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Average: 20 mixed-difficulty Average problems- each with step-by-step (digit-by-digit) solutions

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Average: 20 mixed-difficulty Average problems – each with step-by-step (digit-by-digit) solutions

Worksheet: 20 Average Questions

Basic

  1. Find the average of 12, 15, and 18.
  2. The average of 5 numbers is 24. Four numbers are 20, 25, 30, and 22. Find the 5th number.
  3. Class A has 10 students with an average of 70. Class B has 15 students with an average of 75. Find the combined average.
  4. The average of 6 numbers is 50. A new number 80 is added. What is the new average?
  5. The average of 7 numbers is 40. One number 30 is removed. What is the new average?
  6. Find the weighted average of marks: 80 (weight 3), 70 (weight 2), 60 (weight 1).
  7. Find the average of integers from 14 to 20 inclusive.

Medium

  1. Group A has 12 people with an average of 56. Group B has an average of 62. The combined average of both groups is 58. Find the number of people in Group B.
  2. The average of 6 numbers is 30. One number is replaced by 50, and the new average becomes 32. Find the replaced number.
  3. A person travels 100 km at 50 km/h and another 100 km at 75 km/h. Find the average speed.
  4. Find the weighted average: 92 (credit 4), 85 (credit 3), 76 (credit 2).
  5. A class has 30 students with an average of 70. How many students with an average of 60 must join so that the overall average becomes 68?
  6. Find the average of −5,0,5,10-5, 0, 5, 10−5,0,5,10.
  7. Group 1 has 5 numbers with average 30. Group 2 has 7 numbers with average 36. Find the average of all 12 numbers.

Hard

  1. The average of 9 numbers is 24. After removing one number, the average of the remaining 8 numbers becomes 26. Find the removed number.
  2. Internal marks are 80 (weight 40%), and final exam marks are 70 (weight 60%). Find the final weighted score.
  3. Find the average of the first 50 natural numbers.
  4. There are nnn numbers with average aaa. If one number is removed and the new average becomes bbb, express the removed number in terms of a,b,na, b, na,b,n.
  5. A car travels 120 km at 40 km/h and then 240 km at 60 km/h. Find the average speed for the whole trip.
  6. The average of 25 students is 72. How many students scoring 100 must be added to raise the average to 75?

20 mixed-difficulty Average problems — each with step-by-step (digit-by-digit) solutions

Easy

1) Average of three numbers

Problem: Find the average of 12, 15 and 18.
Solution:

  1. Sum the numbers stepwise:
    • 12+15=27
    • 27+18=45.
  2. Number of observations =3
  3. Average =45/3=15

Answer: 15

2) Missing number (one unknown)

Problem: Average of 5 numbers is 24. Four numbers are 20, 25, 30, 22. Find the 5th.
Solution:

  1. Total required =24×5= 24 Compute: 24×5=(20+4)×5=100+20=120.
  2. Sum of known four:
    • 20+25=45.
    • 45+30=75.
    • 75+22=97.
  3. Missing number =120−97=23.
    Answer: 23

3) Combined groups (two classes)

Problem: Class A: 10 students, avg 70. Class B: 15 students, avg 75. Find combined average.
Solution:

  1. Sum A =70×10=700.
  2. Sum B: compute 75×15 stepwise:
    • 75×10=750.
    • 75×5=375.
    • Sum B =750+375=1125.
  3. Combined sum =700+1125=1825
  4. Total students =10+15=25
  5. Combined average =1825/25=73 (since 25×73=1825).
    Answer: 73

4) Adding a number

Problem: Average of 6 numbers is 50. A new number 80 is added. What is the new average?
Solution:

  1. Original total =50×6=300
  2. New total =300+80=380
  3. New count =7
  4. New average =380/7​. Long division: 7×54=378 remainder 2. So average =54×2/7≈54.285714….
    Answer: 380/7≈54.2857

5) Removing a number

Problem: Average of 7 numbers is 40. One number 30 is removed. What is the new average?
Solution:

  1. Original total =40×7=280
  2. After removal total =280−30=250.
  3. New count =6
  4. New average =250/6​. Compute: 6×41=246 remainder 4. So =41×4/6=41×2/3=41.666666

Answer: 250/6=41×2/3≈41.6667

6) Weighted average (simple)

Problem: Marks: 80 (weight 3), 70 (weight 2), 60 (weight 1). Find weighted average.
Solution:

  1. Weighted sums:
    • 80×3=240
    • 70×2=140
    • 60×1=60
  2. Total weighted sum =240+140+60=440.
  3. Total weights =3+2+1=6.
  4. Weighted average =440/6=73×1/3=73.333..
    Answer: 440/6≈73.3333

7) Average of consecutive integers

Problem: What is the average of integers from 14 to 20 (inclusive)?
Solution:

  1. Sum stepwise: 14+15=29 ; 29+16=45; 45+17=62; 62+18=80: 80+19=99; 99+20=119.
  2. Count = 7.
  3. Average =119/7=17 (Also for consecutive integers average = middle term.)
    Answer: 17

Medium

8) Find group size (algebra)

Problem: Group A: 12 people avg 56. Group B avg 62. Combined avg is 58. Find size of Group B.
Solution:

  1. Sum A =56×12. Compute: 56×10=560, 56×2=112; 560+112=672.
  2. Let Group B size be y. Total sum =672+62y . Total people =12+y.
  3. Equation: 672+62y/12+y =58. Multiply both sides: 672+62y=58(12+y). Compute RHS: 58×12=696; 58y.
  4. So 672+62y=696+58y. Rearrange: 62y−58y=696−672 → 4y=24.
  5. So y=6y = 6y=6.
    Answer: 6

