AP State Syllabus AP Board 8th Class Maths Solutions Chapter 2 Linear Equations in One Variable Ex 2.5 Textbook Questions and Answers.
AP State Syllabus 8th Class Maths Solutions 2nd Lesson Linear Equations in One Variable Exercise 2.5
Question 1.
Solve the following equations.
i) [latex]\frac{n}{5}-\frac{5}{7}=\frac{2}{3}[/latex]
Solution:
ii) [latex]\frac{x}{3}-\frac{x}{4}=14[/latex]
⇒ [latex]\frac{4 x-3 x}{12}[/latex] = 14
⇒ [latex]\frac{x}{12}[/latex] = 14
⇒ x = 12 × 14 = 168
∴ x = 168
iii) [latex]\frac{z}{2}+\frac{z}{3}-\frac{z}{6}=8[/latex]
iv) [latex]\frac{2 p}{3}-\frac{p}{5}=11 \frac{2}{3}[/latex]
v) [latex]9 \frac{1}{4}=y-1 \frac{1}{3}[/latex]
vi) [latex]\frac{x}{2}-\frac{4}{5}+\frac{x}{5}+\frac{3 x}{10}=\frac{1}{5}[/latex]
vii) [latex]\frac{x}{2}-\frac{1}{4}=\frac{x}{3}+\frac{1}{2}[/latex]
viii) [latex]\frac{2 x-3}{3 x+2}=\frac{-2}{3}[/latex]
⇒ 3(2x – 3) = – 2(3x + 2)
⇒ 6x – 9 = -6x – 4
⇒ 6x + 6x = -4 + 9
⇒ 12x = 5
∴ x = [latex]\frac{5}{12}[/latex]
ix) [latex]\frac{8 p-5}{7 p+1}=\frac{-2}{4}[/latex]
Solution:
⇒ [latex]\frac{8 p-5}{7 p+1}=\frac{-2}{4}[/latex]
⇒ 2(8p – 5) = – (7p + 1)
⇒ 16p – 10 = – 7p – 1
⇒ 16p + 7p = – 1 + 10
⇒ 23p = 9
∴ x = [latex]\frac{9}{23}[/latex]
x) [latex]\frac{7 y+2}{5}=\frac{6 y-5}{11}[/latex]
⇒ 11 (7y + 2) = 5 (6y-5)
⇒ 77y + 22 = 30y – 25
⇒ 77y – 30y = – 25 – 22
⇒ 47y = – 47
∴ y = [latex]\frac{-47}{47}[/latex]
∴ y = -1
xi) [latex]\frac{x+5}{6}-\frac{x+1}{9}=\frac{x+3}{4}[/latex]
⇒ 4(x + 13) = 18 (x + 3)
⇒ 4x + 52 = 18x + 54
⇒ 4x – 18x = 54-52
⇒ – 14x = 2
⇒ x = [latex]\frac{2}{-1}[/latex] = [latex]\frac{-1}{7}[/latex]
∴ x = [latex]\frac{-1}{7}[/latex]
xii) [latex]\frac{3 t+1}{16}-\frac{2 t-3}{7}=\frac{t+3}{8}+\frac{3 t-1}{14}[/latex]
Solution:
⇒ -11t + 55 = 2(19t + 17) = 38t + 34
⇒ -11t – 38t = 34 – 55
⇒ -49t = – 21
⇒ [latex]\frac{-21}{-49}[/latex] = [latex]\frac{3}{7}[/latex]
∴ t = [latex]\frac{3}{7}[/latex]
Question 2.
What number is that of which the third part exceeds the fifth part by 4?
Solution:
Let the number be ‘x’ say.
[latex]\frac{1}{3}[/latex] rd of a number = [latex]\frac{1}{3}[/latex] x x = [latex]\frac{x}{3}[/latex]
[latex]\frac{3}{7}[/latex] th of a number = [latex]\frac{1}{5}[/latex] x x = [latex]\frac{x}{5}[/latex]
According to the sum
∴ The required number is 30.
