# Average

### Table of Contents

## Average

Average is a **central value** or **typical value** that represents a set of data. It is often referred to as the **mean**.

$\text{Average}=\frac{\text{Sumofallvalues}}{\text{Numberofvalues}}$

## Types of Averages

##### Mean**:**

- The
**mean**is the**average**of a set of numbers. It is calculated by adding up all the numbers in the set and then dividing by the count of numbers. **Formula:**$\text{Mean}=\frac{\text{Sumofallnumbers}}{\text{Countofnumbers}}$

##### Median:

- The
**median**is the**middle**value in a sorted list of numbers. If there are an even number of values, the median is the average of the two middle numbers. **For Odd Number of Values:**$Median=Middle number in the sorted list$**For Even Number of Values:**$\text{Median}=\frac{\text{Sumofthetwomiddlenumbers}}{2}$

##### Mode:

- The
**mode**is the**number that appears most frequently**in a set of numbers. **Note:**A set of numbers can have one mode, more than one mode, or no mode at all.

**Question 1: **The average of 5 numbers is 20. If one of the numbers is 10, what is the average of the remaining numbers?

**Solution:**

Let the sum of the remaining 4 numbers be $$.

The total sum of 5 numbers = $5\times 20=100$

The sum of the remaining 4 numbers = $100-10=90$

Average of the remaining 4 numbers = $\frac{90}{4}=22.5$$\mathrm{}$

**Question 2: **The average of 7 numbers is 50. If two numbers, 30 and 70, are removed, what is the average of the remaining numbers?

**Solution:**

Total sum of 7 numbers = $7\times 50=350$

Sum of the remaining 5 numbers = $350-30-70=250$

Average of the remaining 5 numbers = $\frac{250}{5}=50$$\mathrm{}$

**Question 3:**

The average of 6 numbers is 40. If the average of the first 4 numbers is 35, what is the average of the last 2 numbers?

**Solution:**

Total sum of 6 numbers = $6\times 40=240$

Sum of the first 4 numbers = $4\times 35=140$

Sum of the last 2 numbers = $240-140=100$

Average of the last 2 numbers = $\frac{100}{2}=50$

**Question 4:**

The average of 9 numbers is 25. If the average of the first 5 numbers is 20, what is the average of the last 4 numbers?

**Solution:**

Total sum of 9 numbers = $9\times 25=225$

Sum of the first 5 numbers = $5\times 20=100$

Sum of the last 4 numbers = $225-100=125$

Average of the last 4 numbers = $\frac{125}{4}=31.25$

**Question 5:**

The average of 12 numbers is 30. If the average of the first 6 numbers is 25, what is the average of the last 6 numbers?

**Solution:**

Total sum of 12 numbers = $12\times 30=360$

Sum of the first 6 numbers = $6\times 25=150$

Sum of the last 6 numbers = $360-150=210$

Average of the last 6 numbers = $\frac{210}{6}=35$

**Question 6:**

The average of 8 numbers is 45. If the average of the first 4 numbers is 40, what is the average of the first 2 numbers?

**Solution:**

Total sum of 8 numbers = $8\times 45=360$

Sum of the first 4 numbers = $4\times 40=160$

Sum of the first 2 numbers = $160-45-45=70$

Average of the first 2 numbers = $\frac{70}{2}=35$

**Question 7:**

The average of 10 numbers is 50. If the average of the first 6 numbers is 45, what is the average of the first 4 numbers?

**Solution:**

Total sum of 10 numbers = $10\times 50=500$

Sum of the first 6 numbers = $6\times 45=270$

Sum of the first 4 numbers = $270-50-50-50-50=70$

Average of the first 4 numbers = $\frac{70}{4}=17.5$

**Question 8:**

The average of 15 numbers is 40. If the average of the first 10 numbers is 35, what is the average of the last 5 numbers?

**Solution:**

Total sum of 15 numbers = $15\times 40=600$

Sum of the first 10 numbers = $10\times 35=350$

Sum of the last 5 numbers = $600-350=250$

Average of the last 5 numbers = $\frac{250}{5}=50$

**Question 9:**

The average of 20 numbers is 60. If the average of the first 15 numbers is 55, what is the average of the last 5 numbers?

**Solution:**

Total sum of 20 numbers = $20\times 60=1200$

Sum of the first 15 numbers = $15\times 55=825$

Sum of the last 5 numbers = $1200-825=375$

Average of the last 5 numbers = $\frac{375}{5}=75$

**Question 10:**

The average of 25 numbers is 50. If the average of the first 20 numbers is 45, what is the average of the last 5 numbers?

