Average
- Among three numbers, the first is twice the second and thrice the third. If the average of the three numbers is 49.5, then the difference between the first and the third number is
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Let the second number be p.
∴ First number = 2p∴ Third number = 2p 3 ∴ 2p + p + 2p = 49.5 × 3 3
⇒ 6p + 3p + 2p = 49.5 × 9 = 445.5
⇒ 11p = 445.5⇒ p = 445.4 = 40.5 11 ∴ Required difference = 2p - 2p = 4p 3 3 Required difference = 4 × 40.5 = 54 3
Aliter : Using Rule 15,Here, a = 2, b = 3 , X = 49.5 2 First Number = 3ab X 1 + b + ab First Number = 3 × 2 × 3 × 49.5 2 1 + 3 + 2 × 3 2 2
Correct Option: A
Let the second number be p.
∴ First number = 2p∴ Third number = 2p 3 ∴ 2p + p + 2p = 49.5 × 3 3
⇒ 6p + 3p + 2p = 49.5 × 9 = 445.5
⇒ 11p = 445.5⇒ p = 445.4 = 40.5 11 ∴ Required difference = 2p - 2p = 4p 3 3 Required difference = 4 × 40.5 = 54 3
Aliter : Using Rule 15,Here, a = 2, b = 3 , X = 49.5 2 First Number = 3ab X 1 + b + ab First Number = 3 × 2 × 3 × 49.5 2 1 + 3 + 2 × 3 2 2 First Number = 18 × 49.5 2 11 2 First Number = 18 × 49.5 = 18 × 4.5 11 First Number = 18 × 45 = 81 10 Third Number = 3 X 1 + b + ab Third Number = 3 × 49.5 1 + 3 + 2 × 3 2 2 Third Number = 3 × 49.5 11 2
Third Number = 6 × 4.5 = 27
Difference = 81 – 27 = 54
- Of the three numbers, the first number is twice of the second and the second is thrice of the third number. If the average of these 3 numbers is 20, then the sum of the largest and smallest numbers is
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Let the third number be p.
∴ Second number = 3p
and first number = 6p
∴ 6p + 3p + p = 3 × 20
⇒ 10p = 60
⇒ p = 6
∴ Required sum = 6p + p = 7p = 7 × 6 = 42
Second method to solve this question with the help of given formulas :
Here, a = 2, b = 3, X = 20Largest Number = 3ab X 1 + b + ab Largest Number = 3 × 2 × 3 × 20 1 + 3 + 2 × 3
Correct Option: B
Let the third number be p.
∴ Second number = 3p
and first number = 6p
∴ 6p + 3p + p = 3 × 20
⇒ 10p = 60
⇒ p = 6
∴ Required sum = 6p + p = 7p = 7 × 6 = 42
Second method to solve this question with the help of given formulas :
Here, a = 2, b = 3, X = 20Largest Number = 3ab X 1 + b + ab Largest Number = 3 × 2 × 3 × 20 1 + 3 + 2 × 3 Largest Number = 18 × 20 = 36 10 Smallest Number = 3 X 1 + b + ab Smallest Number = 3 × 20 1 + 3 + 2 × 3 Smallest Number = 3 × 20 = 6 10
Sum = 36 + 6 = 42
- Of three numbers, the first is 4 times the second and 3 times the third. If the average of all the three numbers is 95, what is the third number ?
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Let the third number be p,
∴ First number = 3p∴ Second number = 3p 4 According to the question, 3p + 3p + p = 3 × 95 4 ⇒ 12p + 3p + 4p = 285 4
⇒ 19p = 285 × 4⇒ 285 × 4 = 60 19
Aliter : Using Rule 15,Here, a = 4, b = 3 , X = 95 4
Correct Option: B
Let the third number be p,
∴ First number = 3p∴ Second number = 3p 4 According to the question, 3p + 3p + p = 3 × 95 4 ⇒ 12p + 3p + 4p = 285 4
⇒ 19p = 285 × 4⇒ 285 × 4 = 60 19
Aliter : Using Rule 15,Here, a = 4, b = 3 , X = 95 4 Third Number = 3 X 1 + b + ab Third Number = 3 × 95 1 + 3 + 4 × 3 4 4 Third Number = 3 × 4 × 95 = 60 4 + 3 + 12
- A and B have their annual average income Rs. 80,000. B and C have their annual average income Rs. 75,000. C and A have their annual average income Rs. 78,000. The annual income of A is
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According to question ,
Average of annual income of (A + B) = Rs. 80000
Total annual income of (A + B) = Rs. (2 × 80000) = Rs. 160000 ..... (i)
Average of annual income of (B + C) = Rs. 75000
Total annual income of (B + C) = Rs. (2 × 75000) = Rs. 150000 ..... (ii)
Average of annual income of (C + A) = Rs. 78000
Total annual income of (C + A) = Rs. (2 × 78000) = Rs. 156000 ..... (iii)
On adding all three equations , we get
Total annual income of 2 (A + B + C) = Rs. (160000 + 150000 + 156000) = Rs. 466000
∴ Total annual income of (A + B + C) = Rs. 233000 ..... (iv)Correct Option: C
According to question ,
Average of annual income of (A + B) = Rs. 80000
Total annual income of (A + B) = Rs. (2 × 80000) = Rs. 160000 ..... (i)
Average of annual income of (B + C) = Rs. 75000
Total annual income of (B + C) = Rs. (2 × 75000) = Rs. 150000 ..... (ii)
Average of annual income of (C + A) = Rs. 78000
Total annual income of (C + A) = Rs. (2 × 78000) = Rs. 156000 ..... (iii)
On adding all three equations , we get
Total annual income of 2 (A + B + C) = Rs. (160000 + 150000 + 156000) = Rs. 466000
∴ Total annual income of (A + B + C) = Rs. 233000 ..... (iv)
∴ A’s annual income = Equation (iv) – (ii)
∴ A’s annual income = Rs. (233000 – 150000) = Rs. 83000
- The average of a collection of 20 measurements was calculated to be 56 cm. But later it was found that a mistake had occurred in one of the measurements which was recorded as 64 cm., but should have been 61 cm. The correct average must be
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Total length of 20 measurements = 56 × 20 = 1120 cm
Correct length of 20 measurements = 1120 – 64 + 61 = 1117
Correct average = 1117 ÷ 20 = 55.85 cm
Second method to solve this question with the help of given formula :
Here, n = 20, m = 56 , a = 61, b = 64Correct Average = m + (a - b) n
Correct Option: C
Total length of 20 measurements = 56 × 20 = 1120 cm
Correct length of 20 measurements = 1120 – 64 + 61 = 1117
Correct average = 1117 ÷ 20 = 55.85 cm
Second method to solve this question with the help of given formula :
Here, n = 20, m = 56 , a = 61, b = 64Correct Average = m + (a - b) n Correct Average = 56 + 
61 - 64 
20 Correct Average = 56 - 3 20
Correct Average = 56 – 0.15 = 55.85 cm
