Given:
Identify the correct mathematical
Statements:
A. The expression (1)/(5 × 6) + (1)/(6 × 7) + (1)/(7 × 8) + … + (1)/(24 × 25) is equal to (4)/(25)
B. Of the three numbers, second is twice the first and is also thrice the third. If the average of three numbers is 55, then largest number is 90.
C. If (a+b):(b+c):(c+a) = 6:7:8 and a+b+c=14, then value of c is 8
D. (0.2 × 0.2 + 0.02 × 0.02 - 0.4 × 0.02)/(0.18) is equal to 0.09
Formula used:
For A: Use the property of telescoping series.
For B: Average = (Sum of numbers)/3, and use given ratios.
For C: Ratios and sum relation.
For D: Simplify the numerator and divide by the denominator.
Calculations:
Statement A:
1/(5×6) + 1/(6×7) + … + 1/(24×25)
Use identity:
1 / [n(n+1)] = 1/n − 1/(n+1)
So the series becomes telescopic:
(1/5 − 1/6) + (1/6 − 1/7) + … + (1/24 − 1/25)
= 1/5 − 1/25
= (5 − 1)/25
= 4/25
Statement A is Correct.
Statement B:
Let first number = x
Second = 2x
Also second = 3 × third
So third = 2x/3
Average = 55 ⇒ Sum = 55 × 3 = 165
x + 2x + 2x/3 = 165
(3x + 2x/3) = 165
(11x/3) = 165
x = 45
Numbers: 45, 90, 30
Largest = 90
Statement B is Correct.
Statement C:
(a+b):(b+c):(c+a) = 6:7:8
Let a+b = 6k, b+c = 7k, c+a = 8k
Adding:
2(a+b+c) = 21k
Given a+b+c = 14
2×14 = 21k
28 = 21k
k = 4/3
Now,
a+b = 6k = 8
b+c = 7k = 28/3
c+a = 8k = 32/3
Using:
c = [(b+c) + (c+a) − (a+b)] / 2
= (28/3 + 32/3 − 8)/2
= (60/3 − 8)/2
= (20 − 8)/2
= 12/2
= 6
So c ≠ 8
Statement C is Incorrect.
Statement D:
Numerator:
0.2×0.2 = 0.04
0.02×0.02 = 0.0004
0.4×0.02 = 0.008
0.04 + 0.0004 − 0.008 = 0.0324
Now divide:
0.0324 ÷ 0.18 = 0.18
So not 0.09
Statement D is Incorrect.
Final Answer:
Only A and B are correct.
∴ The Correct option is 4