Question 3.
The difference between two positive integers is 36. The quotient when one integer is
divided by other is 4. Find the integers.
(Hint: If one number is ‘X’, then the other number is ‘x – 36’)
Solution:
Let the two positive numbers be x, (x – 36) say.
If one number is divided by second tten the quotient is 4.
∴ [latex]\frac{x}{x-36}=4[/latex]
⇒ x = 4(x – 36) = 4x – 144
⇒ 4x – x = 144
3x = 144
x = 48
∴ x – 36 = 48 – 36 = 12
∴ The required two positive intgers are 48, 12.
Question 4.
The numerator of a fraction is 4 less than the denominator. If 1 is added to both its
numerator and denominator, it becomes 1/2 . Find the fraction.
Solution:
Let the denominator of a fractin be x.
The numerator of a fraction is 4 less than the denominator.
∴ The numerator = x – 4
∴ Fraction [latex]\frac{x-4}{x}[/latex]
If ‘1’ is added to both, its numerator and denominator, it becomes [latex]\frac{1}{2}[/latex]
∴ [latex]\frac{1+x-4}{1+x}=\frac{1}{2}[/latex]
2 + 2x – 8 = 1 + x
2x – x = 1 + 6 = 7
x = 7
∴ The denominator = 7
The numerator = 7 – 4 = 3
∴ Fraction = [latex]\frac{3}{7}[/latex]
Question 5.
Find three consecutive numbers such that if they are divided by 10, 17, and 26 respectively,
the sum of their quotients will be 10.
(Hint: Let the consecutive numbers = x, x+ 1, x+ 2, then [latex]\frac{x}{10}+\frac{x+1}{17}+\frac{x+2}{26}=10[/latex])
Solution:
Let the three consecutive numbers be assume that x, (x + 1), (x + 2) respectively.
Given that x, (x + 1), (x + 2) are divided by 10, 17, 26 respectively, the sum of the quotients is 10. Then
⇒ [latex]\frac{x}{10}+\frac{x+1}{17}+\frac{x+2}{26}=10[/latex]
⇒ [latex]\frac{x \times 221+130(x+1)+85(x+2)}{2210}=10[/latex]
⇒ 221x + 130x + 85x + 130 + 170 = 22,100
⇒ 436x + 300 = 22,100
⇒ 436x = 22,100 – 300
⇒ 436x = 21,800
⇒ [latex]\frac{21800}{436}[/latex]
∴ x = 50
∴ The required three consecutive num-bers are x = 50
x + 1 =50+ 1 = 51
x + 2 = 50 + 2 = 52
Question 6.
In class of 40 pupils the number of girls is three-fifths of the number of boys. Find the
number of boys in the class.
Solution:
Let the number of boys = x say.
Total number of students = 40
Number of girls = [latex]\frac{3}{5}[/latex] × x = [latex]\frac{3x}{5}[/latex]
According to the sum 3x
∴ x = 25
∴ Number of boys in the class room = 25
Question 7.
After 15 years , Mary’s age will be four times of her present age. Find her present age.
Solution:
Let the present age of Mary = x years say.
After 15 years Mary’s age = (x + 15) years
According to the sum
(x + 15) = 4 x x
⇒ x + 15 = 4x
⇒ 4x – x =15
⇒ 3x = 15
⇒ x = 5
∴ The present age of Mary = 5 years.
Question 8.
Aravind has a kiddy bank. It is full of one-rupee and fifty paise coins. It contains 3 times
as many fifty paise coins as one rupee coins. The total amount of the money in the bank is
₹ 35. How many coins of each kind are there in the bank?
Solution:
Number of 1 rupee coins = x say.
