Problem 48 - Fundamentals of Mathematics

Question

Find each value. Check each result with a calculator. \(18+\frac{7+2}{9}\)

Step-by-Step Solution

Answer

The value is 19.

Step 1: Simplify the Numerator

First, simplify the numerator of the fraction within the expression. The fraction is \( \frac{7+2}{9} \). Add \( 7 \) and \( 2 \) to get \( 9 \). The fraction now is \( \frac{9}{9} \).

Step 2: Simplify the Fraction

Now, evaluate the fraction \( \frac{9}{9} \). Since the numerator is equal to the denominator, the fraction simplifies to \( 1 \).

Step 3: Add to 18

Now, add the result from Step 2 to 18, which gives us \( 18 + 1 \). This equals \( 19 \).

Step 4: Verification with a Calculator

Use a calculator to check the result by calculating \( 18 + \frac{7+2}{9} \), which confirms the computation is correct and the result of the expression is indeed \( 19 \).

Key Concepts

Simplifying Fractions

When we talk about simplifying fractions, what we mean is expressing a fraction in its simplest form. This makes it easier to understand and work with. For example, if we have the fraction \( \frac{9}{9} \), we can simplify it because both the numerator and denominator are the same. In such cases, the fraction simplifies to \( 1 \) since any number divided by itself equals \( 1 \).

Simplifying fractions involves:

• Dividing the numerator and the denominator by their greatest common divisor (GCD).
• Recognizing when the numerator equals the denominator, as this simplifies directly to \( 1 \).

Simplifying fractions makes arithmetic operations like addition and subtraction much easier and clearer.

Numerator and Denominator

Every fraction is made up of two parts: the numerator and the denominator. The numerator is the top number, and the denominator is the bottom number. These terms are crucial when dealing with fractions. The numerator indicates how many parts we have, while the denominator tells us how many parts make up a whole.

In our exercise, we started with the fraction \( \frac{7+2}{9} \).

• The numerator was \( 7+2 \), which we simplified to \( 9 \).
• The denominator was \( 9 \), indicating the whole was divided into 9 equal parts.

Understanding the roles of both the numerator and the denominator helps in correctly simplifying and computing fractions.

Addition

Addition is one of the basic arithmetic operations, often used to combine numbers to find the total. It's crucial when dealing with an expression that includes both whole numbers and fractions. In our exercise, after simplifying the fraction, we added the result to 18.

Here’s how it works:

• After simplifying \( \frac{9}{9} \) to \( 1 \), we are left adding it to 18.
• This results in a new total: \( 18 + 1 = 19 \).

Adding fractions to whole numbers usually involves similar steps: simplify and then sum up. This straightforward process ensures our final answer is both simple and accurate.

Calculator Verification

Calculator verification is a practical step to ensure the accuracy of your computations. It involves using a calculator to re-calculate the expressions or operations you've performed manually. In our example, after reaching the result 19 from the expression \( 18+\frac{7+2}{9} \), using a calculator reassures us.

Here are some tips for calculator verification:

• Always re-enter the expression exactly as it appears in your calculations.
• If your manual and calculator results match, it confirms the accuracy of your answer.
• Using this step is especially useful when dealing with more complex operations, reducing mental computation errors.

It's always wise to double-check, ensuring the integrity of your work with a quick electronic verification.