8th maths :8th Linear Equations of one or two Variables Solved questions

8th Mathematics

JSUNIL TUTORIAL,SAMASTIPUR

 What is an equation?

A statement which contains the equal sign = is known as an equation. eg; 2x -4x=2x , 45=0

1. The sum of the ages of anup and his father is 100. When anup is as old as his father now, he will be 5 times as old as his son anuj is now. Anuj will be eight years older than anup is now, when anup is as old as his father. What are their ages now?

Let the present ages (in years) of Anup’s father, Anup and Anuj be x, (100 – x) and respectively.

Difference between Anup’s father and Anup’s ages (in years) = x – (100 – x) = x – 100 + x = 2x – 100

2x – 100 years later, Anup will be x years old and 5 times old as Anuj is now.

X = 5y  

y = x/5 -----------(i)

When Anup is x yr old,then Anuj will be y +(2x-100)yrs old and 8yrs older than Anup is now

Therefore y + (2x-100) = (100 – x ) + 8      -------------(ii)

On putting y = x/5 from (i) we get

X = 65 yrs

So Anup’s fathers age = 65 yrs,        Anup’s age = 100-65 = 35 yrs   ,         Anuj ages = x/5 = 65/5 = 13 yrs

2.  I am currently 5 times as old as my son. In 6 years of time I will be three times as old he will be then. What are our ages now?

Let the Age Of Me Be 5x And Sons Be x.     

After 6 Years  5x+6 And x+6

It is Given That I Would Be Three Times The Age Of My Son

5x+6 = 3(x+6)  Þ  5x+6 = 3x+18 Þ 5x-3x = 18-6Þ 2x = 12Þx = 6

So My Age = 6x5 = 30     

Answer:  My Age- 30 Years, Sons Age 6 Years

3. If the length and breadth of a rectangular field are in the ration 6:4. Find the length and breadth if the cost of fencing the rectangular field ar the rate of Rs 80 per meter is Rs 16000.

Given, the ratio of length and breadth of rectangular field = 6:4 = 6x/4x

∴ Perimeter of field = 2(l+b) = 2(6x+4x) = 20x

It is given that cost of fencing the field = Rs 16000      ⇒ 20x × 80 = 16000   ⇒ x = 10

∴ Length of rectangular field = 6x = 6 × 10 = 60 m  and breadth of rectangular field = 4x = 4 × 10 = 40 m

4.The present age of Prabha 's mother is three times the present age of Prabha. After 5 years, there ages will add to 66 years. Find there present ages.

Let the present age of prabha = x years

∴ present age of prabha's mother = 3x years

After 5 years, age of prabha = (x + 5) years

After 5 years, age of prabha's mother = 3x + 5 years

According to the given condition,

3x+ 5 + x+ 5 = 66     ⇒ 4x = 56          ⇒ x = 14

5. Ramesh has three rimes as many two - rupee coins as he has 5 rupee coins. If he has in all a sum of Rs 77, then how many coins each denomination does he have?

Let the number of 5 rupee coins = x      The number of 2 rupee coins = 3x

According to the given condition, 

2×3x + 5× x = 77     ⇒ 11x = 77    ⇒ x = 7    Hence, Ramesh has 7 five rupee coins and 21 two rupee coins.

6.  One of the two digits of tow digit number is three times the other. if we interchange the digit and add the resulting number to the original number we get 132. Find the number.

Let the two digit number is 10 x + y

Now, y = 3x

now, when we interchange the digits, we get, 10y + x

According to the question, 10 x + y + 10 x + y = 132

Put:  y = 3x ,  10x + 3x + 30x + x = 132

=> 44x = 132 Þ x = 132/44 = 3

Hence first digit = x = 3  and second digit = y = 3x = 3 X 3 = 9  Hence the number  = 39

7.  'A ' is twice as old as 'B '. Five years ago A 's age was 3 times B 's age. Find there present ages.

let "A" be 'x ' years old.  and "B" be 'y ' years old.

now, x = 2y .................. ........ .(1)

"A" 's age 5 years ago = x - 5

"B" 's age 5 years ago = y-5

now, it is given

(x- 5) = 3(y - 5)

Substituting (1) we get, 2y - 5 = 3(y - 5)

=> 2y - 5 = 3y – 15  => 3y – 2y = 10 =>y = 10 a = 2b = 20

Therefore their presents ages are 15 and 5 respectively.

Now you can solve:

8.  A digit of a two - digit number differs by 3. If the digits are interchanged and the resulting number is added to the original number, we get 121. Fond the original number.

9.  The sum of the digits of a two - digit number is 13. If the digits are interchanged and the resulting number is added to the original number, then we get 143. What is the original number?

10. five years ago amrita’s age was thrice as old as his brother. now the difference of their ages is 16. What are their present ages?

Let the Amrita age is x yrs then his brothers age will be (x -16)yrs

5 Years Ago their ages will be  x-5 5nd x-21

It is given that,

3(x-21) = x-5             

3x-63 = x-5        

3x-x = -5+63        

2x = 58   

x = 29 

So ,Her Brothers age 29-16 = 13

11. The sum of the digits of a two digit number is 12.if the new number formed by reversing the digits is greater than the original number by 18, find the original number.

