To combine rational numbers, first we have to check whether the denominators are same or not.
(i) If the denominators are same, we can put one denominator and combine the numerators.
(ii) If the denominators are not same, we will take least common multiple and combine the numerators.
(iii) Keep the unknowns in one side of the equal sign.
(iv) Do the possible simplification and get the value of unknown.
If we see only one fraction on both sides of the equal sign, by doing cross multiplication we can start the problem.
Problem 1 :
(x + 1) / 4 = (x - 2) / 3
Solution :
(x + 1) / 4 = (x - 2) / 3
Doing cross multiplication, we get
3(x + 1) = 4(x - 2)
Using distributive property, we get
3x + 3 = 4x - 8
Subtract 4x on both sides.
3x - 4x + 3 = -8
-x + 3 = -8
Subtract 3 on both sides.
-x = -8 - 3
-x = -11
Divide by -1 on both sides.
x = 11
Problem 2 :
(2x - 1) / 5 = (3x + 1) / 3
Solution :
(2x - 1) / 5 = (3x + 1) / 3
On both sides of the equal sign, we have fractions. By doing cross multiplication, we get
3(2x - 1) = 5(3x + 1)
Using distributive property, we get
6x - 3 = 15x + 5
Subtracting 15x on both sides and add 3 on both sides, we get.
6x - 15x = 5 + 3
-9x = 8
x = -8/9
Problem 3 :
3x - (x - 2)/3 = 4 - (x - 1)/4
Solution :
3x - (x - 2)/3 = 4 - (x - 1)/4
(9x - x + 2) / 3= (16 - x + 1) / 4
(8x + 2) / 3 = (17 - x) / 4
Cross multiplying both sides.
4(8x + 2) = 3(17 - x)
32x + 8 = 51 - 3x
32x + 3x = 51 -8
35x = 43
x = 43/35
So, the answer is x= 43/35.
Problem 4 :
(3t + 5/4) - 1 = (4t - 3) / 5
Solution :
In LHS,LCM is 4.
(3t + 5 - 4) / 4= (4t - 3) / 5
(3t + 1) / 4 = (4t - 3) / 5
Doing cross multiplication.
5(3t + 1) = 4(4t - 3)
15t + 5 = 16t - 12
15t - 16t = -12 - 5
-t = -17
t = 17
Hence value of t is 17.
Problem 5 :
(2y - 3)/4 - (3y - 5)/2 = y + 3/4
Solution :
Find common denominator.
LCM of 4, 2 is 4.
[(2y - 3) - 2(3y - 5)] / 4= y + 3/4
By distributing
(2y - 3 - 6y + 10) / 4 = y + 3/4
Combine like terms.
(-4y + 7) / 4 = y + 3/4
Subtract y from both sides.
(-4y + 7)/4 - y = y + 3/4-y
(-4y + 7 - 4y) / 4 = 3/4
(-8y + 7) / 4 = 3/4
Multiplying by 4 on both sides, we get
-8y + 7 = 3
-8y = 3 - 7
-8y = -4
y = 1/2
So, the answer is y = 1/2.
Problem 6 :
(3t - 2) / 3 + (2t + 3) / 2 = t + 7/6
Solution :
Find common denominator.
LCM of 3, 2 is 6.
(3t - 2) / 3 + (2t + 3) / 2 = t + 7/6
Least common multiple for 2 and 3 is 6.
[2(3t - 2) + 3(2t + 3)] / 6 = t + 7/6
[(6t - 4)+(6t + 9)] / 6 = t + 7/6
(12t+5) / 6 = t + 7/6
Subtract t from both sides.
((12t+5)/6) - t = t + 7/6 - t
(12t + 5 - 6t)/6 = 7/6
(6t + 5) / 6 = 7/6
Multiplying by 6 on both sides, we get
6t + 5 = 7
6t = 7 - 5
6t = 2
t = 1/3
So, the answer is t = 1/3.
Problem 7 :
m - (m - 1)/2 = 1 - (m - 2) / 3
Solution :
[m - (m - 1)/2 = 1 - (m - 2) / 3
(2m - m + 1) / 2 = (3 - m + 2) / 3
Simplify like terms.
(m + 1) / 2 = (5 - m) / 3
Doing cross multiplication, we get
3(m + 1) = 2(5 - m)
3m + 3 = 10 - 2m
3m + 2m = 10 - 3
5m = 7
m = 7/5
So, the answer is m = 7/5.
May 21, 24 08:51 PM
May 21, 24 08:51 AM
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