Problem 66 - A Graphical Approach to College Algebra

Question

Approximate each expression to the nearest hundredth. $$\frac{34+25}{23}$$

Step-by-Step Solution

Answer

2.57

Step 1: Simplify the Numerator

First, we need to simplify the expression inside the numerator of the fraction. Add 34 and 25 together.\[34 + 25 = 59\]

Step 2: Perform the Division

Next, divide the result from Step 1 by the denominator 23.\[\frac{59}{23} = 2.5652\]

Step 3: Round to the Nearest Hundredth

Finally, round the quotient from Step 2 to the nearest hundredth. The number 2.5652 rounded to the nearest hundredth will be 2.57.

Key Concepts

Fraction Simplification

When simplifying fractions, the goal is to make it easier to understand and work with. This typically involves combining elements in the numerator or the denominator of a fraction.
In the given exercise, the first step is to simplify the numerator, which consists of adding two numbers: 34 and 25. By performing this addition, we get 59.

• The simplification of the numerator does not involve further reducing, as no common factor is involved.
• It simply means combining or calculating values to make further operations easier.

Simplifying before dividing ensures that the calculation in subsequent steps is straightforward. This step is crucial for accurate results when dealing with more complex arithmetic expressions.

Division

Division is one of the fundamental operations in arithmetic. It involves splitting a number into a specified number of equal parts. In this exercise, after simplifying the numerator, we perform division by dividing this simplified numerator by the denominator.
In the expression, we have 59 in the numerator and 23 in the denominator. To find out how many times 23 can fit into 59, you perform the division:

• Calculate \( \frac{59}{23} \).
• This division yields approximately 2.5652.

Understanding division is important because it helps determine the size of equal parts or groups and is used in many applications related to measurements, construction, and finance. It is imperative to do this step accurately to ensure the rest of the problem solving can happen smoothly.

Numerical Approximation

Numerical approximation is the process of finding an approximate value for a number when an exact value is difficult or impossible to obtain. It simplifies complex values, making them easier to interpret and work with.
In this case, after finding the division result 2.5652, we need to round this number to the nearest hundredth.

• The hundredths place is the second digit to the right of the decimal point.
• Rounding involves looking at the third digit to decide whether to round up or keep it the same.

In the number 2.5652, the third digit is 5, which means we round up. This makes the rounded value 2.57. Rounding makes the number simpler and more manageable for reporting and presentation, without sacrificing too much accuracy. Often, in real-world applications, fully precise numbers aren't practical, and approximations give a sufficiently close value for decision-making.