Average word problems use the relationship between the total, the number of values, and their mean. A useful formula is \(\displaystyle \text{Average}=\frac{\text{Sum}}{\text{Number of Values}}\), which can be rearranged to find a missing total, missing value, or new average. These problems include basic averages, combined groups, missing scores, changing averages, and integer constraints.
Notes
Practice Problems
\(\textbf{1)}\) The average of seven numbers is 12. What is the sum of the numbers? The sum is 84
\(\,\,\,\,\,\,\displaystyle\text{Average}=\frac{\text{Sum}}{\text{Number of Values}}\)
\(\,\,\,\,\,\,\displaystyle12=\frac{\text{Sum}}{7}\)
\(\,\,\,\,\,\,12(7)=\text{Sum}\)
\(\,\,\,\,\,\,84=\text{Sum}\)
The sum is 84.
\(\,\,\,\,\,\,\displaystyle\text{Average}=\frac{\text{Sum}}{\text{Number of Values}}\)
\(\,\,\,\,\,\,\displaystyle12=\frac{\text{Sum}}{7}\)
\(\,\,\,\,\,\,12(7)=\text{Sum}\)
\(\,\,\,\,\,\,84=\text{Sum}\)
The sum is 84.
\(\textbf{2)}\) Joe has had 9 quizzes so far in his class. His average score is 80%. He scored a 100% on his tenth quiz. What is his average now? His average is now 82%
\(\,\,\,\,\,\,9(80)=720\)
\(\,\,\,\,\,\,720+100=820\)
\(\,\,\,\,\,\,\displaystyle\frac{820}{10}=82\)
His average is now 82%.
\(\,\,\,\,\,\,9(80)=720\)
\(\,\,\,\,\,\,720+100=820\)
\(\,\,\,\,\,\,\displaystyle\frac{820}{10}=82\)
His average is now 82%.
\(\textbf{3)}\) The average of three positive integers a, b, and c is 20. If a\(\lt\)b\(\lt\)c, what is the largest possible value for c? 57 is the largest possible value for c
\(\,\,\,\,\,\,3(20)=60\)
\(\,\,\,\,\,\,a+b+c=60\)
\(\,\,\,\,\,\,\text{To maximize }c\text{, minimize }a\text{ and }b\)
\(\,\,\,\,\,\,a=1,\quad b=2\)
\(\,\,\,\,\,\,1+2+c=60\)
\(\,\,\,\,\,\,c=57\)
57 is the largest possible value for c.
\(\,\,\,\,\,\,3(20)=60\)
\(\,\,\,\,\,\,a+b+c=60\)
\(\,\,\,\,\,\,\text{To maximize }c\text{, minimize }a\text{ and }b\)
\(\,\,\,\,\,\,a=1,\quad b=2\)
\(\,\,\,\,\,\,1+2+c=60\)
\(\,\,\,\,\,\,c=57\)
57 is the largest possible value for c.
\(\textbf{4)}\) The average of six numbers is 100, and the average of four of these numbers is 80. What is the average of the other two numbers? The average is 140
\(\,\,\,\,\,\,6(100)=600\)
\(\,\,\,\,\,\,4(80)=320\)
\(\,\,\,\,\,\,600-320=280\)
\(\,\,\,\,\,\,\displaystyle\frac{280}{2}=140\)
The average is 140.
\(\,\,\,\,\,\,6(100)=600\)
\(\,\,\,\,\,\,4(80)=320\)
\(\,\,\,\,\,\,600-320=280\)
\(\,\,\,\,\,\,\displaystyle\frac{280}{2}=140\)
The average is 140.
\(\textbf{5)}\) The sum of 5 numbers is 45. What is the average of the numbers? The average is 9
\(\,\,\,\,\,\,\displaystyle\text{Average}=\frac{\text{Sum}}{\text{Number of Values}}\)
\(\,\,\,\,\,\,\displaystyle\text{Average}=\frac{45}{5}\)
\(\,\,\,\,\,\,\text{Average}=9\)
The average is 9.
\(\,\,\,\,\,\,\displaystyle\text{Average}=\frac{\text{Sum}}{\text{Number of Values}}\)
\(\,\,\,\,\,\,\displaystyle\text{Average}=\frac{45}{5}\)
\(\,\,\,\,\,\,\text{Average}=9\)
The average is 9.
\(\textbf{6)}\) The average of x,y,z, and w is 14. The average of x and z is 10, what is the average of y and w? The average is 18
\(\,\,\,\,\,\,4(14)=56\)
\(\,\,\,\,\,\,x+y+z+w=56\)
\(\,\,\,\,\,\,2(10)=20\)
\(\,\,\,\,\,\,x+z=20\)
\(\,\,\,\,\,\,y+w=56-20=36\)
\(\,\,\,\,\,\,\displaystyle\frac{36}{2}=18\)
The average is 18.
