Practice question on Algebra

ratio and proportion

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Question no. 1

The product of the ages of Ankit and Sachin is 240. If twice the age of Sachin is more than Ankit’s age by 4 years, what is Sachin’s age?

looks_one 24

looks_two 27

looks_3 40

looks_4 None of these

Option looks_4 None of these

Solution:
let Ankit age is x , Sachin is y
According to the Question
xy = 240
2y = x+4 ⇒ x =2y - 4
xy = 240 ⇒ (2y - 4)(y) = 240
2y²-4y - 240 = 0 ⇒ y² -2y -120 = 0
y² -12y +10y -120 =0
(y -12)(y + 10) = 0
y = 12 , -10
y =12 (age can't be negative)

Question no. 2

Two numbers are in the ratio 3:5. If 9 is subtracted from each of the numbers, their ratio become 12:23. Find the greater of the two numbers.

looks_one 55

looks_two 65

looks_3 45

looks_4 None of these

Option looks_one 55

Solution :
let first number is a , second number is b
According to question
a/b = 3/5 ⇒ 5a = 3b
(a-9)/(b-9) = 12/23 ⇒ 23a - 9*23 = 12b -12*9
23a = 12b +9(23-12) ⇒ 23a = 12b + 99
23*3b/5 = 12b + 99 ⇒ 69b = 60b + 99*5
9b = 99* 5 ⇒ b = 55
a = 3b/a = 3*55/5 = 33

Question no. 3

If three numbers are added in pairs, the sums equal 10, 19 and 21. The numbers are

looks_one 4,6,10

looks_two 6,4,15

looks_3 3,5,10

looks_4 none of these

option looks_two 6,4,15

Solution :
let first number is a , second number is b , third is c
According to question
a+b = 10 (1)
b+c = 19 (2)
a + c = 21 (3)
By Equation 2 and 3 (subtract (3) from (2))
a - b = 2 (4)
now solve (1) and (4) equation(ADD (1) in (4) )
2a = 12 ⇒ a =6
put this a = 6 in (1) and (3) ⇒ b =4 , c = 15

Question no. 4

Two numbers are such that if the first be added to 5 times the second, their sum becomes 52, and if the second be added to 8 times the first, their sum becomes 65. The two numbers are:

looks_one 9,7

looks_two 3,7

looks_3 7,9

looks_4 None of these

Option looks_3 7,9

solution :
let first number is a , second number is b
According to Question
a+ 5b = 52 ⇒ a =52 - 5b
8a+b = 65 ⇒ 8(52 - 5b) + b = 65
416 -40b + b = 65
-39b = -351 ⇒ b = 9
a= 52- 5b = 52 -5*9 = 7

Question no. 5

A number is doubled and 9 is added. If the resultant is tripled, it becomes 75. What is that number?

looks_one 3.5

looks_two 6

looks_3 8

looks_4 None of these

Option looks_3 8

Solution :
let Number is x
According to question
3(2x+ 9) = 75
2x+9 = 25 ⇒ x = 8

Question no. 6

In a two-digit number, if it is known that its unit’s digit exceeds its ten’s digit by 2 and that the product of the given number and the sum of its digits is equal to 144, then the number is:

looks_one 24

looks_two 26

looks_3 42

looks_4 46

Option looks_one 24

Solution :
let Unit place digit is x , ten place digit is y
Number = 10y + x
According to Question
x = y+ 2
(10y +x)(x+y)=144 ⇒ (10y + y + 2)(y+2 +y) = 144
(11y + 2)(2y + 2) = 144 ⇒ (11y + 2)(y+1)= 72
11y² + 13y - 70 = 0
11² + 35y - 22y -70 = 0
(11y + 35)(y-2) = 0 ⇒ y = 2
x =y+ 2 = 2 + 2 = 4
Number = 10y + x = 10*2 + 4 = 24

Question no. 7

If the sum of two numbers is 42 and their product in 437, then find the absolute difference between the numbers.

