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21. The digit in the unit's place of a numbers is equal to the digit in the ten's place of half of that number and the digit in the ten's place of that number is less than the digit in unit's place of half of the number by 1. If the sum of the digits of the number is 7, then what is the number ?

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Solution:
Let the ten's digit be x and unit's digit be y
Then,
⇔10x+y2=10y+(x+1)⇔10x+y=20y+2x+2⇔8x−19y=2.....(i)And⇔x+y=7.....(ii)
Solving (i) and (ii), we get :
x = 5, y = 2
Hence, required number = 52
22. Of the three number, the sum of the first two is 73, the sum of the second and the third is 77 and the sum of the third and thrice the first is 104. The third number is ?

Discuss
Solution:
Let the numbers be x, y and z
Then,
x+y=73⇔y=73−xy+z=77⇔z=77−y⇔z=77−(73−x)⇔z=4+x3x+z=104⇔3x+4+x=104⇔4x=100⇔x=25∵y=(73−25)⇔y=48∵z=(4+25)⇔z=29
∴ Third number = 29
23. The difference between a number and its three-fifths is 50, What is the number ?

Discuss
Solution:
Let the number be x
Then,
⇔x−35x=50⇔25x=50⇔x=(50×52)⇔x=125
24. Two-third of a positive number and 25216 of its reciprocal are equal, The number is :

Discuss
Solution:
Let the number be x
Then,
23x=25216×1x⇔x2=25216×32⇔x2=25144⇔x=25144⇔x=512
25. The difference between two integers is 5. Their product is 500. Find the numbers.

Discuss
Solution:
Let the integers be x and (x + 5)
Then,
⇔x(x+5)=500⇔x2+5x−500=0⇔(x+25)(x−20)=0⇔x=20
So, the numbers are 20 and 25
26. What is the sum of two consecutive even numbers, the difference of whose squares is 84 ?

Discuss
Solution:
Let the numbers be x and x + 2
Then,
(x + 2)2 - x2 = 84
⇔ 4x + 4 = 84
⇔ 4x = 80
⇔ x = 20
∴ Required sum
= x + (x + 2)
= 2x + 2
= 42
27. A certain number of two digits is three times the sum of its digits and if 45 be added to it, the digits are reversed. The number is ?

Discuss
Solution:
Let the ten's digit be x and unit's digit be y
Then,
⇒10x+y=3(x+y)⇒7x−2y=0.....(i)10x+y+45=10y+x⇒y−x=5.....(ii)
Solving (i) and (ii), we get :
x = 2 and y = 7
∴ Required number = 27
28. If the product of two numbers is 5 and one of the number is 32, then sum of two numbers is :

Discuss
Solution:
Let two numbers be a and b
Given ab = 5 and a = 32
⇒b=5a⇒b=532⇒b=5×23⇒b=103
Required sum of :
⇒ a + b = 32 + 103
L.C.M. of 2 and 3 = 6
⇒ a + b = 9+206
⇒ a + b = 296
⇒ a + b = 456
29. If a number is added to two-fifths of itself, the value so obtained is 455. What is the number ?

Discuss
Solution:
Let the number be x
Then,
⇔x+25x=455⇔75x=455⇔x=(455×57)⇔x=325
30. Find the whole number which when increased by 20 is equal to 69 times the reciprocal of the number.

Discuss
Solution:
Let the required number be x
Then,
⇔x+20=69x⇔x2+20x−69=0⇔x2+23x−3x−69=0⇔x(x+23)−3(x+23)=0⇔(x+23)(x−3)=0⇔x=3[∵x≠−23]

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