\(\,\,\,\,\,\,4(14)=56\)
\(\,\,\,\,\,\,x+y+z+w=56\)
\(\,\,\,\,\,\,2(10)=20\)
\(\,\,\,\,\,\,x+z=20\)
\(\,\,\,\,\,\,y+w=56-20=36\)
\(\,\,\,\,\,\,\displaystyle\frac{36}{2}=18\)
The average is 18.
\(\textbf{7)}\) On your first three tests you scored 89, 91 and 95. What do you have to score on the fourth and final test to average a 90% in the class? You have to score 85% on the final.
\(\,\,\,\,\,\,4(90)=360\)
\(\,\,\,\,\,\,89+91+95=275\)
\(\,\,\,\,\,\,275+x=360\)
\(\,\,\,\,\,\,x=85\)
You have to score 85% on the final.
\(\,\,\,\,\,\,4(90)=360\)
\(\,\,\,\,\,\,89+91+95=275\)
\(\,\,\,\,\,\,275+x=360\)
\(\,\,\,\,\,\,x=85\)
You have to score 85% on the final.
\(\textbf{8)}\) The average of five numbers is 18. What is the sum of the five numbers? The sum is 90.
\(\,\,\,\,\,\,5(18)=90\)
The sum is 90.
\(\,\,\,\,\,\,5(18)=90\)
The sum is 90.
\(\textbf{9)}\) The sum of seven numbers is 84. What is their average? The average is 12.
\(\,\,\,\,\,\,\displaystyle\text{Average}=\frac{84}{7}\)
\(\,\,\,\,\,\,\text{Average}=12\)
The average is 12.
\(\,\,\,\,\,\,\displaystyle\text{Average}=\frac{84}{7}\)
\(\,\,\,\,\,\,\text{Average}=12\)
The average is 12.
\(\textbf{10)}\) Maya wants an average of 84 on four tests. Her first three scores are 72, 85, and 91. What score does she need on the fourth test? She needs an 88.
\(\,\,\,\,\,\,4(84)=336\)
\(\,\,\,\,\,\,72+85+91=248\)
\(\,\,\,\,\,\,248+x=336\)
\(\,\,\,\,\,\,x=88\)
She needs an 88.
\(\,\,\,\,\,\,4(84)=336\)
\(\,\,\,\,\,\,72+85+91=248\)
\(\,\,\,\,\,\,248+x=336\)
\(\,\,\,\,\,\,x=88\)
She needs an 88.
\(\textbf{11)}\) Eight students have an average score of 76. A ninth student with a score of 94 joins the group. What is the new average? The new average is 78.
\(\,\,\,\,\,\,8(76)=608\)
\(\,\,\,\,\,\,608+94=702\)
\(\,\,\,\,\,\,\displaystyle\frac{702}{9}=78\)
The new average is 78.
\(\,\,\,\,\,\,8(76)=608\)
\(\,\,\,\,\,\,608+94=702\)
\(\,\,\,\,\,\,\displaystyle\frac{702}{9}=78\)
The new average is 78.
\(\textbf{12)}\) The average of 12 numbers is 15. If one of the numbers, 26, is removed, what is the average of the remaining 11 numbers? The new average is 14.
\(\,\,\,\,\,\,12(15)=180\)
\(\,\,\,\,\,\,180-26=154\)
\(\,\,\,\,\,\,\displaystyle\frac{154}{11}=14\)
The new average is 14.
\(\,\,\,\,\,\,12(15)=180\)
\(\,\,\,\,\,\,180-26=154\)
\(\,\,\,\,\,\,\displaystyle\frac{154}{11}=14\)
The new average is 14.
\(\textbf{13)}\) Six students have an average score of 14, and four other students have an average score of 24. What is the average score of all ten students? The average is 18.
\(\,\,\,\,\,\,6(14)=84\)
\(\,\,\,\,\,\,4(24)=96\)
\(\,\,\,\,\,\,84+96=180\)
\(\,\,\,\,\,\,6+4=10\)
\(\,\,\,\,\,\,\displaystyle\frac{180}{10}=18\)
The average is 18.
\(\,\,\,\,\,\,6(14)=84\)
\(\,\,\,\,\,\,4(24)=96\)
\(\,\,\,\,\,\,84+96=180\)
\(\,\,\,\,\,\,6+4=10\)
\(\,\,\,\,\,\,\displaystyle\frac{180}{10}=18\)
The average is 18.
\(\textbf{14)}\) Five consecutive integers have an average of 28. What are the five integers? The integers are 26, 27, 28, 29, and 30.
\(\,\,\,\,\,\,\text{For five consecutive integers, the average is the middle number}\)
\(\,\,\,\,\,\,\text{Middle number}=28\)
\(\,\,\,\,\,\,26,\ 27,\ 28,\ 29,\ 30\)
The integers are 26, 27, 28, 29, and 30.
\(\,\,\,\,\,\,\text{For five consecutive integers, the average is the middle number}\)
\(\,\,\,\,\,\,\text{Middle number}=28\)
\(\,\,\,\,\,\,26,\ 27,\ 28,\ 29,\ 30\)
The integers are 26, 27, 28, 29, and 30.