looks_one 4

looks_two 7

looks_3 9

looks_4 none of these

Option looks_one 4

Solution :
let first number is x , second number is y
according to question
x+ y = 42
xy = 437 ⇒ y = 437/x
x + 437/x = 42 ⇒ x² -42x + 437 = 0
x² -23x+19x + 437 = 0 ⇒ (x-23)(x+ 19) = 0 ⇒ x = 23 , -19
their corresponding value y(23) = 19 , y(-19) = 61
x = -19 , y = 61 can't be the solution
absolute differnce = |x- y| = |23-19| = 4

Question no. 8

In a two-digit number, the digit in the unit’s place is four times the digit in ten’s place and sum of the digits is equal to 10. What is the number?

looks_one 14

looks_two 41

looks_3 82

looks_4 None of these

Option looks_4 None of these

Solution :
let unit digit is x , ten's digit is y
According to Question
x= 4y , x + y = 10
4y + y = 10
5y = 10 ⇒ y = 2
x =4y ⇒ x = 8
Number = 10y + x = 28

Question no. 9

If the sum of a number and its square is 182, what is the number?

looks_one 15

looks_two 26

looks_3 28

looks_4 None of these

Option looks_4 None of these

Solution :
let number is x .
According to Question
x + x² = 182
x² + x -182 =0
x² + 14x - 13x -182 =0
(x+14)(x-13) = 0 ⇒ x= 13 , -14

Question no. 10

If (x–3)(2x+1)=0, then the possible values of 2x+1 are:

looks_one 0 only

looks_two 0 and 3

looks_3 -1/2 and 3

looks_4 0 and 7

Option looks_4 0 and 7

Solution:
(x–3)(2x+1)=0 ⇒ x = 3 , -1/2
there are two values of x
For x = 3 :
(2x+1) = (2*3 + 1 ) = 7
For x = -1/2
(2x+ 1) = (2*(1/2) + 1) = 0

Question no. 11

Ratio of the two numbers is 3:4 and the sum of these two numbers is 420. The sum of their squares is:

looks_one 9000

looks_two 90000

looks_3 9*10⁵

looks_4 None of these

Option looks_two 90000

solution :
first number = a
second number = b
According to question
a/b = 3/4 ⇒ 4a = 3b
a+b = 420
3b/4 + b = 420
7b = 420*4
b = 240
a = 3*240/4 = 180
a² + b² = 180² + 240² = 90000

Question no. 12

Q is as much younger than Ras he is older than T. If the sum of the ages of R and T is 50 years, what is definitely the difference between R and Q’s age?

looks_one 1

looks_two 2

looks_3 25

looks_4 data insufficient

Option looks_4 data insufficient

Solution :
Age of Q < Age of R
Age of Q > Age of T
R +T = 50
R + T < Q+R < 2R
there is no way to calculate the exact difference

Question no. 13

Anuradha’s father was 38 years of age when she was born while her mother was 36 years old when her brother four years younger to her was born. What is the difference between the ages of her parents?

looks_one 2

looks_two 6

looks_3 4

looks_4 None of these

Option looks_two 6

Solution :
Anuradha's father age when she born = 38 years
Her mother age when her age is 4 year(when her brother is born , he is 4 years younger than her) = 36 years
Her mother age when she born = 36 -4 = 32
difference between their parents age = 38 -32 = 6 years

Question no. 14

The total age of A and B is 12 years more than the total age of B and C. C is how many years younger than A?

looks_one 12

looks_two 24

looks_3 C is elder than A

looks_4 can't be determined

Option looks_one 12

Solution :
A + B = 12 + B + C
A = 12 + C
C = A - 12
C is younger than A by 12 years

Question no. 15

Tanya’s grandfather was 8 times older to her 16 years ago. He would be 3 times of her age 8 years from now. Eightyears ago, what was the ratio of Tanya’s age to that of her grandfather?

looks_one 1:2

looks_two 1:5

looks_3 3:8

looks_4 None of these

Option looks_4None of these

Solution :
16 years ago, let T=x years and G = 8x years.
After 8 years from now, T= (x+16+8)years and G=(8x+16+8)years.
8x+24=3(x+24) => 5x=48
8 years ago , (T+8)/(G+8) = (48/5 + 8)/(8*48/5 + 8) = 11/53