Solving Equations Involving Decimals

Approaches to Solve Equations with Decimals:

1. Clear All Decimals: Multiply both sides of the equation by an appropriate power of 10 to eliminate decimals. This makes calculations easier and allows working with whole numbers.

2. Solve the Equation in Decimal Form: If necessary, solve the equation without clearing decimals, but be meticulous in calculating decimal values to avoid errors.

Exercises: Equations to Solve

1. Equation: 0.12t - 2.1 = 0.07t - 0.2Steps to Solve:

   • Move all terms involving (t) to one side by subtracting 0.07t from both sides:0.12t - 0.07t = 2.1 - 0.2

   • Combine like terms:0.05t = 1.9

   • Divide both sides by 0.05 to isolate (t):t = 1.9 / 0.05 = 38

2. Equation: 0.04y + 0.08y = 4.8Steps to Solve:

   • Combine (y) terms:0.12y = 4.8

   • Divide by 0.12 to solve for (y):y = 4.8 / 0.12 = 40

3. Equation: 0.08(x + 200) = 0.07x + 20Steps to Solve:

   • Distribute 0.08 on the left:0.08x + 16 = 0.07x + 20

   • Move 0.07x and constants:0.08x - 0.07x = 20 - 16

   • Combine:0.01x = 4

   • Divide by 0.01:x = 400

4. Equation: 0.07x = 152 - 0.08(2000 - x)Steps to Solve:

   • Distribute 0.08 on the right:0.07x = 152 - 160 + 0.08x

   • Combine:0.07x - 0.08x = -8

   • Solve for (x):-0.01x = -8therefore, x = 800

More Exercises:

1. Equation: 0.09x = 1650 - 0.12(x + 5000)Steps to Solve:

   • Distribute 0.12:0.09x = 1650 - 0.12x - 600

   • Combine like terms, isolate (x):0.21x = 1050therefore, x = 5000

2. Equation: 0.1(x - 0.1) - 0.4(x + 2) = -5.31Steps to Solve:

   • Distribute:0.1x - 0.01 - 0.4x - 0.8 = -5.31

   • Combine:-0.3x - 0.81 = -5.31

   • Isolate (x):-0.3x = -4.5thus, x = 15

Real-Life Applications:

1. Problem: Gabby bought jeans for $45 at a 30% discount.Question: What was the original price?Steps to Solve:

   • Let the original price be (p). The equation is set up as:0.7p = 45

   • Solve for (p):p = 45 / 0.7 ≈ 64.29

2. Problem: A pizza owner makes a 70% profit; cost is $7.50.Question: What should be the selling price?Steps to Solve:

   • Selling price (s):s = 7.50 + 0.7(7.50)

   • Calculate:s = 7.50 + 5.25 = 12.75

Additional Investment Problems:

1. Investment Problem:Scenario: Eva invested at 4% and $1500 more at 6%. Total interest: $390.Steps to Solve:

   • Let initial investment at 4% be (x). Then the investment at 6% is (x + 1500).

   • Set up the interest equation:0.04x + 0.06(x + 1500) = 390

   • Solve for (x).

2. Investment Problem:Scenario: $4000 total invested; some at 5% and remainder at 6%. Total interest: $230.Steps to Solve:

   • Let amount at 5% be (y); the rest is (4000 - y).

   • Set up the interest equation:0.05y + 0.06(4000 - y) = 230

Coin Collection Problem:

Scenario: Sara has nickels, dimes, and quarters worth $15.75.Conditions: 10 more dimes than nickels; twice as many quarters as dimes.Steps to Solve:

• Set variables for each coin type:

  • Let (n) be nickels, (d = n + 10) be dimes, and (q = 2d) be quarters.

• Use equations to represent total value:

  • 0.05n + 0.10(n + 10) + 0.25(2(n + 10)) = 15.75

• Solve for (n), then find (d) and (q).