Question

The sum of four numbers is 80. If we subtract 1, 2, 11, 44 from these numbers in that order. We obtain an arithmetic progression. Find the numbers.

Answer 1

Hint:Here we go through by assuming the four different numbers as a, b, c and d. and then apply the properties of AP as given in question. We know that Three number x, y and z are in an A.P then 2y = x + z.

Complete step-by-step solution:
Let the four different numbers as a, b, c and d respectively.
It is given that a +b +c +d =80
Now according to the question we subtract 1, 2, 11, 44 from these numbers.
I.e. (a−1), (b−2), (c−11), (d−44) are in AP
So by applying the properties of AP that are given in the hint we write
2(b−2) = (c−11) + (a−1)
2b=a+c-8 ……… (1)
And we also write
2(c−11) = (d−44) + (b−2)
2c=b+d-24………. (2)
By adding equation 1 and 2 we get,
b + c = a+ d -32 $ \ldots \ldots $(3)
now it is given that a +b +c + d=80
put the value from equation (3) we get,
2a+2d=112
a + d =56${\kern 1pt} \ldots \ldots $(4)
b + c =24$ \ldots \ldots $(5)
Now we have to solve by hit and trial method that means we assume a value of c and b to proceed further in the question,
Let c=18, b=6
b-2 =4 is second term of AP and c-11 =7 is third term
so, common difference d=7-4=3.
Common difference is 3 and second term is 4 hence first term will be 4-3=1
So, a-1=1, a=2

b-2=4, b=6 and c-11=7 so c=18
Fourth term of Ap will be equal to third term +common difference $ \Rightarrow $7+3 =10
Here d-44 is fourth term so, d-44=10, d= 54.
Hence a=4, b=6, c=18, d=54 are the required numbers.

Note: Whenever you get this type of question the key concept of solving is you have to use the property of arithmetic progression and also use the concept that difference of consecutive terms is constant that is called common difference.