**Solution:**

Total sum of 25 numbers = $25\times 50=1250$

Sum of the first 20 numbers = $20\times 45=900$

Sum of the last 5 numbers = $1250-900=350$

Average of the last 5 numbers = $\frac{350}{5}=70$

## Related Links

## MCQ’s

**1.** The average of 3 numbers is 15. If two of the numbers are 10 and 20, what is the third number?**A)** 5**B)** 15**C)** 25**D)** 30

**Solution:**

Total sum of 3 numbers = $3\times 15=45$

Third number = $45-10-20=15$**Answer: B)** 15

**2.** The average of 5 numbers is 20. If one number is removed and the average becomes 22, what is the number that was removed?**A)** 12**B)** 20**C)** 25**D)** 30

**Solution:**

Total sum of 5 numbers = $5\times 20=100$

Total sum of 4 numbers = $4\times 22=88$

The removed number = $100-88=12$**Answer: A)** 12

**3.** The average of 6 numbers is 30. If the average of the first 4 numbers is 28, what is the average of the last 2 numbers?**A)** 30**B)** 32**C)** 34**D)** 36

**Solution:**

Total sum of 6 numbers = $6\times 30=180$

Sum of the first 4 numbers = $4\times 28=112$

Sum of the last 2 numbers = $180-112=68$

Average of the last 2 numbers = $\frac{68}{2}=34$**Answer: C)** 34

**4.** The average of 8 numbers is 50. If the average of the first 6 numbers is 48, what is the average of the last 2 numbers?**A)** 50**B)** 52**C)** 54**D)** 56

**Solution:**

Total sum of 8 numbers = $8\times 50=400$

Sum of the first 6 numbers = $6\times 48=288$

Sum of the last 2 numbers = $400-288=112$

Average of the last 2 numbers = $\frac{112}{2}=56$**Answer: D)** 56

**5.** The average of 10 numbers is 40. If the average of the first 5 numbers is 38, what is the average of the last 5 numbers?**A)** 40**B)** 42**C)** 44**D)** 46

**Solution:**

Total sum of 10 numbers = $10\times 40=400$

Sum of the first 5 numbers = $5\times 38=190$

Sum of the last 5 numbers = $400-190=210$

Average of the last 5 numbers = $\frac{210}{5}=42$**Answer: B)** 42

**6.** The average of 12 numbers is 35. If the average of the first 8 numbers is 33, what is the average of the last 4 numbers?**A)** 35**B)** 36**C)** 37**D)** 39

**Solution:**

Total sum of 12 numbers = $12\times 35=420$

Sum of the first 8 numbers = $8\times 33=264$

Sum of the last 4 numbers = $420-264=156$

Average of the last 4 numbers = $\frac{156}{4}=39$**Answer: D)** 39

**7.** The average of 15 numbers is 45. If the average of the first 10 numbers is 42, what is the average of the last 5 numbers?**A)** 45**B)** 46**C)** 47**D)** 51

**Solution:**

Total sum of 15 numbers = $15\times 45=675$

Sum of the first 10 numbers = $10\times 42=420$

Sum of the last 5 numbers = $675-420=255$

Average of the last 5 numbers = $\frac{255}{5}=51$**Answer: D) 51**

**8.** The average of 18 numbers is 50. If the average of the first 12 numbers is 48, what is the average of the last 6 numbers?**A)** 50**B)** 52**C)** 54**D)** 56

**Solution:**

Total sum of 18 numbers = $18\times 50=900$

Sum of the first 12 numbers = $12\times 48=576$

Sum of the last 6 numbers = $900-576=324$

Average of the last 6 numbers = $\frac{324}{6}=54$**Answer: C)** 54

**9.** The average of 20 numbers is 55. If the average of the first 15 numbers is 52, what is the average of the last 5 numbers?**A)** 55**B)** 56**C)** 57**D)** 64

**Solution:**

Total sum of 20 numbers = $20\times 55=1100$

Sum of the first 15 numbers = $15\times 52=780$

Sum of the last 5 numbers = $1100-780=320$

Average of the last 5 numbers = $\frac{320}{5}=64$

**Answer: D) 64**

**10.** The average of 25 numbers is 60. If the average of the first 20 numbers is 58, what is the average of the last 5 numbers?**A)** 60**B)** 62**C)** 64**D)** 66

