Problems on averages

The key fact about aver­ages is short: sum = num­ber of obser­va­tions × aver­age. Almost every aver­age prob­lem is solved by turn­ing the aver­ages into sums, adding or tak­ing away those sums, and then divid­ing again if needed. This les­son works through the com­mon kinds of prob­lem: ages, parts of a list, vis­i­tors in a month, cricket scores and run rates, and one per­son replac­ing another.

Ages#

Exam­ple 1: Before the youngest was born#

The aver­age age of a fam­ily of 6 mem­bers is 22 years. The youngest mem­ber is 7 years old. What was the aver­age age of the fam­ily when the youngest mem­ber was born?

Sum of the 6 ages now $= 6 \times 22 = 132$ years.

The youngest is 7, so the other 5 mem­bers add up to $132 - 7 = 125$ years.

When the youngest was born, 7 years ago, each of those 5 was 7 years younger. Their sum then was

$$125 - 7 \times 5 = 125 - 35 = 90$$

At that time the fam­ily had only 5 mem­bers. So the aver­age age then was

$$\displaystyle \frac{90}{5} = 18 \text{ years}$$

Parts of a list#

Exam­ple 2: The miss­ing mid­dle num­ber#

The aver­age of 9 num­bers is 30. The aver­age of the first 5 num­bers is 25, and the aver­age of the last 3 num­bers is 35. What is the 6th num­ber?

$$x_1 + x_2 + \cdots + x_9 = 9 \times 30 = 270$$

$$x_1 + x_2 + x_3 + x_4 + x_5 = 5 \times 25 = 125$$

$$x_7 + x_8 + x_9 = 3 \times 35 = 105$$

The first 5, the 6th, and the last 3 make up all 9:

$$\begin{aligned} 125 + x_6 + 105 &= 270 \\ x_6 &= 270 - 230 = 40 \end{aligned}$$

The 6th num­ber is 40.

Exam­ple 3: The rest of the group#

The aver­age age of 30 girls is 13 years. The aver­age age of the first 18 girls is 15 years. Find the aver­age age of the remain­ing 12 girls.

Sum of all 30 ages $= 30 \times 13 = 390$.

Sum of the first 18 ages $= 18 \times 15 = 270$.

Sum of the other 12 ages $= 390 - 270 = 120$.

$$\displaystyle \text{average of the 12} = \frac{120}{12} = 10 \text{ years}$$

Exam­ple 4: The last two num­bers#

The sum of five num­bers is 555. The aver­age of the first two is 75 and the third num­ber is 115. Find the aver­age of the last two num­bers.

$$x_1 + x_2 = 2 \times 75 = 150$$

$$\begin{aligned} 150 + 115 + x_4 + x_5 &= 555 \\ x_4 + x_5 &= 555 - 265 = 290 \end{aligned}$$

$$\displaystyle \frac{x_4 + x_5}{2} = \frac{290}{2} = 145$$

Exam­ple 5: A num­ber counted twice#

The aver­age of 9 obser­va­tions is 9. The aver­age of the first 5 is 10, and the aver­age of the last 5 is 8. Find the 5th obser­va­tion.

$$x_1 + x_2 + \cdots + x_9 = 9 \times 9 = 81 \quad (1)$$

$$x_1 + x_2 + x_3 + x_4 + x_5 = 5 \times 10 = 50 \quad (2)$$

$$x_5 + x_6 + x_7 + x_8 + x_9 = 5 \times 8 = 40 \quad (3)$$

The "first 5" and the "last 5" over­lap: both con­tain $x_5$. Add (2) and (3). You get all nine obser­va­tions, with $x_5$ counted twice:

$$\begin{aligned} (x_1 + x_2 + \cdots + x_9) + x_5 &= 50 + 40 = 90 \\ 81 + x_5 &= 90 \\ x_5 &= 9 \end{aligned}$$

Vis­i­tors in a month#

Exam­ple 6#

A library has an aver­age of 510 vis­i­tors on Sun­days and 240 on other days. Find the aver­age num­ber of vis­i­tors per day in a month of 30 days that begins with a Sun­day.

