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91. A student was asked to divide the half of a certain number by 6 and the other half by 4 and then to add the two quantities so obtained. Instead of doing so the student divided the number by 5 and the result fell short by 4. The given number was ?

Discuss
Solution:
Let the number be x
Then,
⇔[(x2)6+(x2)4]−x5=4⇔x12+x8−x5=4⇔10x+15x−24x120=4⇔x=4×120⇔x=480
92. If the sum of numbers is 33 and their difference is 15, the smaller number is ?

Discuss
Solution:
Let the numbers be x and y
Then,
x+y=33.....(i)
And,
x−y=15.....(ii)
Solving (i) and (ii), we get :
x = 24 , y = 9
∴ Smaller number = 9
93. In a two-digit positive number, the digit in the unit's place is equal to the square of the digit in ten's place, and the difference between the number and the number obtained by interchanging the digits is 54. What is 40% of the original number ?

Discuss
Solution:
Let ten's digit = x
Then, unit's digit = x2
Then, number = 10x + x2
Clearly, since x2 > x,
So, the number formed by interchanging the digits is greater than the original number.
∴(10x2+x)−(10x+x2)=54⇔9x2−9x=54⇔x2−x=6⇔x2−x−6=0⇔x2−3x+2x−6=0⇔(x−3)(x+2)=0⇔x=3
So. ten's digit = 3, unit's digit = 32 = 9
∴ Original number = 39
Required result :
= 40% of 39
= 15.6
94. 243 has been divided into three parts such that half of the first part, one-third of the second part and one-fourth of the third part are equal. The largest part is :

Discuss
Solution:
Let the three parts be A, B and C
Let A2=B3=C4=xThen, A=2x,B=3x and C=4xSo, A:B:C=2:3:4∴Largest part :=(243×49)=108
95. If the sum of a number and its square is 182, what is the number ?

Discuss
Solution:
Let the number be x
Then,
⇔x+x2=182⇔x2+x−182=0⇔(x+14)(x−13)=0⇔x=13
96. The sum of two numbers is 40 and their difference is 4. The ratio of the numbers is :

Discuss
Solution:
Let the numbers be x and y
Then,
⇔x+yx−y=404⇔x+yx−y=10⇔(x+y)=10(x−y)⇔9x=11y⇔xy=119⇔x:y=11:9
97. The sum of three consecutive odd numbers and three consecutive even numbers together is 231. Also, the smallest odd number is 11 less than the smallest even number. What is the sum of the largest odd number and the largest even number ?

Discuss
Solution:
Let the three odd numbers be x, (x + 2), (x + 4) and
The three even numbers be (x + 11), (x + 13) and (x + 15)
Then,
⇔ x + (x + 2) + (x + 4) + (x + 11) + (x + 13) + (x + 15) = 231
⇔ 6x + 45 = 231
⇔ 6x = 186
⇔ x = 31
∴ Required sum :
= (x + 4) + (x + 15)
= 2x + 19
= 2 × 31 + 19
= 62 + 19
= 81
98. A number consists of 3 digits whose sum is 10. The middle digit is equal to the sum of the other two and the number will be increased by 99 if its digits are reversed. The number is :

Discuss
Solution:
Let the middle digit be x
Then,
2x = 10 or x = 5
So, the number is either 253 or 352
Since the number increases on reversing the digits, so the hundred's digit is smaller than the unit's digit.
Hence, required number = 253
99. The sum of four numbers is 64. If you add 3 to the first number, 3 is subtracted from the second number, the third is multiplied by 3 and the fourth is divided by 3, then all the results are equal. What is the difference between the largest and the smallest of the original numbers ?

Discuss
Solution:
Let the four numbers be , A, B, C and D
Let A + 3 = B - 3 = 3C = D3 = x
Then,
A = x - 3
B = x + 3
C = x3
D = 3x
⇔A+B+C+D=64⇔(x−3)+(x+3)+x3+3x=64⇔5x+x3=64⇔16x=192⇔x=12
Thus, the numbers are 9, 15, 4 and 36
∴ Required difference :
= (36 - 4)
= 32
100. The sum of five consecutive odd numbers is 575. What is the sum of the next set of five consecutive odd numbers ?

Discuss
Solution:
Let the five numbers be x, (x + 2), (x + 4), (x + 6) and (x + 8)
Then,
⇔ x + (x + 2) + (x + 4) + (x + 6) + (x + 8) = 575
⇔ 5x + 20 = 575
⇔ 5x = 555
⇔ x = 111
∴ Required sum :
= (x + 10) + (x + 12) + (x + 14) + (x + 16) + (x + 18)
= 5x + 70
= 5 × 111 + 70
= 555 + 70
= 625

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