9) Replaced number (average increases)

Problem: Average of 6 numbers is 30. One number is replaced by 50 and the new average becomes 32. Find the replaced number.
Solution:

  1. Original total =30×6=180
  2. New total =32×6=192
  3. Increase in total =192−180=12. That increase equals 50−x where x is the replaced number.
  4. Solve 50−x=12 → x=50-12=38.
    Answer: 38

10) Average speed — equal distances (harmonic mean)

Problem: Trip A: 100 km at 50 km/h, Trip B: 100 km at 75 km/h. Find average speed overall.
Solution (best via harmonic mean):

  1. For equal distances d at speeds v1,v2 ​: average speed =2/1/v1+1/v2​.
  2. Compute: 2/1/50+1/75=2/3+2/150=2/5/150=2×150/5=300/5=60 km/h.
    (Alternative: total distance =200 km; time =100/50+100/75=2+4/3=3×1/3-200÷3×1/3=200÷10/3=200×3/10=60.)


Answer: 60 km/h

11) Weighted average — exam with credits

Problem: Marks: 92 (credit 4), 85 (credit 3), 76 (credit 2). Find weighted average.
Solution:

  1. Weighted sums: 92×4=368;; 85×3=255; 76×2=152.
  2. Weighted total =368+255+152=775. (Check: 368+255=623; 623+152=775.)
  3. Total credits =4+3+2=9.
  4. Weighted average =775/9​. Divide: 9×86=774 remainder 1 → 86×1/9≈86.111

Answer: 775/9≈86.1111

12) How many to add? (non-integer result)

Problem: Class has 30 students, average 70. How many students with average 60 must be added to make the combined average 68?
Solution:

  1. Current total =30×70=2100.
  2. Suppose m students with average 60 join. New total =2100+60m. New count =30+m= 30 + m. We want: 2100+60m/30+m=68
  3. Multiply: 2100+60m=68(30+m)=2040+68m.
  4. Rearr.: 60m−68m=2040−2100 → −8m=−60 → m=60/8=7.5.
  5. Interpretation: m=7.5m — not an integer. That means no whole number of students with average exactly 60 will make the combined average exactly 68. (If you add 7 students the combined average is slightly above 68; if you add 8 students it is slightly below 68.)
    Answer: 7.5 (not an integer — so exact 68 impossible with whole students)

13) Average with negative numbers

Problem: Find average of −5,0,5,10-5, 0, 5, 10−5,0,5,10.
Solution:

  1. Sum: −5+0=−5. Then −5+5=0. Then 0+10=10.
  2. Count = 4. Average =10/4=2.5
    Answer: 2.5

14) Average of averages (weighted)

Problem: Group 1: 5 numbers average 30. Group 2: 7 numbers average 36. What is the average of all 12 numbers?
Solution:

  1. Sum1 =5×30=150.
  2. Sum2 =7×36=252
  3. Combined sum =150+252=402. Total count =12.
  4. Combined average =402/12=33.5

Answer: 33.5

Hard

15) One removed makes average increase (find removed number)

Problem: Average of 9 numbers is 24. If one number is removed the new average of the 8 remaining numbers becomes 26. Find the removed number.
Solution:

  1. Original total =9×24=216
  2. New total after removal =8×26=208.
  3. Removed number =216−208=8

Answer: 8

16) Weighted percentages (exam score)

Problem: Internal marks = 80 (weight 40%), final exam = 70 (weight 60%). Final score?
Solution:

  1. Internal contribution =0.40×80=32.0.
  2. Final exam contribution =0.60×70=42.0
  3. Total =32.0+42.0=74.0
    Answer: 74

17) Average of first nnn natural numbers (application)

Problem: What is the average of the first 50 natural numbers 1,2,…,501,2,\dots,501,2,…,50?
Solution:

  1. Sum of first 50 = (50×51)/2=25×51=1275  (Compute: 50/2=25; 25×51=25×50+25=1250+25=1275)
  2. Average =1275/50=25.5 (Or use (1+50)/2 for consecutive integers.)
    Answer: 25.5

18) General formula — express the removed value

Problem: There are nnn numbers with average aaa. If one number xxx is removed the new average becomes bbb. Express xxx in terms of a,b,na,b,na,b,n.
Solution (algebra):

  1. Total initially =a×n. After removal total =b×(n−1).
  2. The removed number x=initial total−new total=an−b(n−1).
    Answer: x=an−b(n−1)​.

19) Average speed — unequal distances (use time)

Problem: A car goes 120 km at 40 km/h and then 240 km at 60 km/h. What is the average speed for the whole trip?
Solution:

  1. Total distance =120+240=360km
  2. Time for first part =120/40=3hours.
  3. Time for second part =240/60=4hours.
  4. Total time =3+4=7 hours.
  5. Average speed =360/7=51×3/7≈51.428571…km/h.
    Answer: 360/7≈51.4286 km/h

20) How many top scores to raise class average?

Problem: Current average 72 for 25 students. How many students scoring 100 are needed so that the new average becomes 75?
Solution:

  1. Current total =25×72=1800
  2. If k students scoring 100 join, new total =1800+100k. New count =25+k. We want: (1800+100k)/(25+k)=75.
  3. Multiply both sides: 1800+100k=75(25+k)=1875+75k.
  4. Rear.: 100k−75k=1875−1800 → 25k=75 → k=3.
    Answer: 3 students
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