Problem 7 - Technical Mathematics with Calculus

Question

Combined Operations with Exact Numbers. Perform each computation by calculator. $$\frac{647+688}{337+108}$$

Step-by-Step Solution

Answer

First add 647 and 688 to get 1335. Then add 337 and 108 to get 445. Finally, divide 1335 by 445 to get 3.

Step 1: Add the Numerator

First, we add the numbers in the numerator. This means we add 647 and 688 together. To do this, simply type 647 + 688 into the calculator.

Step 2: Add the Denominator

Next, we add the numbers in the denominator. This means we add 337 and 108 together. Type 337 + 108 into the calculator to find the sum.

Step 3: Divide the Numerator by the Denominator

After obtaining both sums, divide the sum of the numerator by the sum of the denominator to get the final answer. Use the calculator to divide the result from Step 1 by the result from Step 2.

Key Concepts

Exact Numbers

When performing mathematical operations, understanding the concept of exact numbers is crucial. An exact number is a value that is known with complete precision. It has no uncertainty or approximation involved. For instance, the number of students in a classroom, the number of apples in a basket, or the result of adding 647 and 688 are all exact numbers. They are not estimates, but rather, definite and precise quantities.

In the exercise provided, the numbers involved in the computation of the numerator and denominator are considered exact numbers. The importance of working with such numbers is that when we perform calculations, especially with a calculator, we expect to get precise results. This also implies that the computation we perform can be replicated with the same exact outcome—ensuring consistency in our mathematical operations and the reliability of our results.

Numerator and Denominator

A fraction is made up of two parts: the numerator and the denominator. The numerator, located above the fraction bar, signifies how many parts we have, while the denominator, below the fraction bar, describes the total number of equal parts into which a whole is divided. For example, in the fraction \( \frac{3}{4} \), 3 is the numerator and 4 is the denominator, meaning we have 3 parts out of 4 equal parts of a whole.

In our exercise, the numerator is \(647 + 688\), and the denominator is \(337 + 108\). Understanding these components is essential when it comes to performing the operations correctly. You first evaluate each part separately, as performed in the step-by-step solution, before attempting to divide them. Grasping this concept aids in simplifying complex fractions, ensuring that each step—addition and division—is handled systematically for accurate results.

Calculator Operations

Today's calculators are powerful tools that can perform a wide range of mathematical operations with speed and precision. The operation of a calculator follows a sequence much like manual computation—the order of operations. This ensures that the calculations are carried out correctly.

In solving our fraction computation example, using a calculator greatly simplifies the process. Here are the steps tailored for calculator use:

• Add the numbers specified for the numerator when instructed.
• Do the same for the denominator.
• Once you have both sums, use the division function to divide the numerator's sum by the denominator's sum.

It's paramount to understand how to use the calculator's functionalities to prevent input errors. Always double-check if the numbers displayed match the numbers you intended to input. Various calculators may have different ways to input information, so consulting your device's manual or guide may be helpful to ensure the correct operation sequence and prevent common mistakes, such as entering a wrong number or misinterpreting the result.