Problem 66 - A Graphical Approach to Precalculus with Limits

Question

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

Step-by-Step Solution

Answer

2.57

Step 1: Simplify the Numerator

Add the numbers in the numerator: \(34 + 25\). Calculate \(34 + 25 = 59\).

Step 2: Set Up the Division

Express the division using the simplified numerator: \(\frac{59}{23}\).

Step 3: Perform the Division

Divide 59 by 23 to get approximately 2.565217. Perform the calculation: \(59 \div 23 \approx 2.565217\).

Step 4: Round to the Nearest Hundredth

To round 2.565217 to the nearest hundredth, look at the third decimal place (5). Since it is 5, round up the second decimal place (6) by 1. Thus, the result is 2.57.

Key Concepts

Division

Division is a fundamental mathematical operation where you determine how many times a number (the denominator) fits into another number (the numerator). It is represented by the symbol "÷" or a fraction bar "⁄".
Understanding division is essential because it helps in distributing quantities evenly and finding ratios. For instance, in our exercise, we first simplify the expression by adding 34 and 25 to get 59. This is the new numerator in our division problem.

• The division is set up as \( \frac{59}{23} \), which asks: "How many times does 23 fit into 59?"
• Performing this division gives us approximately 2.565217.

Division can sometimes result in exact whole numbers, but more often it gives us a decimal, which leads to the need for rounding.

Numerator and Denominator

The concepts of numerator and denominator are central to understanding fractions and division problems.

• The numerator is the top part of a fraction that signifies the number of parts being considered.
• The denominator is the bottom part that shows the total number of equal parts into which the whole is divided.

In our exercise, once we add 34 and 25, we get 59 as the numerator, which represents the total quantity we are dividing. The denominator, which remains as 23, indicates how many parts we are dividing the numerator into.
Understanding these roles makes division and fraction problems more straightforward. It helps in setting up the right calculations and eventually solving the exercise correctly. Remember: numerator tells "how many", while the denominator tells "of what".

Decimal Places

Decimals are a simplified way of representing fractional values. When you perform a division and the result isn't a whole number, it often leads to a decimal with several digits.
Each digit following the decimal point is a decimal place. For example, in the result 2.565217, 5 is the first decimal place, 6 is the second, and 5 is the third.

• For precision, sometimes you need to round the number to a certain decimal place, such as nearest hundredth (second decimal place).

In order to round to the nearest hundredth, you look at the thousandth place (third digit after the decimal). If it's 5 or more, round up; otherwise, round down.
So, from our exercise, 2.565217 becomes 2.57 after rounding since the third decimal place is 5. This process is crucial for approximations, ensuring calculations are manageable and values are more practical in real-world applications.