Averages

By Praveen Dwivedi -
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An average in mathematical terms is an arithmetic mean. It is the sum of n different terms divided by n.

If a student scores 7, 6 and 8 in three different tests, each of ten marks, then his average score will be

These are the two main formulae which will be used under this topic

A few illustrations have been given below.

1. The average weight of ten students is increased by 1.8 kg when one student weighing 53 kg is replaced by a new student. Find the weight of the new student.

If we assume the sum of the weights of the other nine students to be x then the earlier average would have been (x + 53) / 10,

Since the weight of those nine students remains constant, and the new average is 1.8 kg more than the earlier one , we have

New Average – Old average = 1.8

(x+ unknown weight)/ 10 – (x + 53) / 10= 1.8

On solving we get

x+ unknown weight – x – 53 = 18

unknown weight – 53 = 18

unknown weight = 18 + 53

Unknown weight = 71 kgs.

To save time during exams we can use this formula directly

  • If the average is increased then

Weight of new comer = weight of the person who left+ total increase in weight

Total increase in weight = number of men * increase in average weight

  • If the average is decreased then

Weight of new comer = weight of the person who left – total decrease in weight

The total decrease in weight is = number of men * decrease in average weight

2. The average of 15 results is 75. If the average of the first seven results is 77 and that of the last seven is 72, find the seventh result.

Let the sum of all fifteen terms be x

x= sum of first seven terms + seventh term + sum of last seven terms – (1)

Using the formula:

“Total = Average * Number of data elements”

We get

  • x = 75 * 15
  • x = 1125 i.e. sum of all fifteen terms = 1125

Using the same formula:

Sum of first seven terms = 7* 77 = 539

Sum of last seven terms = 7 * 72 = 504

Using (1) we get

1125 = 539 + seventh term + 504

  • Seventh term = 82

3. The average of w, x, y, z is 16. Half the sum of x, y, z is 28. What is the value of w?

Since the average of the four terms is 16

  • (w + x + y + z) /4 = 16
  • (w + x + y + z) = 64 – (1)

We also have

  • (x + y + z ) / 2 = 28
  • (x + y + z ) = 56

Using (1) we have

  • w + 56 = 64
  • w = 8

4. The average cost of eight items in a shop is Rs. 5000. Three items costing Rs.4000, Rs. 4200, Rs. 4300 are sold. What is the average cost of the remaining items?

Using the formula:

Total = Average * Number of data elements

  • Total Cost of eight items = 5000 * 8
  • Total Cost of eight items = 40000

Total cost of the three items which are sold = 4000 + 4200 + 4300 = 12500

Now the cost of the remaining five items = Cost of eight items – cost of the three items which are sold

  • 40000 – 12500 =
  • 27500

Using the formula:

Average = Total of the data

Number of data elements

Average = 27500/ 5 = Rs 5500

5. A driver drove at an average speed of 50 km/hour for the first two hours of her journey. For the next three hours she averaged 20 km/ hour. What was the average speed overall?

Distance covered in the first two hours = 50 km/hour * 2 hours = 100 km

Distance covered in the last three hours = 20 km/ hour * 3 hours = 60 km

Total distance = 160 km

Total time= 5 hours

Average speed = 32 km/ hour

The following exercise will help in understanding averages better.

Exercise

  1. Find the average of all prime numbers between 35 and 45.
  2. The average weight of A, B, C is 45 kg. The average weight of A and B is 40 kg, and of B and C is 43 kg. Find the weight of B.
  3. The average salary per month of 30 teachers is Rs 4000. If the Principal’s salary is added, the average salary is increased to Rs 4, 300. What is the Principal’s salary?
  4. The arithmetic mean of 5 numbers is 27. If one of the numbers is excluded their mean is 25. What is the excluded number?
  5. The mean temperature from Monday to Wednesday was 370 C and Tuesday to Thursday was 340 C. if the temperature on Thursday was 4/5th of Monday, what was the temperature on Thursday?
  6. The average age of 3 children in a family is 20% of the average age of the father and the eldest child. the total age of the mother and the youngest child is 39 years. if the father’s age is 26 years, what is the age of the second child?
  7. The mean weight of 120 students in the first year class of a college is 56 kg. If the mean weight of boys and that of girls in the class are 60kg and 50 kg respectively, then what is the number of boys and girls in the class?
  8. There are 35 students in a hostel. Due to the admission of 7 new students, the expenses of the mess were increased by Rs 42 per day while the average expenditure per head diminished by Re. 1. What was the original expenditure by the mess?
  9. The average age of 30 students in a class is 15 years. If 10 more boys join the class, the average age of the whole class is reduced by one year. What is the average age of the new comers?
  10. The average age of a family of six members is 25 years. After two years, a person dies at the age of 36 years and a new baby is born. What would be the average age of the family after eight years?

