Problem 26 - Algebra 1: Concepts and Skills

Question

Solve the equation. Round the result to the nearest hundredth. Check the rounded solution. $$ 99=21 t+56 $$

Step-by-Step Solution

Answer

The solution to the equation, rounded to the nearest hundredth, is \(t \approx 2.05\).

Step 1: Rearrange the equation

The first step is to isolate the variable \(t\). This can be done by subtracting 56 from both sides of the equation. This results in the equation: \(99 - 56 = 21t\). Simplifying the left-hand side gives \(43 = 21t \).

Step 2: Solve for \(t\)

Divide both sides of the equation by 21 to solve for \(t\). This gives \(t = \frac{43}{21}\).

Step 3: Round the result

Now the result should be rounded to the nearest hundredth. As a decimal, \(t \approx 2.05\) when rounded to the nearest hundredth.

Step 4: Check the result

Substitute the result back into the original equation to confirm. Thus the check statement will be \(99 \stackrel{?}{=} 21*2.05+56\). Simplify the right-hand side gives sort of \(99\). Since the result is almost 99, we can therefore say that the result is correct, bearing in mind that there is some discrepancy due to the rounding we did during the calculation. However, this discrepancy is in acceptable range.

Key Concepts

Rounding Numbers

When solving equations, rounding numbers is commonplace, especially when dealing with decimals that go beyond the necessary precision. In mathematics, rounding involves adjusting a number to a nearby, more convenient value. The common practice is to round to the nearest ten, hundred, or in this exercise, the nearest hundredth.

Here's how you can perform such rounding:

• Identify the digit at the place to which you are rounding, in this case, the hundredth.
• Look at the digit immediately to the right of that place.
• If this adjacent digit is 5 or more, round up the digit in the place you are focusing on.
• If it is less than 5, keep the digit the same, and adjust the digits to the right to zero (or simply do not write them in a decimal).

For example, rounding the number 2.0476 to the nearest hundredth involves examining the third decimal digit, which is 7. Since 7 is greater than 5, you round up the second decimal digit from 4 to 5, resulting in 2.05. Thus, you ensure that the number is precise enough for the context it is used in.

Verifying Solutions

Verifying solutions in mathematics ensures that the solution obtained accurately satisfies the original equation. This step is crucial to avoid errors and confirm the solution's correctness.

To verify a solution:

• Substitute the obtained solution back into the original equation wherever the variable appears.
• Perform the necessary calculations to simplify the equation.
• Check if both sides of the equation match or are almost equal due to rounding discrepancies.

For our problem, after rounding, the solution for the variable was substitute back as follows: Plugging our rounded value of 2.05 into the equation led to the expression: \( 21 \times 2.05 + 56 \). Simplifying this expression results in a number very close to 99, affirming the correctness of our solution. The proximity to the original value and acceptance of slight rounding differences ensure our method's accuracy.

Decimal Representation

Decimal representation is a system of numbers expressed in a base-10 format, which is widely used due to its convenience and simplicity. It provides a uniform way to represent fractions and can often require rounding to maintain simplicity when dealing with long decimals.

When a number is expressed in decimal form, it allows:

• Closer approximation of complex fractions.
• Precision representation when exactness is needed, illustrated through scientific and financial calculations.
• Easy comparison and operations among numbers with different values.

In many mathematical problems, converting results into a simple decimal form, as shown in our exercise with \( t = 2.05 \), makes the solution accessible and understandable. While certain calculations might lead to infinite decimals, rounding ensures that these values remain practical. The simplicity and universality of decimals make them a first choice in everyday calculations.