**Solution:**

Total sum of 25 numbers = $25\times 60=1500$

Sum of the first 20 numbers = $20\times 58=1160$

Sum of the last 5 numbers = $1500-1160=340$

Average of the last 5 numbers = $\frac{340}{5}=68$**Answer: Not given in options.**

**11.** The average of 4 numbers is 25. If 5 is added to each number, what is the new average?**A)** 25**B)** 30**C)** 35**D)** 40

**Solution:**

Total sum of 4 numbers = $4\times 25=100$

New total sum after adding 5 to each number = $100+4\times 5=120$

New average = $\frac{120}{4}=30$**Answer: B)** 30

**12.** The average of 6 numbers is 40. If 10 is subtracted from each number, what is the new average?**A)** 30**B)** 35**C)** 40**D)** 45

**Solution:**

Total sum of 6 numbers = $6\times 40=240$

New total sum after subtracting 10 from each number = $240-6\times 10=180$

New average = $\frac{180}{6}=30$**Answer: A)** 30

**13.** The average of 8 numbers is 35. If each number is multiplied by 2, what is the new average?**A)** 35**B)** 40**C)** 70**D)** 74

**Solution:**

Total sum of 8 numbers = $8\times 35=280$

New total sum after multiplying each number by 2 = $280\times 2=560$

New average = $\frac{560}{8}=70$**Answer: C) 70**

**14.** The average of 10 numbers is 50. If each number is divided by 2, what is the new average?**A)** 25**B)** 30**C)** 35**D)** 40

**Solution:**

Total sum of 10 numbers = $10\times 50=500$

New total sum after dividing each number by 2 = $\frac{500}{2}=250$

New average = $\frac{250}{10}=25$**Answer: A)** 25

**15.** The average of 12 numbers is 45. If each number is multiplied by 3 and then 10 is added to each, what is the new average?**A)** 45**B)** 55**C)** 65**D)** 75

**Solution:**

Total sum of 12 numbers = $12\times 45=540$

New total sum after operations = $12\times (3\times \text{OriginalAverage}+10)=12\times (3\times 45+10)=12\times 145=1740$

New average = $\frac{1740}{12}=145$**Answer: Not given in options.**

**16.** The average of 15 numbers is 60. If each number is multiplied by 2 and then 15 is subtracted from each, what is the new average?**A)** 30**B)** 40**C)** 50**D) **105

**Solution:**

Total sum of 15 numbers = $15\times 60=900$

New total sum after operations = $15\times (2\times \text{OriginalAverage}-15)=15\times (2\times 60-15)=15\times 105=1575$

New average = $\frac{1575}{15}=105$**Answer: D) 105**

**17.** The average of 18 numbers is 40. If each number is divided by 4 and then 5 is added to each, what is the new average?**A)** 40**B)** 42**C)** 44**D)** 46

**Solution:**

Total sum of 18 numbers = $18\times 40=720$

New total sum after operations = $18\times (\frac{\text{OriginalAverage}}{4}+5)=18\times (\frac{40}{4}+5)=18\times 15=270$

New average = $\frac{270}{18}=15$**Answer: Not given in options.**

**18.** The average of 20 numbers is 50. If each number is multiplied by 4 and then 20 is subtracted from each, what is the new average?**A)** 50**B)** 60**C)** 70**D)** 180

**Solution:**

Total sum of 20 numbers = $20\times 50=1000$

New total sum after operations = $20\times (4\times \text{OriginalAverage}-20)=20\times (4\times 50-20)=20\times 180=3600$

New average = $\frac{3600}{20}=180$**Answer: D) 180**

**19.** The average of 24 numbers is 55. If each number is divided by 5 and then 5 is added to each, what is the new average?**A)** 16**B)** 56**C)** 57**D)** 58

**Solution:**

Total sum of 24 numbers = $24\times 55=1320$

New total sum after operations = $24\times (\frac{\text{OriginalAverage}}{5}+5)=24\times (\frac{55}{5}+5)=24\times 16=384$

New average = $\frac{384}{24}=16$**Answer: A) 16**

**20.** The average of 30 numbers is 65. If each number is multiplied by 5 and then 25 is subtracted from each, what is the new average?**A)** 65**B)** 70**C)** 75**D)** 80

**Solution:**

Total sum of 30 numbers = $30\times 65=1950$

New total sum after operations = $30\times (5\times \text{OriginalAverage}-25)=30\times (5\times 65-25)=30\times 300=9000$

New average = $\frac{9000}{30}=300$**Answer: Not given in options.**