Solve the following equations.
1. (8x – 3)/3x = 2
Solution:
Given, (8x – 3)/3x = 2
Now,
8x/3x – 3/3x = 2
8/3 – 1/x = 2
8/3 – 2 = 1/x
(8 – 6)/3 = 1/x
2/3 = 1/x
x = 3/2
2. 9x/(7 – 6x) = 15
Solution:
Given, 9x/(7 – 6x) = 15
Now,
9x = 15(7 – 6x)
9x = 105 – 90x
9x + 90x = 105
99x = 105
x = 105/99 = 35/33
3. z/(z + 15) = 4/9
Solution:
Given, z/(z + 15) = 4/9
Now,
z = 4/9 (z + 15)
9z = 4(z + 15)
9z = 4z + 60
9z – 4z = 60
5z = 60
z = 12
4. (3y + 4)/(2 – 6y) = -2/5
Solution:
Given, (3y + 4)/(2 – 6y) = -2/5
Now,
3y + 4 = -2/5 (2 – 6y)
5(3y + 4) = -2(2 – 6y)
15y + 20 = -4 + 12y
15y – 12y = -4 – 20
3y = -24
y = -8
5. (7y + 4)/(y + 2) = -4/3
Solution:
Given, (7y + 4)/(y + 2) = -4/3
Now,
7y + 4 = -4/3 (y + 2)
3(7y + 4) = -4(y + 2)
21y + 12 = -4y – 8
21y + 4y = -8 – 12
25y = -20
y = -20/25 = -4/5
6. The ages of Hari and Harry are in the ratio 5:7. Four years from now the ratio of their ages will be 3:4. Find their present ages.
Solution:
Let us assume the age of Hari be 5x and Harry be 7x.
After 4 years,
Age of Hari = 5x + 4
Age of Harry = 7x + 4
According to the question,
(5x + 4)/(7x + 4) = ¾
⇒ 4(5x + 4) = 3(7x + 4)
⇒ 20x + 16 = 21x + 12
⇒ 21x – 20x = 16 – 12
⇒ x = 4
Hence,
Hari age = 5x = 5 × 4 = 20 years
Harry age = 7x = 7 × 4 = 28 years
7. The denominator of a rational number is greater than its numerator by 8. If the numerator is increased by 17 and the denominator is decreased by 1, the number obtained is 3/2. Find the rational number.
Solution:
Let us assume the numerator be x then denominator will be (x + 8)
According to the question,
(x + 17)/(x + 8 – 1) = 3/2
⇒ (x + 17)/(x + 7) = 3/2
⇒ 2(x + 17) = 3(x + 7)
⇒ 2x + 34 = 3x + 21
⇒ 34 – 21 = 3x – 2x
⇒ 13 = x
Hence, the rational number is x/(x + 8) = 13/21
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