Seven Numbers

a+b+c+d = 20; //4 5 d+e+f+g = 32; // 8 4 a+b+c+d+e+f+g = 46; // (46/7)*7 32+20 = 52; 52-46 = 6;

Let the seven numbers be a, b, c, d, e, f and g.

We need to find the middle term which is d.

Now, according to the question,

the average of all seven numbers is 46/7

(a + b + c+ d+ e+ f+ g)/7 = 46 / 7

a + b + c+ d+ e+ f+ g = 46 ------------------------------------ ( i )

the average of first four numbers is 5

(a + b + c+ d)/4 = 5

a + b + c+ d= 20

d= 20 - a - b - c ------------------------------------ ( ii )

the average of last four numbers is 8

(d + e + f + g)/4 = 8

d + e + f +g= 32

d= 32 - e - f - g ------------------------------------ ( iii )

Substituting the value of d of equation ( ii ) in equation ( i ) we get.

a + b + c+ d+ e+ f+ g = 46

a + b + c+ 20 - a - b - c+ e+ f+ g = 46

e + f + g = 26 ------------------------------------ ( iv )

Substituting the value of d of equation ( iii ) in equation ( i ) we get.

a + b + c+ d+ e+ f+ g = 46

a + b + c+ 32 - e - f - g+ e+ f+ g = 46

a + b + c = 14 ------------------------------------ ( v )

Finally ,

Substituting the value of a,b,c of equation ( v ) and e,f,g of equation ( iv ) in equation ( i ) we get.

a + b + c+ d+ e+ f+ g = 46

14 + d + 26 = 46

d + 40 = 46

d = 46 - 40

Hence, d = 6

Let the seven numbers be a, b, c, d, e, f & g. Now (a+b+c+d)/4 = 5 i.e. a+b+c+d =20........................................(1) and (d+e+f+g)/4 = 7 i.e. d+e+f+g =32.......................................(2) but (a+b+c+d+e+f+g)/7 =47/7 i.e. a+b+c+d+e+f+g =46...........(3) Now (1) + (2) - (3) {(a+b+c+d)+(d+e+f+g)-(a+b+c+d+e+f+g)} = 20+32-46 or d = 6