The average of five consecutive odd numbers is 61. What is the difference between the highest and the lowest numbers? \[ (a){\text{ 2}} \\ (b){\text{ 5}} \\ (c){\text{ 8}} \\ (d){\text{ Can not be determined}} \\ (e){\text{ None of these}} \\ \]
ANSWER
Verified
Hint – In this problem let the first odd number be x, then use the concept that the next consecutive odd number will be the addition of the previous odd number with 2, thus odd numbers are x,x+2,x+4,x+6……… . Use this along with questions constraints to get the answer. Complete step-by-step answer: As we know, an odd number is not divisible by 2. So let the first odd number be x. Now if we add 1 in this odd number it becomes even for example (3 + 1 = 4) 4 is an even number so we have to add 2 to get the next consecutive odd number. So the next consecutive odd number is (x + 2). Now we have to again add 2 in this odd number to get the next consecutive odd number. So the third consecutive odd number is = ((x + 2) + 2) = x + 4. Similarly fourth consecutive odd numbers = x + 6. And the fifth consecutive odd number = x + 8. So the five consecutive odd numbers are (x), (x + 2), (x + 4), (x + 6) and (x + 8) Now it is given that the average of five consecutive odd numbers is 61. Now as we know that the average is calculated as the sum of numbers divided by the total numbers. \[ \Rightarrow \dfrac{{x + \left( {x + 2} \right) + \left( {x + 4} \right) + \left( {x + 6} \right) + \left( {x + 8} \right)}}{5} = 61\] Now simplify the above equation we have, \[ \Rightarrow 5x + \left( {2 + 4 + 6 + 8} \right) = 61 \times 5\] $ \Rightarrow 5x = 305 - 20 = 285$ Now divide by 5 we have, $ \Rightarrow x = \dfrac{{285}}{5} = 57$ So the first odd number which is the smallest odd number is 57. And the highest odd number which is the fifth odd number = (x + 8) = 57 + 8 = 65. So the difference of highest and lowest odd number is $ \Rightarrow 65 - 57 = 8$ So this is the required answer. Hence option (C) is correct.
Note – The trick part here was the average concept, average is a number expressing the central or typical value in a set of data, and it is calculated by dividing the sum of the values in the set by their number. An odd number is any integer (not fraction) that cannot be divided exactly by 2.