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Systems of Equations Word Problems


Systems of Equations Word Problems
  • Page 1
 1.  
A fair ticket for children is $\$$1.50 and for adult is $\$$4.50. If in a day 2000 people enter and total collection is $\$$4050. Find how many children and adult enter on that day?     
a.
Adult = 1600, Children = 350
b.
Adult = 350, Children = 1650
c.
Adult = 400, Children = 1650
d.
Adult = 1700, Children = 300


Solution:

 
Let number of adult = x
and number of children = y

Total number: x + y = 2000.....(1)
Total collection: 4.5x + 1.5y = 4050.....(2)

Now we have 2 equations and 2 variables
From equation (1)
y = 2000 - x 
substitute y value in equation (2)
4.5x + 1.5(2000-x) = 4050
x = 350
substitute x value in equation
350 + y = 2000
y = 2000 - 350
y = 1650

Adult = 350
children = 1650

 


Correct answer : (2)
 2.  
If the difference between two numbers is 49 and the sum is 123. Find the numbers?
a.
86 and 35
b.
80 and 37
c.
86 and 37
d.
80 and 30


Solution:

 
Let the numbers are x and y
as given 
x + y = 123.......(1)
x - y = 49......(2)

add equation (1) and (2)
2x = 172
x = $\frac{172}{2}$
x = 86
substitute x value in equation (1)
86 + y = 123
y = 123 - 86
y = 37

The numbers are 86 and 37

 


Correct answer : (3)
 3.  
In a farmhouse total animals are 20, some of them are hens and some are cows. Total legs of these animals are 66. Find how many hens and cows are in the farmhouse?
a.
Hens = 7, cows = 13
b.
Hens = 10, cows =  10
c.
Hens = 5, cows = 15
d.
Hens = 6, cows = 14


Solution:

 
Let total hen = x
total cow = y 

By given data
x + y = 20......(1)
as we know each cow have 4 legs and each hen have 2 legs
2x + 4y = 66....(2) 
 
multiply equation (1) with -4 and add with equation (2)
-2x = -14
x = 7
substitute x value in equation (1)
7 + y = 20
y = 20 - 7
y = 13

hence, total hens = 7 
and total cows = 13    

 


Correct answer : (1)
 4.  
Maria bought 5 tee shirts in $\$$18. Each half sleeve tee shirt cost is $\$$3 and full tee sleeve shirt cost is $\$$4. Find how many each type of tee she bought?
a.
Half sleeve tee = 1 ,Full sleeve tee = 4
b.
Half sleeve tee = 5 ,Full sleeve tee = 0
c.
Half sleeve tee = 2 ,Full sleeve tee = 3
d.
Half sleeve tee = 3 ,Full sleeve tee = 2


Solution:

 
Let half sleeve tee = x 
and full sleeve tee = y

by given data,
x + y = 5....(1)
3x + 4y = 18...(2)

multiply by -4 in equation (1) and add with equation (2)
x = 2
substitute x value in equation (1)
2 + y = 5
y = 3

hence, half sleeve tee = 2
and full sleeve tee = 3

 


Correct answer : (3)
 5.  
A group of 100 students went to a trip. They used cars and buses altogether 7 vehicles. If in a car 4 students can sit and in a bus 40 students can sit. Find how much each car and bus they use?
a.
2 cars and 5 buses
b.
5 cars and 2 buses
c.
4 cars and 3 buses
d.
6 cars and 1 bus


Solution:

 
Let car = a and bus =b
as given 
a + b = 7......(1)
4a + 40b = 100.....(2)

multiply by -4 in equation (1) and add with equation (2)
36b = 72
b = 2
substitute b value in equation (1)
a + 2 = 7
a = 5

Answer is 5 cars and 2 buses   

 


Correct answer : (2)
 6.  
12 men and 6 women can paint a wall in 12 hours. 6 men and 8 women can paint it in 15 hour. Find the time to paint the same wall by 1 men alone and 1 women alone.
a.
 Men = 210 hours, Women = 200 hours
b.
 Men = 225 hours, Women = 225 hours
c.
 Men = 200 hours, Women = 225 hours
d.
 Men = 225 hours, Women = 200 hours


Solution:

 
Let m = the part of the job by 1 men in 1 hour
and w = the part of the job by 1 women in 1 hour 

By given we have,
12(12m + 6w) = 1.....(1)
15(6m + 8w) =1.....(2)
12m + 6w = $\frac{1}{12}$
6m + 8w = $\frac{1}{15}$

use elimination: 
-8(12m + 6w) = $\frac{1}{12}$ (-8)
6(6m+8w) = $\frac{1}{15}$ (6)

m = $\frac{1}{225}$
substitute m value in equation (1)
12(12($\frac{1}{225}$) + 6w) = 1
w = $\frac{1}{200}$

Hence 225 hours to paint the wall by 1 men and 200 hours by a women 

 


Correct answer : (4)
 7.  
In a test you have 30 questions worth 100 points. Some questions are 2 points and some are 4 points. Find how many each type of questions are in the test?
a.
2 points = 10, 4 points = 20
b.
2 points = 15, 4 points = 15
c.
2 points = 20, 4 points = 10
d.
2 points = 25, 4 points = 5


Solution:

 
Let  2 points questions = x
and 4 points questions = y
as given 

x + y = 30...(1)
2x + 4y = 100...(2)  
Multiply by -4 in equation (1) and add with equation (2)
-2x = -20
x = 10
substitute x value in equation(1)
10 + y = 30
y = 30 - 10
y = 20

Answer is 2 points questions = 10 
and 4 points questions = 20 

 


Correct answer : (1)
 8.  
Let two angles are supplementary. The measure of one angle is 40 degrees smaller than 3 times the other angle. Find the angles.
a.
50 degrees and degrees 130
b.
55 degrees and degrees 125
c.
30 degrees and degrees 150
d.
55 degrees and degrees 125


Solution:

 
Let two angles are x and y 
By given, 
x + y = 180...(1)
x = 3y - 40....(2)

substitute the x value from equation (2) to equation (1)
3y - 40 + y = 180
4y = 180 + 40
4y = 220
y = 55
again substitute y value in equation (2)
x = 3(55) - 40
x = 125 

Answer is 55 degrees and 125 degrees 

 


Correct answer : (2)
 9.  
The equations 6x + 3y = 38 and 2x + 3y = 26 represent the money collected from a fair tickets. if each adult ticket cost is $\$$x and each child ticket cost is $\$$y. Find the exact cost for each adult.
a.
$\$$5
b.
$\$$6
c.
$\$$3
d.
$\$$2


Solution:

 
Given x = ticket for each adult
and y = ticket for each child

6x + 3y = 38....(1)
2x + 3y = 26....(2)

Multiply by -1 in equation(1) and add equation (2)
-4x = -12
x = 3

Answer :
Ticket for each adult = $\$$3 

 


Correct answer : (3)
 10.  
Three small containers and one large container hold 10 cups of water. 1 large container minus 2 small containers hold 5 cups of water. How many cups of water can each container hold?
a.
Small container = 2 cup, Large container = 8 cups
b.
Small container = 0 cup, Large container = 10 cups
c.
Small container = 1 cup, Large container = 7 cups
d.
Non of them


Solution:

 
Let x = small container
and y = large container

3x + y = 10...(1)
y - 2x = 5...(2)

multiply by -1 in equation (2) and add with equation (1)
5x = 5x
x = 1

substitute x value in equation (1)
3 + y = 10
y = 7

Answer: 
Small container = 1 cup
Large container = 7 cups


 


Correct answer : (3)

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