Problem 28 - Algebra 1

Question

Solve the equation. Round the result to the nearest hundredth. Check the rounded solution. $$ 3(31-12 t)=82 $$

Step-by-Step Solution

Answer

The solution to the equation, when rounded to the nearest hundredth, is \(t = 0.31\).

Step 1: Simplifying the equation

Firstly, apply distributive property to multiply the 3 into the brackets. The equation then becomes \[3*31 - 3*12t = 82\] which further simplifies to \[93 - 36t = 82\].

Step 2: Isolating the term with 't'

Next, isolate the term with 't' by moving '93' to the other side of the equation. This can be done by subtracting '93' from both sides, giving \[-36t = 82 - 93\] simplifying to \[-36t = -11\].

Step 3: Solving for 't'

Now, solve for 't' by dividing both sides of the equation by -36. This gives \(t = \frac{-11}{-36}\).

Step 4: Rounding to the nearest hundredth

On calculating this division, the answer is approximately 0.30556. However, we're asked to round to the nearest hundredth. Hence, we round the calculated answer to give us \(t = 0.31\).

Step 5: Checking the solution

Finally, we substitute \(t = 0.31\) back into the original equation to confirm if it is the solution. Substituting, we get \[3(31 - 12 * 0.31) = 82\]. On calculating the left-hand side, it rounds to approximately 82, which is equal to the right-hand side. Hence, the calculated 't' is the correct solution.

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra that helps simplify expressions. It involves multiplying a term outside a parenthesis by each term inside the parenthesis individually. This is particularly useful when dealing with equations that contain parentheses, as it allows us to "distribute" or "spread" the multiplication over the addition or subtraction inside the brackets.
For example, in the provided equation \(3(31 - 12t) = 82\), the distributive property is used as follows:

• Multiply 3 by 31, which is 93.
• Multiply 3 by \(-12t\), which is \(-36t\).

After applying the distributive property, the equation simplifies to \(93 - 36t = 82\). This simplifies our equation by removing the parentheses, allowing us to isolate the variable we are solving for.

Rounding Numbers

Rounding is the process of adjusting numbers to make them simpler while retaining their overall value to a useful degree of accuracy. In most cases, we round numbers to a specific place value such as the nearest whole number, tenth, hundredth, etc.
In this exercise, we calculate \(t = \frac{-11}{-36}\), which approximates to about 0.30556. Since the problem instructs us to round to the nearest hundredth:

• Look at the third digit after the decimal, which is 5 in 0.30556.
• If this digit is 5 or greater, we round the second digit up. So, 0.30556 rounds to 0.31.

Rounding helps provide a more manageable and interpretable number, particularly when dealing with decimals in practical problems.

Checking Solutions

Checking your solution is an essential step in solving equations to ensure accuracy. It involves substituting your calculated value back into the original equation to verify that both sides are equal.
For this exercise, we found \(t = 0.31\). To check if this is correct, substitute back into the original equation \(3(31 - 12t) = 82\):

• Substitute \(t = 0.31\) into the equation: \(3(31 - 12 \times 0.31)\).
• Calculate the inside of the parentheses: \(31 - 3.72 = 27.28\).
• Multiply by 3: \(3 \times 27.28 = 81.84\).

Since 81.84 rounds to the original right-hand side, 82, we confirm the solution is approximately correct. This step is critical as it confirms the solution within the context of the problem's requirements.