SOLVE LINEAR EQUATIONS WORD PROBLEMS

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. 

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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