The month starts on a Sun­day, so the Sun­days fall on days 1, 8, 15, 22 and 29. That is 5 Sun­days, and $30 - 5 = 25$ other days.

Vis­i­tors on Sun­days $= 5 \times 510 = 2550$.

Vis­i­tors on other days $= 25 \times 240 = 6000$.

$$\displaystyle \text{average per day} = \frac{2550 + 6000}{30} = \frac{8550}{30} = 285$$

Cricket scores#

Exam­ple 7: Rais­ing the aver­age#

A bats­man's aver­age over 10 innings is 32. How many runs must he score in the next innings to raise his aver­age by 4?

Runs in 10 innings $= 10 \times 32 = 320$.

After 11 innings he wants an aver­age of $32 + 4 = 36$, so a total of $11 \times 36 = 396$.

$$\text{runs needed} = 396 - 320 = 76$$

Exam­ple 8: The 16th innings#

A bats­man scores 64 runs in his 16th innings, and this raises his aver­age by 3. Find his aver­age after the 16th innings.

Let his aver­age after 15 innings be $P$. Then his total after 15 innings is $15P$.

After the 16th innings the total is $15P + 64$, and the aver­age is $P + 3$:

$$\displaystyle \begin{aligned} \frac{15P + 64}{16} &= P + 3 \\ 15P + 64 &= 16P + 48 \\ 64 - 48 &= 16P - 15P \\ P &= 16 \end{aligned}$$

His aver­age after the 16th innings is $16 + 3 = 19$.

Exam­ple 9: Run rate#

In the first 10 overs of a cricket match, the run rate was only 3.2. What run rate is needed in the remain­ing 40 overs to reach a tar­get of 282 runs?

The run rate is the aver­age num­ber of runs per over.

Runs in the first 10 overs $= 3.2 \times 10 = 32$.

Runs still needed $= 282 - 32 = 250$.

$$\displaystyle \text{run rate needed} = \frac{250}{40} = 6.25$$

One per­son replaced by another#

Exam­ple 10#

The aver­age weight of 8 peo­ple goes up by 2.5 kg when a new per­son comes in place of one of them who weighs 65 kg. What is the weight of the new per­son?

The total weight of the 8 peo­ple goes up by $8 \times 2.5 = 20$ kg.

The only change is that 65 kg went out and the new per­son came in. So the new per­son weighs 20 kg more than 65 kg:

$$N = 65 + 20 = 85 \text{ kg}$$

With let­ters: if the old aver­age is $P$, the old total is $8P$. The new total is $8P - 65 + N$, and it equals $8(P + 2.5) = 8P + 20$. So $N - 65 = 20$ and $N = 85$.

Prac­tice#

1. The aver­age of 7 num­bers is 20. The aver­age of the first 4 is 17. Find the aver­age of the last 3.
2. The aver­age of 11 num­bers is 50. The aver­age of the first 6 is 49 and the aver­age of the last 6 is 52. Find the 6th num­ber.
3. A bats­man scores 87 runs in his 17th innings and raises his aver­age by 3. Find his aver­age after the 17th innings.
4. The aver­age age of a fam­ily of 5 is 24 years. The youngest mem­ber is 4 years old. What was the aver­age age of the fam­ily when the youngest was born?
5. The aver­age weight of 10 peo­ple goes up by 1.5 kg when a per­son weigh­ing 60 kg is replaced by a new per­son. Find the weight of the new per­son.
6. In the first 20 overs, the run rate was 4.5. What run rate is needed in the remain­ing 30 overs to reach a tar­get of 300?
7. A park has an aver­age of 300 vis­i­tors on Sun­days and 150 on other days. Find the aver­age num­ber of vis­i­tors per day in a month of 30 days that begins with a Sun­day.
8. A play­er's aver­age over 12 innings is 40. How many runs must he score in the 13th innings to raise his aver­age to 42?

Answers#