the 29th jump reaches 450 cm = 4.5 m. Answer (b).

Time and work, pipes

Give the whole job an easy number of units, such as the LCM of the days. Then each person’s daily work is units ÷ days, and people working together simply add their units. Pipes are the same: a tank of 60 units.

CSAT 2015, Q53: Two pipes A and B can independently fill a tank completely in 20 and 30 minutes respectively. If both the pipes are opened simultaneously, how much time will they take to fill the tank completely? Prelims 2015 Tank = 60 units. A fills 3 units a minute, B 2 units. Together 5 a minute: 60 ÷ 5 = 12 minutes. Answer (b).

CSAT 2016, Q60: Ram and Shyam work on a job together for four days and complete 60% of it. Ram takes leave then and Shyam works for eight more days to complete the job. How long would Ram take to complete the entire job alone? Prelims 2016 Shyam alone did the last 40% in 8 days: 5% a day. Together they did 60% in 4 days: 15% a day. So Ram does 10% a day and needs 10 days. Answer (c).

CSAT 2020, Q71: A person X can complete 20% of work in 8 days and another person Y can complete 25% of the same work in 6 days. If they work together, in how many days will 40% of the work be completed? Prelims 2020 X: 20% in 8 days. Y: 25% in 6 days. Take 24 days as a base: X does 60% in 24 days, Y 100% in 24 days, so together 160% in 24 days, that is 40% in 6 days. Answer (a).