In this section, you will learn how to solve word problems using linear equations.
To solve word problems using linear equations, we have follow the steps given below.
If you find it difficult to convert the given statement into equation or if you wish to know the stuff on translating word problems into equations in detail,
Example 1 :
The sum of two numbers is 150. If 70 percent of one number is equal to 80 percent of the other, find the two numbers.
Solution :
Let x and y be the two numbers.
Given : The sum of two numbers is 150.
Then, we have
x + y = 150 -----(1)
Given : 70 percent of one number is equal to 80 percent of the other.
Then, we have
70% of x = 80% of y
0.7x = 0.8y
To get rid of decimal, multiply each side by 10.
7x = 8y
Divide each side by 7.
x = 8y / 7 -----(2)
Substitute 8y/7 for x in (1).
(1)-----> 8y/7 + y = 150
8y/7 + 7y/7 = 150
(8y + 7y) / 7 = 150
15y / 7 = 150
Multiply each
side by 7/15.
7 / 15 ⋅ (15y / 7) = (150) ⋅ 7 / 15
y = 70
Substitute 70 for y in (2).
(2)-----> x = 8(70) / 7
x = 80
So, the two numbers are 80 and 70.
Example 2 :
P and Q each have certain number of apples. P says to Q, “if you give me 10 of your apples, I will have twice the number of apples left with you”. Q replies, “if you give me 10 of your apples, I will the same number of apples as left with you”. Find the number of apples with P and Q separately.
Solution :
Let p and q be the number of apples that P and Q have originally.
Given : P says to Q, “if you give me 10 of your apples, I will have twice the number of apples left with you”.
Then, we have
p + 10 = 2(q - 10)
p + 10 = 2q - 20
Subtract 2q and 10 from each side.
p - 2q = - 30 -----(1)
Given : Q says to P, “if you give me 10 of your apples, I will the same number of apples as left with you.”
Then, we have
q + 10 = p - 10
Subtract p and 10 from each side.
- p + q = - 20 -----(2)
In order to eliminate p in (1) and (2), add (1) and (2).
- q = - 50
Multiply each side by (-1).
q = 50
Substitute 50 for q in (1).
(1)-----> p - 2(50) = - 30
p - 100 = - 30
Add 100 to each side.
p = 70
So, P and Q originally have 70 and 50 mangoes respectively.
Example 3 :
A and B are friends and their ages differ by 2 years. A's father D is twice as old as A and B is twice as old as his sister C. The age of D and C differ by 40 years. Find the ages of A and B.
Solution :
Let a, b, c and d be the ages of A, B, C and D respectively.
Given : The ages of A and B differ by 2 years.
a - b = 2 -----(1)
Given : A's father D is twice as old as A.
d = 2a -----(2)
Given : B is twice as old as his sister C.
b = 2c -----(3)
Given : The age of D and C differ by 40 years.
d - c = 40 -----(4)
From (3), substitute 2c for b in (1).
a - 2c = 2 -----(5)
From (2), substitute 2a for d in (4).
2a - c = 40 -----(6)
In order to eliminate a in (5) and (6), subtract (6) from 2 times of (5).
- 3c = - 36
Divide each side by (-3).
c = 12
Substitute 12 for c in (3).
(3)-----> b = 2(12)
b = 24
Substitute 24 for b in (1).
(1)-----> a - 24 = 2
Add 24 to each side.
a = 26
So, the ages of A and B are 26 years and 24 years respectively.
Example 4 :
A takes 3 hours more than B to walk 30 miles. But if A doubles his pace, he is ahead of B by one and half hrs. Find their speed of walking.
Solution :
Let a and b be the speeds of A and B respectively.
Formula to find time when distance and speed are known :
Time = Distance / Speed
Time taken by A and B to cover 30 miles :
Time_{A} = 30 / a
Time_{B} = 30 / b
Given : A takes 3 hours more than B to walk 30 miles.
Then, we have
Time_{A} - Time_{B} = 3
30/a - 30/b = 3
Let 1/a = x and 1/b = y.
30x - 30y = 3
Divide each side by 3.
10x - 10y = 1 -----(1)
Given : If A doubles his pace, he is ahead of B by one and half hrs.
If A doubles his pace, the speed of A will be 2a and the time taken by A to cover 30 miles will be
Time_{A }= 30 / 2a
Then, we have
Time_{B} - Time_{A} = 1 1/2
30/b - 30/2a = 3/2
30/b - 15/a = 3/2
30y - 15x = 3/2
Multiply each side by 2.
60y - 30x = 3
Divide each side by 3.
20y - 10x = 1
-10x + 20y = 1 -----(2)
In order to eliminate x in (1) and (2), add (1) and (2).
10y = 2
Divide each side by 10.
y = 1 / 5
Substitute 1/5 for y in (1).
(1)-----> 10x - 10(1/5) = 1
10x - 2 = 1
Add 2 to each side.
10x = 3
Divide each side by 10.
x = 3 / 10
Find the values of a and b from the values of x and y.
x = 3 / 10 1 / a = 3 / 10 a = 10 / 3 a ≈ 3.33 |
y = 1 / 5 1 / b = 1 / 5 b = 5 |
So, the speed of A is about 3.33 miles per hour and the speed of B is 5 miles per hour.
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