Answers

  1. 40.33
  2. 31kg
  3. Rs. 13,300
  4. 35
  5. 360 C
  6. Cannot be determined from the information given
  7. 72 boys and 48 girls
  8. Rs. 420
  9. 11 years
  10. 29 years

Arti Mohan

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37 COMMENTS

  2. Let the temperature of Monday, Tuesday, Wednesday be x degree celsius, y degree celsius and z degree celsius respectively
    The temperature of Thursday is 4/5 * x degree celsius
    The average of the temperatures of Monday, Tuesday, Wednesday gives us
    (x +y + z)/3 = 37

    The average of the temperatures of Tuesday, Wednesday, Thursday gives us
    (y + z + 4/5 x) /3= 34

    We get
    (x +y + z) = 37 * 3
    y + z = (37 * 3) -x
    using the value of y+z in the second equation we get
    (37 * 3) -x + 4/5 x = 34 *3
    (37 * 3) – (34 * 3) = x – 4/5 x
    3 ( 37-34) = 1/5x
    9= 1/5 x
    x = 45

    The temperature on Monday is 45 degree celsius
    The temperature on Thursday is 4/5x = 4/5 * 45 = 36 degree celsius

  3. Hi, This really helped me, thanks a lot! 🙂
    But I’m a lill confused about the 8th and 10th one. Can you please explain the working?
    Thanks. 🙂

    • total w of a b c = 135 ie. 45*3
      total weight of a b = 80 ie. 40*2
      total weight of b c =86 ie. 43*2
      weight of c= 135-80= 55
      weight of a= 135-86 = 49
      weight of b= 135- (49+55)
      = 135-104
      = 31

  5. hey,can someone plz explain m the working of the 10th ques..??
    i think the answer should be 33.
    here s my logic,d initial sum ( let be sum1 )of ages is = 25*6
    after 8 yrs,sum2= 25*6+ 2+6+8*5
    so the avg wil be sum2 / 6 = 33
    help plz !

  7. hey the 10th one will be done like this:-
        total sum of ages of 6 memebers=25*6=150
    after 2 years the sum will be=150+2*6=162
    now a person dies at the age of 36 nd a new baby is born so no. of family members will remain 6 only.
    but the new total sum of ages=162-36=126
    after 8 yrs the total age would become=126+6*8=174
    therefore,avg.age=174/6=29yrs.
    can sum1 tell me how 2 do the 8th one????

  9. Solution to No.8-
    Let the per head expense be Rs.x
    Total expense of 35 students= Rs.35x.
    Now,
    7 more students means total no.of students= (35+7)=42students.
    Per-head expense=Rs.(x-1) [ since per-head expense decreases by Re.1]
    So, total expense of 42 people= Rs.42(x-1)
    Therefore now we get the equation as;
    42(x-1)= 35x+42 [As,total expense increases by 42]
    or, 42x-42= 35x+42
    or,42x-35x=42+42
    or,7x=84
    or,x=12
    So, per-head expense of 35 students was Rs.12
    Total expense= Rs.(35*12)= Rs.420

    Hope this helps, Happy Solving Friends! 🙂

  17. Can you solve a question, i m having a problm in it.
    Average of 50 numbers is 30. If two of them are 35 and 40,find the average of remaining numbers

    • If there are 30 members in the beginning then,

      total salary=number of employees *average salary

      that is => 30*4000 = 1,20,000.

      when the principal is added one employee is also increased

      which means now the total employees are 31.

      hence, after principal joined the average became = 4,300

      now, the average salary of 31 employees are = 31*4,300=1,33,300

      hence, salary = 1,33,300-1,20,000 = 13,300

      and it was a QED(Quiet Easily Done!)

  21. The average age of 30 boys in a class is 15.2 years. if 15 more boys join the class, the average of the whole class is reduced by half a year. The average age of the new comer is

    • 30*15.2=456 ,, reduce by half a year means 0.5 year … so , it will be .. 15 more student … 15+30 =45. ,,,, 45*(15.2 yrs – 0.5 yr) = 45*14.7= 661.5. Now ,,661.5-456=205.5 ….. divide 205.5 by no.of new boys join the class …
      205.5/15=13.7 …Answer=13.7 years.

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