Let the two digit number is 10 x + y

A/Q, x + y =12 so, x = 12 – y

Also,

A/Q, The new number formed by reversing the digits = the original number + 18

10 y + x   = 10 x + y + 18  Putting, x = 12- y

10 y + 12 - y = 10(12 - y) + y + 18

9y + 12 = 120 – 10 y + y +18  Þ  9y + 12 = 138 – 9 y

Þ  9y + 9 y = 138 – 12     Þ     18y = 126  Þ y = 126/18 = 7  Þ x = 12-7 = 5

Original number = 10x 5 + 7 = 57

12. Five years ago, John was twice as old as his brother Jim. Three years from now, the sum of their ages will be 31. How old is Jim now?

Let the present age of Jim = x years   and present age of John = y years

According to the given condition ,  (y – 5) = 2 (x – 5)  Þ ⇒ y – 5 = 2x – 10 Þ ⇒ y = 2x – 5 ...... (1)

Again, according to the given condition

(y + 3) + (x + 3) = 31    ⇒ x + y = 25     ⇒ x + 2x – 5 = 25 [ Using (1) ]   ⇒ 3x = 30 ⇒ x = 10

Hence, the present age of Jim is 10 years

13. A father is three times as old as his son is now, but 15 years from now he will be only twice as old as his son at that same time. How old is the son now?

Let the present age of son = x years     ∴ present age of father = 3x years

so, after 15 years age of son = (x + 15) years      after 15 years age of father = (3x + 15) years

According to the given condition,

(3x + 15) = 2(x + 15)   ⇒ 3x + 15 = 2x + 30 ⇒ x = 15

Hence, the present age of son is 15 years.

14. Kanwar is three years older than Amina. Six years ago, Kanwar’s age was four times Amina’s age. Find their ages.

Let the current age of Amina = x years  So, the current age of Kanwar = (x + 3) years

According to the given condition,

(x + 3 – 6) = 4 (x – 6)    ⇒ x – 3 = 4x – 24 ⇒ – 3x = – 21 ⇒ x = 7

Hence, the current age of Amina = 7 years  and the current age of Kanwar = 10 years

15. Sanjana’s mother gave her rs.245 for buying cards. if she got some 10 rupee cards,2/3 as many 5 rupee cards, and 1/5 as many 15 rupee cards ,how many cards of each kind did she bought.

Let the number of 10 rupee cards be x.

Then,  number of 5 rupee cards  = 2x/3  and number of 15 rupee cards = x/5

Now, total amount given to Sanjana = 245

10x +5(2x/3) + 15(x/5) =245 Þ x = 15

Hence, number of 10 rupee cards = 15  number of 5 rupee cards = 10 number of 15 rupee cards  =3

16. if a worker is engaged for 20 days on a condition that he will be paid rs.60 for each dayhe worked and will be fined rs.5 for the day he is absent.In total he recieved rs.745.so tell how many days was he absent

Let the number of days the worker was absent be x

Then the number of days the worker worked = 20 – x

Now, Money paid for working = Rs 60 × (20 – x) = Rs (1200 – 60x)  and  money fined for absent = Rs 5x

Thus money received = money paid – money fined

⇒ 745 = 1200 – 60x – 5x       ⇒ 65x = 1200 – 745 = 455 ⇒ x =455/65=7

Hence, the worker was absent for 7 days

17. A motorboat covers a certain distance downstream in a river in 5 hours.it covers the same distance upstream in 5 hours and a half. the speed of the stream is 1.5km/hr. What is the speed of motorboat in still water?

speed of the stream is 1.5km/hr

distance covered by boat downstream in a river in 5 hours

it covers the same distance upstream in 5.5 hours

let the speed of the motorboat in still water be x

speed of the motorboat down stream = (x+ 1.5)km/h

speed of the motorboat upstream = (x- 1.5)km/h

since distance covered by the boat is same in both cases.

therefore , 5(x+ 1.5) = 5.5(x- 1.5) [ distance = speed* time]

=> 5x + 7.5 = 5.5x - 8.25

=> 5.5x- 5x = 7.5+ 8.25

=> 0.5x = 15.75

=>x = 31.5 km/h

Therefore the speed of motorboat in still water is 31.5km/h

18. Of the three angles of a triangle , the second one is one third of the first and the third angle is 26 degrees more than the first angle. Find all the three angles of the triangle

Let the first angles of the triangle be x. then, second angle =1/3 of first angle x /3

and third angle = first angle + 26 = x + 26 

Now we know that sum of angles of a triangle is 180°

X + x/3  + x + 26 =180 Þ x = 66⁰

 Hence the required angles are = 66°, 22°, 92°

19. The sum of two twin prime numbers is 60. Find the prime numbers?

Let the one prime number be x and therefore, the other twin prime number must be (x + 2).

According to the question,

Sum of two twin prime number = 60

⇒ x + (x + 2) = 60  Þ 2x + 2 = 60 Þ  29     

∴ One prime number = x = 29      Other prime number = (x + 2) = (29 + 2) = 31

20. Sunita is as twice as old as Ashima .if six years is subtracted from Ashima 's age and 4 years added to Sunitas age, then Sunita will be four times Ashima’ s age . How old were they two years ago ?

Let the Ashima 's age be x then Sunita 's age be 2x

A/q, 4( x - 6) = 2x + 4 Þ     4x - 24 = 2x + 4    Þ  4x - 2x = 4 + 24 Þ x =14

Ashima 's present age is 14 and Sunit 's present age is 14 x 2=28.

Now Asthma 2 years ago = 14 - 2 = 12 years  Sunita = 28 - 2 = 26 years