Question

The sum of five numbers in AP is 75,the product of the greatest and the least is 161,find the numbers

Answer 1

Step-by-step explanation:

Suppose the first and the last term of an A.P. are aand l respectively.

Since the product of the greatest and the least is 161, so we have;

al = 161       ...(i)

Also the sum of five numbers of an A.P. is 75, so we have;

s5 = 52(a+l) = 75⇒(a+l) = 75×25⇒a+l = 30 ...(ii)

So from (i) and (ii) we have;

a+161a = 30⇒a2+161a = 30⇒a2+161 = 30a⇒a2−30a+161 = 0⇒a2−23a−7a+161 = 0⇒a(a−23)−7(a−23) = 0⇒(a−7)(a−23) = 0⇒(a−7) = 0 and (a−23) = 0⇒a = 7 and  a = 23

So from (i) we get, when a = 23, then l = 7 and when a= 7, then l = 23

Therefore the greatest number is 23 and the least number is 7.

Suppose d is the commomdn difference of an A.P.

And we know the formula for the fifth term = t5 = a+(5−1)d

Considering 7 as the first term and 23 as the last term i.e. 5th term we have;

⇒23 = 7+(5−1)d⇒23−7 = 4d⇒16 = 4d

Answer 2

Answer:

36:90 = 36 = 36 ÷ 18 = 2 (As the H.C.F. of 36 and 90 is 18.)

90 90 ÷ 18 5

Since the H.C.F. of 2 and 5 is 1, the simplest form of 36:90 is 2:5.

(ii) 324:144 = 324 = 324 ÷ 36 = 9 (As the H.C.F. of 324 and 144 is 36.)

144 144 ÷ 36 4

Since the H.C.F. of 9 and 4 is 1, the simplest form of 324:144 is 9:4.

(iii) 85:561 = 85 = 85 ÷ 17 = 5 (As the H.C.F. of 85 and 561 is 17.)

561 561 ÷ 17 33

Since the H.C.F. of 5 and 33 is 1, the simplest form of 85:561 is 5:33.

(iv) 480:384 = 480 = 480 ÷ 96 = 5 (As the H.C.F. of 480 and 384 is 96.)

384 384 ÷ 96 4

Since the H.C.F. of 5 and 4 is 1, the simplest form of 480:384 is 5:4.

(v) 186:403 = 186 = 186 ÷ 31 = 6 (As the H.C.F. of 186 and 403 is 31.)

403 403 ÷ 31 13

Since the H.C.F. of 6 and 13 is 1, the simplest form of 186:403 is 6:13.

(vi) 777:1147 = 777 ÷ 37 = 21 (As the H.C.F. of 777 and 1147 is 37.)

1147 ÷ 37 31

Since the H.C.F. of 21 and 31 is 1, the simplest form of 777:1147 is 21:31