EQUATIONS WITH RATIONAL NUMBERS WORKSHEET

Problem 1 :

Solve for x :

Problem 2 :

Solve for y :

0.3(y + 3) = 0.15 - 0.2y

Problem 3 :

Solve for k :

4(k - ⅔) = -18

Problem 4 :

Solve for x :

0.4(0.3x - 1) = 2 + 0.15x

Problem 5 :

Solve for z :

Problem 6 :

Solve for p :

Problem 7 :

Solve for y :

Problem 8 :

Solve for x :

Problem 9 :

Lawrence sold at his store. He spent $28.50 on supplies. He sold his muffins for 0.75 each and made a profit of $36.75. Write and solve an equation to find out how many muffins Lawrence sold.

Problem 10 :

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?

Answers

1. Answer :

In the equation above, there is only one fraction with the denominator 7.

So, multiply both sides of the equation by 7 to get rid of the fraction.

3x = 42

Divide both sides by 3.

x = 14

2. Answer :

0.3(y + 3) = 0.15 - 0.2y

Use the distributive property on the right side.

0.3(y) + 0.3(3) = 0.15 - 0.2y

0.3y + 0.9 = 0.15 - 0.2y

Add 0.2y to both sides.

0.5y + 0.9 = 0.15

Subtract 0.9 from both sides.

0.5y = -0.75

Divide both sides by 0.5.

y = -1.5

3. Answer :

4(k - ⅔) = -18

Use the distributive property on the right side.

4(k) - 4(⅔) = -18

4k - ⁸⁄₃ = -18

In the equation above, there is only one fraction with the denominator 3.

So, multiply both sides of the equation by 3 to get rid of the fraction.

12k - 8 = -72

Add 8 to both sides.

12k = -64

Divide both sides by 12.

4. Answer :

0.4(0.3x - 1) = 2 + 0.15x

Use the distributive property on the right side.

0.4(0.3x) - 0.4(1) = 2 + 0.15x

0.12x - 0.4 = 2 + 0.15x

Subtract 0.12x from both sides.

-0.4 = 2 + 0.03x

Subtract 2 from both sides.

- 2.4 = 0.03x

Divide both sides by 0.03.

-80 = x

5. Answer :

Least common multiple of the denominators (3, 6) = 6.

Multiply both sides of the equation by 6 to get rid of the fractions.

1(z + 6) = 2(-2)(z + 2)

z + 6 = -4(z + 2)

z + 6 = -4z - 8

Add 4z to both sides.

5z + 6 = -8

Subtract 6 from both sides.

5z = -14

Divide both sides by 5.

z = ⁻¹⁴⁄₅

6. Answer :

Least common multiple of the denominators (5, 20) = 20.

Multiply both sides of the equation by 20 to get rid of the fractions.

8(p - 2) = 3(p - 4)

8p - 16 = 3p - 12

Subtract 3p from both sides.

5p - 16 = -12

Add 16 to both sides.

5p = 4

Divide both sides by 5.

p = ⁻⁴⁄₅

7. Answer :

Least common multiple of (3, 9, 6) = 18.

Multiply both sides of the equation by 18 to get rid of the fractions.

6y + 4 = 3(y - 4)

6y + 4 = 3y - 12

Subtract 3y from both sides.

3y + 4 = -12

Subtract 4 from both sides.

3y = -16

Divide both sides by 3.

y = ⁻¹⁶⁄₃

8. Answer :

Least common multiple of (2, 4, 8) = 8.

Multiply both sides of the equation by 8 to get rid of the fractions.

-4(x + 9) + 46x = 3x

-4(x) - 4(9) + 46x = 3x

-4x - 36 + 46x = 3x

42x - 36 = 3x

Subtract 3x from both sides.

39x - 36 = 0

Add 36 to both sides.

39x = 36

Divide both sides by 39.

x = ¹²⁄₁₃

9. Answer :

Let x be the number of muffins sold.

Since Lawrence sold muffins for $0.75 each, amount of money received by sales is 0.75x.

money received - money spent = profit

0.75x - 28.50 = 36.75

Subtract 28.50 from both sides.

0.75x = 65.25

Divide both sides by 0.75.

x = 87

Therefore, Larence sold 87 muffins.

10. Answer :

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 10² ( = 100).

100(0.75 + 0.2r) = 100(0.5r)

100(0.75) + 100(0.2r) = 50r

75 + 20r = 50r

Step 6 :

75 + 20r = 50r

Subtract 20r from both sides.

75 = 30r

Divide each side by 30.

2.5 = r

So, the brothers’ walked at a rate of 2.5 miles per hour.

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