Number of 50 – paise coins = 3 x x = 3x
The value of total coins = [latex]\frac{3x}{2}[/latex] + x
[∵50 paisa coins of 3x = ₹[latex]\frac{3x}{2}[/latex]
According to the sum
⇒ [latex]\frac{3x}{2}[/latex] + x = 35
⇒ [latex]\frac{3 x+2 x}{2}[/latex] = 35
⇒ 5x = 2 × 35
⇒ x = 2 × [latex]\frac{35}{5}[/latex]
∴ x = 14
∴ Number of 1 rupee coins = 14
Number of 50 paisa coins = 3 × x = 3 × 14 = 42
Question 9.
A and B together can finish a piece of work in 12 days. If ‘A’ alone can finish the same work in 20days , in how many days B alone can finish it?
Solution:
A, B can do a piece of work in 12 days.
(A + B)’s 1 day work = [latex]\frac{1}{12}[/latex] th part.
A can complete the same work in 20 days.
Then his one day work = [latex]\frac{1}{20}[/latex]
B’s one day work = (A+B)’s 1 day work – A’s 1 day work
[latex]=\frac{1}{12}-\frac{1}{20}=\frac{5-3}{60}=\frac{2}{60}=\frac{1}{30}[/latex]
∴ Number of days to take B to com¬plete the whole work = 30.
Question 10.
If a train runs at 40 kmph it reaches its destination late by 11 minutes . But if it runs at 50 kmph it is late by 5 minutes only. Find the distance to be covered by the train.
Solution:
Let the distance to be reached = x km say. Time taken to travel ‘x’ km with speed x
40 km/hr = [latex]\frac{x}{40}[/latex] hr.
Time taken to travel ‘x’ km with speed 50 km/hr = [latex]\frac{x}{50}[/latex] hr.
According to the sum the difference between the times
∴ The required distance to be trav¬elled by a train = 20 kms‘.
Question 11.
One fourth of a herd of deer has gone to the forest. One third of the total number is
grazing in a field and remaining 15 are drinking water on the bank of a river. Find the total
number of deer.
Solution:
Number of deer = x say.
Number of deer has gone to the forest
= [latex]\frac{1}{4}[/latex] × x = [latex]\frac{x}{4}[/latex]
Number of deer grazing in the field
= [latex]\frac{1}{3}[/latex] × x =[latex]\frac{x}{3}[/latex]
Number of remaining deer =15
According to the sum
∴ x = 36
∴ The total number of deer = 36
Question 12.
By selling a radio for ₹903, a shop keeper gains 5%. Find the cost price of the radio.
Solution:
The selling price of a radio = ₹ 903
Profit % = 5%
C.P = ?
C.P = [latex]\frac{\mathrm{S.P} \times 100}{(100+\mathrm{g})}[/latex]
= [latex]\frac{903 \times 100}{(100+5)}[/latex]
= [latex]\frac{903\times 100}{105}[/latex]
C.P. = 8.6 × 100 = 860
∴ The cost price of the radio = ₹ 860
Question 13.
Sekhar gives a quarter of his sweets to Renu and then gives 5 sweets to Raji. He has 7 sweets left. How many did he have to start with?
Solution:
Number of sweets with Sekhar = x say.
Number of sweets given to Renu
= [latex]\frac{1}{4}[/latex] × x = [latex]\frac{x}{4}[/latex]
Number of sweets given to Raji = 5
Till he has 7 sweets left.
x – ( [latex]\frac{x}{4}[/latex] + 5) = 7
⇒ x – [latex]\frac{x}{4}[/latex] – 5 = 7
⇒ x – [latex]\frac{x}{4}[/latex] = 7 + 5 = 12
⇒ [latex]\frac{4 x-x}{4}[/latex] = 12
⇒ [latex]\frac{3x}{4}[/latex] = 12
⇒ x = 12 × [latex]\frac{4}{3}[/latex] = 16
∴ x = 4 × 4 = 16
∴ Number of sweets with Sekhar at the beginning = 16