SOLVING LINEAR EQUATIONS WITH DECIMALS WORKSHEET

Problem 1 :

Solve for n :

0.7n + 0.33  =  0.3n + 0.5

Solution :

Step 1 :

In the second term 0.33 on the left side, we have two digits (more number of digits) after the decimal.

So, multiply both sides of the equation by 100.

100(0.7n + 0.33)  =  100(0.3n + 0.5)

100(0.7n) + 100(0.33)  =  100(0.3n) + 100(0.5)

70n + 33  =  30n + 50

Step 2 :

Subtract 33 from each side.

70n  =  30n + 17

Subtract 30n from each side.

40n  =  17

Divide each side by 40.

40n/40  =  17/40

n  =  0.425

Problem 2 :

David walks from his house to the zoo at a constant rate. After walking 0.75 mile, he meets his brother Daniel, and they continue walking at the same constant rate. When they arrive at the zoo, David has walked for 0.5 hour and Daniel has walked for 0.2 hour. What is the rate in miles per hour at which the brothers walked to the zoo ?

Solution :

Step 1 : 

Write an expression for the distance from the brothers’ house to the zoo, using the fact that distance equals rate times time.

Let r  be the walking rate of both David and his brother Daniel. 

Distance from the brothers’ house to the zoo

  =  0.2r

Step 2 : 

Write an expression for the distance from the David's house to the zoo, using the distance from his brother's house to the zoo. 

Distance from Davids’ house to the zoo 

=  0.75 + 0.2r -------(1)

Step 3 :

Write an expression for the distance from the David's house to the zoo, using David's total walking time 0.5 hour. 

Distance from Davids’ house to the zoo 

=  0.5r -------(1)

Step 4 : 

Both (1) and (2) represent the distance from David's house to the zoo. 

So, we have

(1)  =  (2)

0.75 + 0.2r  =  0.5r

Step 5 :

In the first term 0.75 on the left side, we have two digits (more number of digits) after the decimal.

So, multiply both sides of the equation by 100.

100(0.75 + 0.2r)  =  100(0.5r)

100(0.75) + 100(0.2r)  =  50r

75 + 20r  =  50r

Step 6 :

Subtract 20r from both sides.

75  =  30r

Divide both sides by 30. 

75 / 30  =  30r / 30 

2.5  =  r

So, the brothers’ constant rate of speed was 2.5 miles per hour.

Problem 3 :

Alex has two aquariums. One aquarium contains 1.3 cubic feet of water and the other contains 1.9 cubic feet of water. The water in the larger aquarium weighs 37.44 pounds more than the water in the smaller aquarium. Write an equation with a variable on both sides to represent the situation. Then find the weight of 1 cubic foot of water.

Solution :

Step 1 : 

Let 'x' pounds be the weight of 1 cubic feet of water.

Then, the weight of 1.3 cubic feet of water : 

1.3x

Weight of 1.9 cubic feet of water : 

1.9x

Step 2 : 

It is given that the water in the larger aquarium weighs 37.44 pounds more than the water in the smaller aquarium. 

1.9x  =  1.3x + 37.44

The above equation with a variable 'x' on both sides represents the given situation.  

Step 3 :

To find the weight of 1 cubic foot of water, solve the above equation for x. 

1.9x  =  1.3x + 37.44

Subtract 1.3x from each side. 

0.6x  =  37.44

Divide each side by 0.6. 

x  =  62.4

So, the weight of 1 cubic foot of water is 62.4 lb.

Problem 4 :

The table shows the balance of Rob's checking account at the end of the day. This is $95.50 less than the amount he had at the beginning of the day. Write and solve an equation to determine the balance at the beginning of the day.

Solution :

Let x be the amount he has at the beginning of the day.

x - 95.50 = 4.50

x = 4.50 + 95.50

x = 100

So, at the start of the day he has $100.

Problem 5 :

Monica earned twice as much as Samuel walking dogs. The amount Samuel earned was $7 more than Kara earned. Suppose Monica earned $48.50. Write and solve an equation to find the amount Kara earned. What is the difference between greatest amount earned and the least amount earned walking dogs ?

Solution :

Let x be the amount Samuel earned.

Monica earned = 2x

Amount earned by Kara = x - 7

2x = 48.50

x = 48.50/2

x = 24.25

Earning of Kara = 24.25 - 7

= 17.25

Greatest amount = 48.50

Least amount = 17.50

Difference = 48.50 - 17.50

= 31

So, the required difference is $31

Problem 6 :

Natalie is participating in a jog a thon. Her goal is to raise $125. She has collected $80 and will receive $3 for each lap around the track she jogs. How many laps does she need to jog to reach her goal ?

Solution :

Let x represent the number of laps Natalie needs to jog around the track to reach her goal. 

125 = 80 + 3x

Subtracting 80

125 - 80 = 3x 

3x = 45

x = 45/3

x = 15

So, the number of laps is 15.

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