\(\textbf{15)}\) The average of four positive integers a, b, c, and d is 17. If a\(\lt\)b\(\lt\)c\(\lt\)d, what is the largest possible value of d? The largest possible value of d is 62.
\(\,\,\,\,\,\,4(17)=68\)
\(\,\,\,\,\,\,a+b+c+d=68\)
\(\,\,\,\,\,\,\text{To maximize }d\text{, use }a=1,\ b=2,\ c=3\)
\(\,\,\,\,\,\,1+2+3+d=68\)
\(\,\,\,\,\,\,d=62\)
The largest possible value of d is 62.
\(\,\,\,\,\,\,4(17)=68\)
\(\,\,\,\,\,\,a+b+c+d=68\)
\(\,\,\,\,\,\,\text{To maximize }d\text{, use }a=1,\ b=2,\ c=3\)
\(\,\,\,\,\,\,1+2+3+d=68\)
\(\,\,\,\,\,\,d=62\)
The largest possible value of d is 62.
\(\textbf{16)}\) A student has scores of 45, 50, 55, and 60. What score must the student earn on a fifth assignment to have an average of 56? The student must score 70.
\(\,\,\,\,\,\,5(56)=280\)
\(\,\,\,\,\,\,45+50+55+60=210\)
\(\,\,\,\,\,\,210+x=280\)
\(\,\,\,\,\,\,x=70\)
The student must score 70.
\(\,\,\,\,\,\,5(56)=280\)
\(\,\,\,\,\,\,45+50+55+60=210\)
\(\,\,\,\,\,\,210+x=280\)
\(\,\,\,\,\,\,x=70\)
The student must score 70.
\(\textbf{17)}\) A student averaged 80 on the first three quizzes and 95 on the next two quizzes. What is the average of all five quizzes? The average is 86.
\(\,\,\,\,\,\,3(80)=240\)
\(\,\,\,\,\,\,2(95)=190\)
\(\,\,\,\,\,\,240+190=430\)
\(\,\,\,\,\,\,\displaystyle\frac{430}{5}=86\)
The average is 86.
\(\,\,\,\,\,\,3(80)=240\)
\(\,\,\,\,\,\,2(95)=190\)
\(\,\,\,\,\,\,240+190=430\)
\(\,\,\,\,\,\,\displaystyle\frac{430}{5}=86\)
The average is 86.
\(\textbf{18)}\) After five tests, a student’s average is 70. After the sixth test, the average becomes 74. What did the student score on the sixth test? The student scored 94.
\(\,\,\,\,\,\,5(70)=350\)
\(\,\,\,\,\,\,6(74)=444\)
\(\,\,\,\,\,\,444-350=94\)
The student scored 94.
\(\,\,\,\,\,\,5(70)=350\)
\(\,\,\,\,\,\,6(74)=444\)
\(\,\,\,\,\,\,444-350=94\)
The student scored 94.
\(\textbf{19)}\) The average of \(x\), \(x+4\), and \(x+8\) is 26. Find the three numbers. The numbers are 22, 26, and 30.
\(\,\,\,\,\,\,\displaystyle\frac{x+(x+4)+(x+8)}{3}=26\)
\(\,\,\,\,\,\,3x+12=78\)
\(\,\,\,\,\,\,3x=66\)
\(\,\,\,\,\,\,x=22\)
\(\,\,\,\,\,\,x+4=26\)
\(\,\,\,\,\,\,x+8=30\)
The numbers are 22, 26, and 30.
\(\,\,\,\,\,\,\displaystyle\frac{x+(x+4)+(x+8)}{3}=26\)
\(\,\,\,\,\,\,3x+12=78\)
\(\,\,\,\,\,\,3x=66\)
\(\,\,\,\,\,\,x=22\)
\(\,\,\,\,\,\,x+4=26\)
\(\,\,\,\,\,\,x+8=30\)
The numbers are 22, 26, and 30.
\(\textbf{20)}\) The average of eight numbers is 20. One of the numbers, 14, is replaced with 30. What is the new average? The new average is 22.
\(\,\,\,\,\,\,8(20)=160\)
\(\,\,\,\,\,\,160-14+30=176\)
\(\,\,\,\,\,\,\displaystyle\frac{176}{8}=22\)
The new average is 22.
\(\,\,\,\,\,\,8(20)=160\)
\(\,\,\,\,\,\,160-14+30=176\)
\(\,\,\,\,\,\,\displaystyle\frac{176}{8}=22\)
The new average is 22.
See Related Pages\(\)
\(\bullet\text{ Word Problems- Linear Equations}\)
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\(\bullet\text{ Word Problems- Averages}\)
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\(\bullet\text{ Word Problems- Consecutive Integers}\)
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\(\bullet\text{ Word Problems- Distance, Rate and Time}\)
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\(\bullet\text{ Word Problems- Break Even}\)
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\(\bullet\text{ Word Problems- Ratios}\)
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\(\bullet\text{ Word Problems- Age}\)
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\(\bullet\text{ Word Problems- Mixtures and Concentration}\)
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